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MAT^Wte inlMhl., 5931 LECTURES ON THE THEORY OF FUNCTIONS OF REAL VARIABLES Volume II BY JAMES PIERPONT, LL.D. Professor of Mathematics in Yale University GINN AND COMPANY BOSTON • NEW YORK • CHICAGO ■ LONDON COPYRIGHT, 1912, BY JAMES PIERPONT ALL RIGHTS RESERVED 912.3 CINN AND COMPANY • PRO- PRIETORS • BOSTON • U.S.A. TO ANDREW W. PHILLIPS THESE LECTURES ARE INSCRIBED WITH AFFECTION AND ESTEEM PREFACE The present volume has been written in the same spirit that animated tlie first. The author has not intended to write a treatise or a manual ; he has aimed rather to reproduce his uni- versity lectures with necessary modifications, hoping that the freedom in the choice of subjects and in the manner of presenta- tion allowable in a lecture room may prove helpful and stimulating to a larger audience. A distinctive feature of these Lectures is an attempt to develop the theory of functions with reference to a general domain of definition. The first functions to be considered were simple combinations of the elementary functions. Riemann in his great paper of 1854, " Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe," was the first to consider seriously functions whose singularities ceased to be intuitional. The re- searches of later mathematicians have brought to light a collection of such functions, whose existence so long unsuspected has revolu- tionized the older notion of a function and made imperative the creation of finer tools of research. But while minute attention was paid to the singular character of these functions, practically none was accorded to the domain over which a function may be defined. After the epoch-making discoveries inaugurated in 1874 by G. Cantor in the theory of point sets, it was no longer neces- sary' to consider a function of one variable as defined in an in- terval, a function of two variables as defined over a field bounded by one or more simple curves, etc. The first to make use of this new freedom was C. Jordan in his classic paper of 1892. He has had, however, but few imitators. In the present Lectures the author has endeavored to develop this broader view of Jordan, persuaded that in so doing he is merely carrying a step farther the ideas of Dirichlet and Riemann. Often such an endeavor leads to nothing new, a mere statement for any n of what is true for n = 1, or 2. A similar condition vi PREFACE prevails in the theory of determinants. One may prefer to treat only two and three rowed determinants, but he surely has no ground of complaint if another prefers to state his theorems and •demonstrations for general n. On the other hand, the general case may present unexpected and serious problems. For example, Jordan has introduced the notion of functions of a single variable having limited variation. How is this notion to be extended to two or more variables ? An answer is far from simple. One was given by the author in Volume I ; its serviceableness has since been shown by B. Camp. Another has been essayed by Lebesgue. The reader must be warned, however, against expecting to find the development always extended to the general case. This, in the first place, would be quite impracticable without greatly increasing the size of the present work. Secondly, it would often be quite beyond the author's ability. Another feature of the present work to which the author would call attention is the novel theory of integration developed in Chapter XVI of Volume I and Chapters I and II of Volume II. It rests on the notion of a cell and the division of space, or in fact any set, into unmixed partial sets. The definition of improper multiple integrals leads to results more general in some respects than yet obtained with Riemann integrals. Still another feature is a new presentation of the theory of measure. The demonstrations which the author has seen leave much to be desired in the way of completeness, not to say rigor. In attempting to find a general and rigorous treatment, he was at last led to adopt the form given in Chapter XI. The author also claims as original the theory of Lebesgue integrals developed in Chapter XII. Lebesgue himself considers functions such that the points e at which a 1 where L'ai lim 5^ = lim 2mA = f fd'Hi. (2 Let us prove 1) ; the relation 2) may be demonstrated in a similar manner. In the first place we show in a manner entirely analo- gous to I, 722, that ^A ffd%-e, 8<8o, (4 we shall modify the proof as follows. Let ^ be a cubical division of space of norm e < e^. We may take e^ so small that If- S, <|. (5 PROPER INTEGRALS 3 The cells of J^ containing points of 31 fall into two classes. 1° the cells e„ containing points of the cell S, but of no other cell of A ; 2° the cells e[ containing points of two or more cells of A. Thus we have _ where M,^, M[, are the maxima of / in e,^, e[. Then as above we have /S^<2aKA« + |, e iS^ — — , which with 5) proves 4). 4. Let fQc-^ ■ ■ • x^) he limited in the limited field 31. Let A he an unmixed division of%,qf norm S, into cells Sj, Sj • • • . Let S^ = -LmA, S^ = 'EMA, where as usual m,,, Mt are the minimum, and maximum of f in S.. Then — Cfd% = Max S^, Cfdn = Min S^. The proof is entirely -similar to I, 723, replacing the theorem there used by 2, 3. 5. In connection with 4 and the theorem I, 696, 723 it may be well to caution tlie reader against an error which students are apt to make. The theorems I, 696, i, 2 are not necessarily true if / 4 POINT SETS AND PROPER INTEGRALS has both signs in 31- For example, consider a unit square S whose center call C. Let us effect a division U oi S into 100 equal squares and let 21 be formed of the lower left-hand square « and of O. Let us define /as follows : /= 1 within 8 = - 100 at O. For the division U, Hence, ^^^ So<- H^- On the other hand, lim^^ = a^. The theorems I, 723, and its analogue 4 are not necessarily true for unmixed divisions of space. The division A employed must be unmixed divisions of the field of integration 21. That this is so, is shown by the example just given. 6. In certain cases the field 21 may contain no points at all. In such a case we define ^ r/=o. 7. From 4 we have at once : Let A he an unmixed division of% into cells Sj, B^, ••• Then S = Min 2S„ toith respect to the class of all divisions A. 8. We also have the following : Let I) he an unmixed division of space. Let c?j, d^,--- denote those cells containing points of 21- Then i = Min 2^„ with respect to the class of th£ divisions D.' _ For if we denote' by S^ the points of 21 in d^ we have obviously B, g in 31 except possibly at the points of a discrete set ©. Then Jj^fj- ^51 »/s For let 21 = ^+©. Then But in A, f>g, hence X^=X^' X^=X^' lf>Sj, by I, 729. The theorem now follows at once. 6 POINT SETS AND PROPER INTEGRALS »«-/-». X^=.X/, Sjf.ejj. For in any cell d^ Max • 6/^=6 Max/; Min-c/=eMin/ when c > ; while Max • c/ = e Min /; Min -0^=0 Max/ when e < 0. 4. 7/" t;' is integrahle in 21, For from Max/+Min^' which is the first half of 1). The other half follows from the relation Min/ + Min^ •-'a i-a •'a which is 1). 8 POINT SETS AND PJiOPER INTEGRALS To prove 3), we use the relation Henoe ^ r r j -\f\ be limited in the limited fields SS, S. Let 21 be the aggregate formed of the points in either 35 or S. Then Jf^g3 »/£ This is obvious since the sums IMA , 'S.MA may have terms in common. Such terms are therefore counted twice on the right of 1) and only once on the left, before passing to the limit. Remark. The relation 1) may not hold when /is not > 0. Example. Let 21 = (0, 1), 53 = rational points, and S = irra- tional points in 21. Let/= 1 in S, and — 1 in g. Then ^% "/as *^e and 1) does not now hold. 2. Let 21 be an unmixed partial aggregate of the limited field ©. ie«g = «-2l. If g=f in 21 = in g, then -7" 7» PROPER INTEGRALS 9 For But and obviously -r 3. The reader should note that the above theorem need not be true if 31 is not an unmixed part of SS- Example. Let 21 denote the rational points in the unit square ^^'* f = 9=-l in 21. Then r r r 4. Let % he a part of the limited field ^. Let f>.^ he limited in %. Let g=f in % and = w g = S - 21- Then Jf=J 9-' (1 /"/> r^- (2 For let ilf[, N^ be the maxima of/, g in the cell c?,. Then 'LNA = ^Nd, + ^NA a Passing to the limit we get 1). To prove 2) we note that in an}'^ cell containing a point of 21 Min/> Min^. 6. 1. Letf(x-^^ ■•■ a;„) be limited in the limited field 21. Let ^,, be an unmixed part of 21 such that ©„ = 21 as u = 0. Then Tf=\imCf. (1 10 POINT SETS AND PROPER INTEGRALS Forlet |/[by 4, 1), 6). Hence passing to the limit m = in 2) we get 1). 2. We note that 1 may be incorrect if the 53« are not unmixed. For let 31 be the unit square. Let ^„ be the rational points in a concentric square whose side is 1 — m. Let/= 1 for the rational points of 31 and = 2 for the other points. Then If-'- -X/=i- 7. In I, 716 we have given a uniform convergence theorem when each IBa < 31. A similar theorem exists when each ©„ > 31, viz.: Let ^^ 0, there exists a pair u^, d^ such that e For SS^ < 21 + -, Uq sufficiently small Also for any division I) of norm d < some d^ e "2 3.„,z.<««.+ H 0- But Hence 8. 1. Let 21 be a point set in m = r + s way space. Let us set certain coordinates as x^+i ••■x^=0 in each point of 31. The resulting points SS we call a projection of 31. The points of 31 PROPER INTEGRALS 11 belonging to a gi^en point h of :SB, we denote by gj or more shortly by £. We write 1 = » • e, and call ^B, E components of 31. We note that the fundamental relations of I, 733 Ja •^— J© Jis. —Jm'' ©tie' hold not only for the components j, ^, etc., as there given, but also for the general components 31, 48. In what follows we shall often give a proof for two dimensions for the sake of clearness, but in such cases the form of proof will admit an easy generalization. In such cases SQ will be taken as the ^-projection or component of 31. 2. if 31 = S • S IS limited and Sd is discrete, 31 is also discrete. For let 31 lie within a cube of edge |-(7>1 in m=r + s way space. Then for any d < some d^. Then i^ < 6"«^ < e. 3. That the converse of 2 is not necessarily true is shown by the two following examples, which we shall use later : Example 1. Let 31 denote the points a;, y in the unit square determined thus : For a; = ^, ji=l, 2, 3, -, moddand<2», 2" let . — "^ — 2" Here 21 is discrete, while S = 1, where S denotes the projection of 31 on the a;-axis. 4. Example 2. Let 21 denote the points x, y in the unit square determined thus : 12 POINT SETS AND PROPER INTEGRALS For ^ a; = — , »w, ra relatively prime, n let . n Then, © denoting the projection of 21 on the rc-axis, we have f=0, «j=l. 9. 1. Let 21= 59 ■ S Se a limited point set. Then %< f'^<%. (1 — ?£SB For let/=l in 91. Let ^' = 1 at each point of 21 and at the other points of a cube A = B- Q containing 21, let ff = 0. Then By I, 733, But by 5, 4, _ _ _ JBjc^ — JmJct' Thus -_ which gives 1), since 2. In cane 3t is metric we have ^=Jm, (2 and S is an integrable function over 53- This follows at once from 1). PROPER INTEGRALS 13 3. In this coflpection we should note, however, that the converse of 2 is not always true, i.e. if S is integrable, then 21 has content and 2, 2) holds. This is shown by the following : Example. In the unit square we define the points a;, «/ of 21 thus : For rational a;, n ^ , ^ i For irrational x, i ^ ^ i Then S = | for every x in S3 Hence But 3[ = 0, 21 = 1. 10. 1. Letf{x^ ■■■ x„) be limited in the limited field 21 = i8 • S- ///>o, r r < r i lff<0, Jffl Jg3«^e Let us first prove 1). Let 21, SS, S lie in the spaces 9?,„, 'Sir, ^si r + s = m. Then any cubical division D divides these spaces into cubical cells d„ d[, d" of volumes c?, d\ d" respectively. Ob- viously d = d'd". J) also divides ® and each E into unmixed cells 8', S". Let M, = Max/ in one of the cells d,, while Jt/;' = Max/ in the corresponding cell h'!. Then by 2, 4, ( f<^M'!l'l<'LMd'!, since ilf;, if/ > 0. Hence SS: f/ < l,d',l.M4" = 2ilC(i. SB ilg SB Letting the norm of D converge to zero, we get 1). We get 2) by similar reasoning or by using 3, 3 and 1). 14 POINT SETS AND PROPER INTEGRALS 2. To illustrate the necessity of making/ > in 1), let us take 21 to be the Pringsheim set of I, 740, 2, while / shall = - 1 in 31. Then // = -!■ On the other hand 7; Hence 71 7: and the relation 1) does not hold here. Iterdble Fields 11. 1. There is a large class of limited point sets which do not have content and yet _ — 31= fs. (1 Any limited point set satisfying the relation 1) we call iterable, or more sijecifically iterable with respect to 53. Example 1. Let 31 consist of the rational points in the unit square. Obviously _ — _ -^_ 21= 16= I «=1, so that 3t is iterable both with respect to SQ and g. Example 2. Let 31 consist of the points x, y in the unit square defined thus : For rational a; let ^ of norm B divides 3?„ into cells of volume d and 9?^ and SR^ into cells of volume d,., d^ where d = d^d,. Let h be any point of 53, lying in a cell d^. Let Id^ denote the sum b of all the cells d„ containing points of 21 whose projection is h. Let ^d, denote the sum of all the cells containing points of 21 dr whose projection falls in d,, not counting two d, cells twice. We have now the following theorem : If % is iterahle with respect to SQ, \iml.dAl.d.-td.\=0. (1 6=0 SB ''r l> ^°^ li < 2d. < -Ld.. b d^ a SB » S .'r Let now S = 0. The first and third members = 21, using I, 699, since 21 is iterable. Thus, the second and third members have the same limit, and this gives 1). 2. //■ 21 is iterable with respect to 4B, lim Idrld. = i. a=o 35 b This follows at once from 1). 3. Let 21 = © • S &e 0 there exist a pair of points, 6j, b^, distinct or coincident in any cell d^ such that as b ranges over this cell, C,^ = MinC, + ^', C,^=M:ixO, + 0", |/3'|, |^"1 d=D ^ r,s > where the sum on the right extends over those cells containing no point of A. Also 31= rg = lim|2d,g + 2(i,g\, (2 where the second sum on the right extends over those cells d^ containing no point of B. Subtracting 1), 2) gives = lim { Aj, - l,dM] + lim | ^d,ds - l.dM]- 18 POINT SETS AND PROPER INTEGRALS As each of the braces is > we have 14. We can now generalize the fundamental inequalities of I, 733 as foUows : Let /(^i ••• x^) he limited in the limited field 31 = S3 • S? iterable with respect to ©. Then For let us choose the positive constants A, B such that f + A>0, f-B<0, inSl. Let us effect a cubical division of the space of 9t„, of norm S into cells d. As in 13, this divides 9?^, 9?^ into cells which we denote, as well as their contents, by d„ d^. Let b denote any point of SB. As usual let m, M denote the minimum and maximum of / in the cell d containing a point of 21. Let m', M' be the corresponding extremes of /when we consider only those points of 21 in d whose projection is h. Let |/| < J' in 21. Then for any 6, we have by I, 696, - B(l.d, - g) + ^md, < ff, (2 or since m>m'. In a similar manner f^f<:LMd, + ACEd,-^). (3 Thus for any b in SS>, 2), 3) give - BC2d, - £) + ^md, < ff< -LMd, + ^(2rf, - g). (4 b b ^(£ b b Let jS > be small at pleasure. There exist two points Jj, b^ dis- tinct or coincident in the cell d„ for which /./=^'+^^' X/=''+^^ ITERABLE FIELDS 19 where | /Sj [, | /Sj [ -i j8 and S^, and gj stand for Sj^, gj^, and finally where j = Min r /, J-= Max (/ for all points 6 in d^. Let c = Min E in c?^, then 4) gives - 5(2d, - c) + Sm(i, there exists a d^ such that for any rectangular divisions D', D" of norms < d^ ■=\s -s <2li, 2l2--)=0. ^'* 2ii=a,ii), 2i,=a,i|), 2i3=a,iD- '^^^'^ Cr(2li, 2l2-) = (0*, 2*),- i)i;(2li, 2t2-) = 2li. 3. Let 2l>2li>3r2>2l3>" (1 ^^* © = i>H2l,li, 2l2,-)- L^* 2l = 2tj + gj, 2ll = 2(2 + (52,■••• ^^^° 9l = ® + gj + g2+ ... Let us first exclude the = sign in 1). Then every element of 21 which is not in 35 is in some 2l„ but not in 2l„+i. It is therefore in S„+i but not in 2l2 — ^ts ••• ie a set of limited complete point aggregates. Then SS = I)v0. Moreover 53 is complete. Let a„ be a point of 2l„, w = 1, 2, ■•• and 31 = «i, «2' '''3 "■ Any limiting point a of 31 is in every 3l„. l^^or it is a limit- ing point of But all these points lie in 3l„, which is complete. Hence a lies in 3lmi and therefore in every 3[j, 3l2i ••• Hence a lies in -93, and «>0. 53 is complete. For let /3 be one of its limiting points. Let ^11 *2' *3' ■■■ = ^• As each 6„ is in each 3l„5 and 3In is complete, /3 is in 2l„. Hence /3 is in 53. 2. iei 31 Je a limited point set of the second species. Then i>f (31', 21", 31'", -) > 0, and is complete. For 3l<"' is complete and > 0. Also 3[("' >^3l("+i). 19. Let 3(i, 2I2 - lie in SQ;let'Si=U\'!HJ. Let A„ be the com- plement of 3l„ with respect to SS, so that J.„ + 2l„ = 53. Let A = I>v\A^\. Then A and 31 are complementary., so that J. + 31 = 53. POINT SETS 25 For each pointy of -33 lies in some 2l„, or it lies in no 2l„, and hence in every j4.„. In the first case 6 lies in 31, in the second in A. Moreover it cannot lie in both A and 21. 20. 1. Let %<%<% - (1 be an infinite sequence of point sets whose union call 31. This fact may be more briefly indicated by the notation ^=U(_^,liml„. (2 That the inequality may hold as well as the equality in 2) is shown by the following examples. Example 1 . Let 3l„ = the segment f - , 1 J • '^^^^ 3l = i7J2t„i = (0M). i = L 3l„ = ^^^ = l. n Example Z. Let o„ denote the points in the unit interval whose abscissae are given by a; = — , wi < w = 1, 2, 3, ••■ m, w relatively prime. n Let 2l„ = ai + - + a„. is the totality of rational numbers in (0*, 1*). ^^ 1=1 and 2l„ = 0, we see t > lim 1„. 2. Let «i>S2>- (S Let S be their divisor. This we may denote briefly by «8 = 2>K«Si>«2 >■•■). Obviously when S5i is limited, B < lim «„. 26 POINT SETS AND PROPER INTEGRALS Example 1. Let ©„ = the segment (0, - 1 ■ Then :g3^2)?;f«B„| =(0), the origin. ^^""^ « = 0. limS„ = lim- = 0, n ^"^^ « = lim «„. Example 2. Let 3l„ be as in 1, Example 2. Let b„ = 31 — 3l„. ^^^ S„ = (l, 2) + b„. ^^'■^ SB = the segment (1, 2) and «„ = 2. 3. Let 53i < ©2 < ■■■ ^^ unmixed parts of 31. ie^ 53„ = 31- -te* $8 = Z7 {93„} . Then S = 31 - 48 w discrete. For let 31 = 53n + S» ; then g„ is an unmixed part of 31. Hence Passing to the limit w = oo, this gives limg„=0. Hence S is discrete by 2. 4. We may obviously apply the terms monotone increasing, monotone decreasing sequences, etc. [Cf. 1, 108, 211] to sequences of the type 1), 3). 21. ie( e = 31 + ©. if 31, « are complete, S = S + S. (1 ^"^^ S = Dist(3l, S)>0, since 31, S are complete and have no point in common. Let D be a cubical division of space of norm d. If d is taken sufficiently small 31/), Sfl have no cells in common. Hence Letting d = vi^e get 1). POINT SETS 27 22. 1. -5'' 21, SS^ire complete, so are also g=(2I, «B), S) = i>«(3l, S). Let us first show that £ is complete. Let c be a limiting point of g. Let Cj, Cj, ••• be points of S which = c. Let us separate the e„ into two classes, according as they belong to 21, or do not. One of these classes must embrace an infinite number of points which = c. As both 21 and © are complete, c lies in either 21 or ^8. Hence it lies in g. To show that J) is complete. Let c?j, d^, ■■■ be points of 35 which = d. As each d„ is in both 21 and SS, their limiting point d is in 31 and SS, since these are complete. Hence d is in J)- 2. J/ 21, 95 are metric so are e = (2l, «) S) = 2)i>(2l, S). For the points of Front © lie either in Front 21 or in Front ©, while the points of Front © < Front 2[ and also < Front Sd. But Front 21 and Front SB are discrete since 21, SB are metric. 23. Let the complete set 2t have a complete part SB- Then how- ever small € > is taken, there exists a complete set S in 21, having no point in common with SB such that g>I-«-e. (1 Moreover there exists no complete set S, having no point in common with SB such that _ _ — g>2l-«. The second part of the theorem follows from 21. To prove 1) let 2) be a cubical division such that i^=2i + e', «„ = « + 6", 0H2I, «). ^^^^ 2i + « = U + ©. (1 ^°^^«* U = 5!l + A Then ^ contains complete sets C, such that ^>U-a-£, (2 but no complete set such that a>U-i, (3 by 23. On the other hand, « = ^ + T). Hence A contains complete sets C, such that ^>«-^-e, (4 but no complete set such that 0>^-T). (6 From 2), 3), and 4), 5) we have 1), since e is arbitrarily small. 25. Let :j) = 2)i;(2li>Sa2>2l3>-), oh that S each 2l„ heing complete and such that 2l„ > some constant k. Then POINT SETS 29 For suppose . l^k-^>Q. Let 7 , ^ n Then by 23 there exists in 3lj a complete set Sj, having no point in common with 3)) such that ei>i]-©-e; or as 91] > k, such that ^ ^ Then by 24, _^ _^ _ ^ ^ _ _^ -^_ >3l2 + (Si -©-€)- Si = i2-^-e >V- Thus 3t2 contains the non-vanishing complete set Q^ having no point in common with 2). In this way we may continue. Thus 3lj, IH^-i ■■■ contain a non-vanishing complete component not in 35, which is absurd. Corollary. Let 21 = (Slj < Slg < •• *« complete. Then 1„ = S. This follows easily from 23, 25. CHAPTER II IMPROPER MULTIPLE INTEGRALS 26. Up to the present we have considered only proper multiple integrals. We take up now the case when the integrand f{x-^ • ■ • x^) is not limited. Such integrals are called improper. When m=l, we get the integrals treated in Vol. I, Chapter 14. An important application of the theory we are now to develop is the inversion of the order of integration in iterated improper integrals. The treatment of this question given in Vol. I may be simplified and generalized by making use of the properties of improper multiple integrals. 27. Let 31 be a limited point set in m-way space 3i„. At each point of 21 let f(x-y ■ ■ • x^) have a definite value assigned to it. The points of infinite discontinuity of / which lie in 91 we shall denote by 3?- In general ^ is discrete, and this case is by far the most important. But it is not necessary. We shall call ^ the singular points. Example. Let % be the unit square. At the point x — ~i r ^ y = -i these fractions being irreducible, let /= ws. At the other points of 31 let / = 1. Here every point of 21 is a point of infinite discontinuity and hence 3< = 21. Several types of definition of improper integrals have been proposed. We shall mention only three. 28. Type I. Let us effect a division A of norm S of 3?^ into cells, such that each cell is complete. Such divisions may be called complete. Let 2ls denote the cells containing points of 31, but no point oi^, while 2lj may denote the cells containing a point of ^. Since A is complete, / is limited in 2lj. Hence / admits an upper and a lower proper integral in 2ls- The limits, when they exist, _ lim r /, lim f /, (1 30 GENERAL THEORY 31 for all possible coiaplete divisions A of norm S, are called the lower and upper integrals of / in 31, and are denoted by or more shortly by ffd%, Cfd% (2 When the limits 1) are finite, the corresponding integrals 2) are convergent. We also say/ admits a lower or an upper improper integral in 31. When the two integrals 2) are equal, we say that / is integrable in 31 and denote their common value by r/^3l or by Cf. (3 ./a «^a We call 3) the improper integral of f in%\ we also say that / admits an improper integral in 31 and that the integral 3) is convergent. The definition of an improper integral just given is an extension of that given in Vol. I, Chapter 14. It is the natural develop- ment of the idea of an improper integral which goes back to the beginnings of the calculus. It is convenient to speak of the symbols 2) as upper and lower integrals, even when the limits 1) do not exist. A similar remark applies to the symbol 3). Let us replace / by |/| in one of the symbols 2), 3). The resulting symbol is called the adjoint of the integral in question. We write — — When the adjoint of one of the integrals 2), 3) is convergent, the first integral is said to be absolutely convergent. Thus if 4) is convergent, the second integral in 2) is absolutely convergent, etc. 29. Type II. Let \, /*>0. We introduce a truncated func- tion f),^ defined as follows : A^ =fQci - ^m) when - X /a. 32 IMPROPER MULTIPLE INTEGRALS We define now the lower integral as ff= lim f f,^. A similar definition holds for the upper integral. The other terms introduced in 28 apply here without change. This definition of an improper integral is due to de la VallSe Poussin. It has been employed by him and M. G-. D. Richardson with great success. 30. Type III. Let «, y8>0. Let 3l.p denote the points of 21 at which -a 2, /3 arbitrarily large. Hence and thus ^ 32. In the following we shall adopt the third type of definition, as it seems to lead to more general results when treating the im- portant subject of inversion of the order of integration in iterated integrals. We note that if /is limited in 21, lim I /= the proper integral I /. a,p="r_ For a, y3 being sufficiently large, 2l„p = Also, if 31 is discrete, r/= r/=o. ^5J For 2l„g is discrete, and hence Hence the limit of these integrals is 0. 33. Let wi=|Min/| , il!f= |Max/| in 31. Then - - lim I /=lim I /, m finite. lim f /=lim f /, M finite. For these limits depend only on large values of a, ^, and when m is finite. „, „, j! n -^^ ^m,^ = Kp ' forall«>r«. Similarly, when iHf is finite a.,8 = 3l„,^ , forall^>ilf. 34 IMPROPER MULTIPLE INTEGRALS Thus in these cases we may simplify our notation by replacing ^y 21-. , 21, , respectively. 2. Thus we have : I / = lim j /, when Min/ is finite. Xf = lim I / , when Max/ is finite. 3. Sometimes we have to deal with several functions /, a. Let D be a cubical division of space of norm d. Then ^ being fixed, J/=lim2 mA, (1 r f^lim-Lmid',, (2 using the notation so often employed before. GENERAL THEORY 35 But each cell d^ of^^p lies among the cells dj of 2la p. Thus we can break up the sum 2), getting lm[d[ = -2m[d[ + ^m'Jd['. Here the second term on the right is summed over those cells not containing points of SI^p. It is thus < 0. In the first term on the right wi/ < »m,. It is thus less than the sum in 1). Hence Thus ^ p. In a similar manner we may prove the second statement ; let us turn to the third. We need only to show that j / is monotone decreasing. Let «'>«. Then -~ I = lim 23f,d.. (3 f =lim2if;U'. (4 ** — a ** — a As before ^M[d[ = ^M[d[ + -EMi'd'J. (5 But in the cells d,, M' = M,. Hence the first term of 5) is the same as 2 in 3). The second term of 5) is < 0. The proof follows now as before. 35. If Max / is finite and if are limited, Cf is convergent and ff< f /• If Min / is finite and i are limited, i f is convergent and ff< ff- ■ *^aa •'a 36 IMPROPER MULTIPLE INTEGRALS For by 34 are limited monotone functions. Their limits exist by I, 277, 8. 36. If M= Ma.x f is finite, and j f is convergent, the correspond- ing upper integral is convergent and where f>_ — a in 3l_„. Similarly, if m= Miii/ is finite and I f is convergent, the corre- sponding lower integral is convergent and Let us prove the first half of the theorem. We have Cf= lim f Now r r r ~ I / < I < < ^31... We have now only to pass to the limit. 37. If J f is convergent, and 53 < 31, does not need to converge. Similarly // does not need to converge, although | / does. Example. Let 21 be the .unit square ; let 55 denote the points for which x is rational. GENERAL THEORY 37 • /= 1 when X is irrational = - when X is rational. y Then ^ ^ I / = 1 ; hence i / = 1. On the other hand, JSB„ Jo Jl w * "^ Hence L = 3 ^ X = lira i =linilog/8= +oo is divergent. 38. 1. In the future it will be convenient to let ^ denote the points of 21 where/ > 0, and 31 the points where/ < 0. We may call them the positive and negative components o/2t. 2. If j f converges, so do I /. If \ f converges, so do I /. For let us effect a cubical division of space of norm d. Let /3' > /8. Let e denote those cells containing a point of 'ip^ ; e' those cells containing a point of 'JP^/ but no point of 'p^ ; 8 those cells containing a point of ^^^ but none of ^p,. Then pi I =limJ2ilf; e + -LM',-- e' + -2M' Sj. Obviously iJf;>ilf, , i!f; = M5 , M,.<0. C - f =\\m\-2(iM',-M,)e + -2My--LM^-e'\. 38 IMPROPER MULTIPLE INTEGRALS We find similarly f -f =lim{2(J!fi-i(f,)e + 2i!fyi. (1 1/ -/ Now /Sg. Hence the same is true of the left side of 1). As corollaries we have : 3. If the upper integral offis convergent in 21, then If the lower integral off is convergent in 31, ff>ff JV<5«. For err 1 < I < etc. 4. Iff > and I / is convergent, so is Moreover the second integral is < the first. This follows at once from 3, as 21 = '^■ 39. If J f <^i^^ J^f converge, so do J^/. We show that I / converges ; a similar proof holds for j . To this end we have only to show that e>0;a, /S>0; f -f I 0. Let tta' » Tt„" denote cells containing only points of Sla'/s' , 5!la-/3" at which/< 0. We have Sil!f,c?. = 2+2; ; 2i!f;d,= 2 + 2 Subtracting, I SiC^;, - 'LMA I < I 'S.MA - Sik?;^?, I + I -LMA - ^MA\ ■ (2 Let if; = Max / for points of 31 in t = Osc/ in d,. Let a' > «, yS' > ^. Then ^a'p' - -4„3 = lim {2ft)[(il - 1(0 A\ ■ 40 IMPROPER MULTIPLE INTEGRALS Now any cell d, of 5S[„p is a cell of St^'ja-, and in d,. m[ > m,. Hence A^-^< > A^p. Thus Aa^ is a monotone increasing function of a, /S. On the other hand lim Aafi = 0, by hypothesis. Hence A^= and thus /is integrable in 21^^. Next let / be limited in SS, then |/|0, = where /<0. h=—f where/<0, = where />0. Thus g, h are both > and \f\ = 9 + ^- We call them the associated non-negative functions. 2. As usual let 3I„p denote the points of 21 at which — a. X*— //* X*=^-//^ <* provided the integral on either side of the equations converges, or provided the integrals on the right side of the inequalities converge. Let us prove 1); the others are similarly established. Effecting a cubical division of space of norm d, we have for a fixed /S, ( g=\im\'LMA + tO-d,\ = lim ^MA = r /• (5 Thus if either integral in 1) is convergent, the passage to the limit /3 = oo in 5), gives 1). 2. If ) f is convergent, j g converge. If I f is convergent, J h converge. This follows from 1 and from 38. GENERAL THEORY 43 3. If I / is convergent, we cannot say that | / is always con- vergent. A similar remark holds for the lower integral. / = 1 at the rational points of 21 = (0, 1) Then = at the irrational points. X 4. That the inequality sign in 2) or 4) may be necessary is shown thus : Let / = — = for rational a: in 31 = (0, 1) Va; = = for irrational x. V; X Then Sff=0 , ff=2. 44. 1. ff=Cg- lim f h, (1 r/= lim fa- fh, (2 provided, 1° the integral on the left exists, or 2° the integral and the limit on the right exist. For let us effect a cubical division of norm d. The cells con- taining points of 31 fall into two classes : a) those in which / is always < 0, 6) those in which/ is >0 for at least one point. In the cells a), since/=^— A, Max/ = Max(^-^)=Max^-MinA, (3 as Max g = 0. In the cells J) this relation also holds as Min A = 0. Thus 3) gives r /=) g-J h. (4 •^aafl -^^a^ ^a<^ 44 IMPROPER MULTIPLE INTEGRALS Let now a, j8 = oo. If the integral on the left of 1) is conver- gent, the integral on the right of 1) is convergent by 43, 2. Hence the limit on the right of 1) exists. Using now 42, 2, we get 1). Let us now look at the 2° hypothesis. By 42, 2, .s^^^ai/ J%^' Thus passing to the limit in 4), we get 1). 2. A relation of the type ^% ^% ^Sl does not always hold as the following shows. m Example. Let/ = n at the points x— ^- = — n tor X = — — - = — 1 at the other points of 21 = (0, 1). Then r/=-l r^ = Ch=0. «^a Jn J% 45. ^ j f is convergent, it is in any unmixed part © o/ 21. Let us consider the upper integral first. By 43, 2, exists. Hence a fortiori, exists. Since 21 = .© + S is an unmixed division, f h= C h+ C h. ^2(a/3 V?!8.3 ^£,3 Hence f ^< f h. GENERAL THEORY 45 As the limit of thg right side exists, that of the left exists also. From this fact, and because 1) exists, exists by 44, i. A similar demonstration holds for the lower integral over 53- 46. If%.i, 3l2 ■■■ '^mfo'Tin an unmixed division of%, then J% Jail ^^m provided the integral on the left exists or all the integrals on the right exist. Yov if 2l„_ j(3 denote the points of 3la3 in 2tm' '^^ have Now if the integral on the left of 1) is convergent, the integrals on the right of 1) all converge by 45. Passing to the limit in 2) gives 1). On the other hypothesis, the integrals on the right of 1) existing, a passage to the limit in 2) shows that 1) holds in this case also. 47. If j f and j f converge, so does i \f\, and J(jj J'ni J% For let A^ denote the points of 21 where 0<|/| 0. Then -p^ 77 r/= r i/i is convergent by 38, 3, since ( |/| is convergent. Similarly, -^ is convergent. The theorem follows now by 39. 2. If \ \f\ converges, so do Jsi _ _ fff , fh. (1 For by 1, /•^ both converge. The theorem now follows by 43, 2. 3. For ff (2 JoM fo converge it is necessary and sufficient that 28 convergent. For if 3) converges, the integrals 2) both converge by 1. On the other hand if both the integrals 2) converge, P ' ff converge by 38, 2. Hence 3) converges by 47. 4. Iffis integraUe in 31, so is \f\. For let Ap denote the points of 21 where < \f\ < ^. Then ri/i=umXi/i, and the limit on the right exists by 3. GENERAL THEORY 47 But by 41, 1,/is jptegrable in A^. Hence |/| is integrable in Ap by I, 720. Thus 49. From the above it follows that if both integrals // converge, they converge absolutely. Thus, in particular, if // converges, it is absolutely convergent. We must, however, guard the reader against the error of sup- posing that onli/ absolutely convergent upper and lower integrals exist. Example. At the rational points of 21 = (0, 1) let At the irrational points let X Here — )/=! r/=-oo. Thus, / admits an upper, but not a lower integral. On the other hand the upper integral of / does not converge absolutely. For obviously r|/| = + oo. 50. We have just noted that if jf{Xi-X^) is convergent, it is absolutely convergent. For m = 1, this result apparently stands in contradiction with the theory developed in Vol. I, where we often dealt with convergent integrals which do not converge absolutely. 48 IMPROPER MULTIPLE INTEGRALS Let us consider, for example, . 1 ,sin- J= I dx= hfdz Jo X ^W If we set a; = -, we get u sin M J aw, I u which converges by I, 667, but is not absolutely convergent by I, 646. This apparent discrepancy at once disappears when we observe that according to the definition laid down in Vol. I, J= R lim I fdx, 11=0 '^a while in the present chapter J= lim I fdx. Now it is easy to see that, taking a large at pleasure but fixed, Xfdx =00 as iS = 00, so that J" does not converge according to our present definition. In the theory of integration as ordinarily developed in works on the calculus a similar phenomenon occurs, viz. only absolutely convergent integrals exist when m > 1. 51. 1. If I \f\ is convergent, Ja IPi0 in^. Let the integral converge. If ff=ff+a, (1 then for any unmixed part S3 < 21. f/=r/+«', (2 where < a' < «. (3 For let 31 = 48 + S. Then 21^ = 48^ + 2^ is an unmixed division. Also r r r J% JSQ J^ = f+a by 1) 50 IMPROPER MULTIPLE INTEGRALS Hence r^f^f+T+,, (4 ji/sB ^e ■i/iU.j ^^^ From 2) " - ^ — IX-XI ''*^ <«, which establishes 3). 53. If the integral I I/I (1 converges, then I ^ ■ 6>0, <7>0, L/ <7. (3 Let us suppose first that/>0. If the theorem is not true, there exists, however small ff>0 is taken, a 53 satisfying 3) such that /! Then there exists a cubical division of space such that those points of 31, call them @, which lie in cells containing a point of S3, are such that @ 0, that also. Let us now take /8 so that /a ^a^ Then ^ ^ and 0 with a, 0. Then lim S„3 = 0. a, ^=00 For if not, let llm ©a(3 = Z, l>0. a, P=x Then for any < X, < Z, there exists a monotone sequence [«„, /8„S such that ©a„p„ > 't f or n > some m. Let /i„=Min(a„, /3„), then |/| > M„ in 5D.„ft., and /i„=oo. Hence ^ I i/|>M„5l=co. (1 But UDm/Sb being a part of 91 by 38, 3. This contradicts 1). 2. Definition. We say J.„, <, is conjugate to 2la3 with respect to/. 55. 1. As usual let -a0, a 0<6„+j<6„, such that if we set f*" dx_p as 6„ = 0. Let now / = 1 for the rational points in 31 = (0, 1), = - for the irrational. X Then r 1 Let ^1^ I ^=(7j, 0<6i0, a>0, \ff\0, and the limit 4) exists, the above shows that /i= lim I h , v= lim I h exist and that X= fJL+V. Then a, b being the same as in 6), and we show that o^v'a, b; then X.*s;.«<| if we take S„, p be a cubical division of space. Let Sap denote the points of !j)ap lying in cells of i?„p, containing no point of the sets 1). Let Sa/3 = 31/, aP — Sa/3- 66 IMPROPER MULTIPLE INTEGRALS Then 2)„p may be chosen so that g«/3 = 0. Now 7: C C since the fields are unmixed. By 56, the second integral on the right = as a. /8 = 00 . Hence ^'"^ £ ^= ^^"^ / •^• Similar reasoning applies to g nndf+g. Again, ^ r r L Cf+ff^2. Thus the upper content of the last set in 1) does not = as a, ^ = 00 and condition 1° is not fulfilled. Also relation 2) does not hold in this case. For Jif + g} = 2 , jr/=0 , fg = 0. 59. Ifc>0, t^enj\f=of^f, (1 {fc<0,thenf^cf = ef^f, (2 provided the integral on either side is convergent. For r cf=cf / ifoO (3 = cr / ifc<0. (4 Let c > 0. Since — « 0. 58 IMPROPER MULTIPLE INTEGRALS Similarly ^^^ ^ ^ ^^ ^ ^ ^^^^ ^ ^ 0_ Thus 3), 4) give '2tc/,aP ''H/- c<0. We now need only to pass to the limit «, /8 = oo . 60. 2/ei one of the integrals converge. If f = g, except at a discrete set !© in 31, 5o(A integrals converge and are equal. A similar theorem holds for the lower integrals. For let us suppose the first integral in 1) converges. Let then •'a '^A *^s '^A Now - - - |„ff = lim I g = lim | or = Hmr / = //. (3 Thus the second integral in 1) converges, and 2), 3) show that the integrals in 1) are equal. / f^ J 9 (1 converge. Let f>g except possibly at a discrete set. Let 7«^ = 0, %afi =0, as a, yS = oo, tAen — — 61. 1. ie« RELATION BETWEEN THE INTEGRALS OF TYPES I, II, 111 59 For let dap be deined as in 58, l. Then Let «, /8 = 00 , we get 2) by the same style of reasoning as in 58. 2. If the integrals 1) converge, and their singular points are dis- crete, the relation 2) holds. This follows by 58, 2. 3. If the conditions of 1 do not hold, the relation 2) may not be true. Uxample. Let 21 denote the rational points in (0*, 1*). Let f=n at a; = — in 21, n g=l in 21. Then /..^ . „, f>g in 21. But ^ f, |/=0 \ g = \. Relation between the Integrals of Types I, II, III 62. Let us denote these integrals over the limited field 21 by On , Vk , Pa respectively. The upper and lower. integrals may be denoted by putting a dash above and below them. When no ambiguity arises, we may omit the subscript 21. The singular points of the inte- grand/, we denote as usual by 3'' 63. If one of the integrals P is convergent, and ^ is discrete, the corresponding integral converges, and both are equal. P% = Pjij + -^31^1 using the notation of 28, ^°^ P%' = as S = by 56. 60 IMPROPER MULTIPLE INTEGRALS Hence Pg=lim^a, 8=0 = 0%, by definition. 64. If C is convergent, we cannot say that P converges. A similar remark holds for the lower integrals. Example. For the rational points in 21 = (0, 1) let 2 Va; for the irrational points let /(^) = - -• X Then _ 7^ _ j Ca = lim I f (x) dx = lim [ Va;] = 1. a=0 ♦'a " On the other hand, _ — Pa = lim f / „, 3=00 ./Sap does not exist. For however lai-ge /S is taken and then fixed, Jf= — 00 as a = 00. Sap 65. If C is absolutely convergent and Q is discrete, then both P converge and are equal to the corresponding integrals. For let I) be any complete division of St of norm S. Then using the notation of 28. Now since 0% 1/1 converges, Ca^ |/| = as S= 0. Again, 2) being fixed, if Kq/Sq are sufficiently large, r f=G%J a>«o, ^>^o- IIELATION BETWEEN THE INTEGKALS OF TYPES I, II, III 61 Hence 1), 2) giv^ I f=G%^ + e' |«'i<- for any S < some 8q. On the other hand, if Sg is sufficiently small, C^^C^, + ^' |e"|<| for80- Let S3 denote the points of 31 where |/|>j8. Then ^>^ for any ;8. Hence as /S = CX5 unless ^ = 0. 67. If Fjt/ i^ absolutely convergent, so is C. For let Z> be a cubical division of space of norm d. Then |/| 0. if Fst/ is convergent, there exists for each e>0, a (7 > SMcA ^Aa^ Fs/)= Jg/; (3 then the integral 2) is - It is important to note at once that although the integrand / is defined for each point in 31, the integrand <^ in 4) may not be. Example. Let 31 consist of the points (x, y') in the unit square : 2; = — , 0. for each point of S3. Thus the integrals 2) are not defined. 64 IMPROPER MULTIPLE INTEGRALS To provide for the case that may not be defined for certain points of SS we give the symbol 2) the following definition. where F = S when the integral 3) is convergent, or in the con- trary case r is such a part of S that -«=f in 21 = in ©. Then Similarly Hence 3. The set S„ being as in 2, we shall write 73. Let B„^ „ denote the points of Sd at which c„ > a-. Then if 21 is iterable, with respect to 53, lim5,,„ = 0. (1 For since 21 is iterable, 21 = 1 £ by definition. Hence S considered as a function of x is an integrable function in «B. Similarly _ ^ _ ^"=X^" and S„ is an integrable function in ©. ITERATED INTEGRALS 69 We have now * 7? 7? , - - -~. /, e = s„ + c„ , c„>o as S„, c„ are unmixed. Hence c„ is an integrable function in 93- But -fc.. As the left side = as n = 00 , lim/^c„=0. (2 But c- r As the left side = 0, we have for a given a- lim B^„ = 0, which is 1). 74. Let 31 = 93 • 6 Je iterable. Let the integral ff , />0 (1 ie convergent and limited in complete 93. Let (5„ denote the points of 93 at which — JfSl-'?/^'- Continuing we get ^^ ^ ^^^^ _ ^^g-. Ti^- s„>^-ruu...+ i)" ^2 4 Let now b = 2)?;|g3„|. Then b>S8-7, (4 by 25. Let 6„ denote those points of b for which 2) does hold. Then b=Jb„S. For let J be any point of 6. Since 1) is convergent, there exists a o-. such that at 5, J/<«' for any c such that c b„. "«"°^ liml„>l, which with 4), gives 3). 75. Let 31 = S3 • S Je iter able. Let the integral be convergent aiid limited in complete 93- lim f f /=0. (1 For let i) be a cubical division of 9tj, of norm c?. r f/ = lim 2ci. Min f/ = lim S^. Let i[ denote those cells of I) containing a point of @„ where S„ is defined as in 74. TUn Then ITERATED INTEGRALS 71 Let d" denote the other cells containing points of S3. Then where — 0< ff , 93 is complete , | I / is convergent. We have by 14, ^ -^ j^ -^ Passing to the limit gives f f < lim f r /. (3 and also r r r lira I I / < I / , finite or infinite. (4 Now € > being small at pleasure, there exists a Ctq such that -e+fCf's,.^e„ ^a^yn ♦^S8i/S« ^Bg^yn By 75, the last term on the right = as w = oo. Thus passing to the limit, r r/0 is small at pleasure. On the other hand, passing to the limit Q- = aa in 7), and then »i = oo, we get lim { {^ < ( ( . riO Thus 3), iO), 9), and 4) give 1). Let us now suppose that the middle term of 1) is divergent. We have as before r r — (r, (7 > 0. Then g=f+a>o, and we can apply 1) to the new function g. ^«^ r^= ff+a% (13 by 58, 6, since Q is discrete. Also by the same theorem, fg= Cf+a\im^y= Cf+ar, (i4 denoting by Sy the points of E where -<^lim6^>e„- ^S •/SB '^SB Hence ^ ^ Hence i=|r, (16 and thus T is integrable in S3. This result in 14) gives, on using 58, 3, From 12), 13), and 17) follows 1). 77. As corollaries of the last theorem we have, supposinff 21 to be as in 76, 1. Iffis integrable in 21 andf'>— G, then Iff < a, then Cf^ r Cf 2. Iff>— G and | is divergent, then -^ _ _ are divergent ITERATED INTEGRALS 75 3. If f >— G- and one of the integrals j \ f is convergent, then IS convergent. 78. Let 31 = SB • S fte iterahle with respect to SB, which last is com- plete or metric. Let the singular points ^ he discrete. If r/> (1 f f/, (2 both converge, they are equal. For let i)j, Dj ••• be a sequence of superimposed cubical divisions as in 72, 2. We may suppose as before that each S8„ = SB- Since 1) is convergent f>0, n,, \Cf-Cf\nQ. (6 We wish now to show that n- ( ( \n.. (7 When this is done, 6) and 7) prove the theorem. To establish 7) we begin by observing that rr=iim r r. 76 IMPROPER MULTIPLE INTEGRALS Now for a fixed w, «, /3 may be taken so that T shall embrace all the points of S„ for every point of 59. Let us set -'aSap'^r '^Bap'^l^n •^»a/3*^Y.. As «„^ = S, lim r r = r f by I, 724. On the other hand, Thus 7) is established when we show that SJjf\n,. (9 To this end we note that |/| is integrable in 21 by 48, 4. Hence by 77, 1, Also by I, 734, f\f\=ff\f\- (11 From 10), 11) we have for m > w^, since the left side = 0. But as in 8) Passing to the limit Cr = oo gives XX'-^!-XX.i^i+XX'^|- This in 12) gives 9). CHAPTER III SERIES Preliminary Definitions and Theorems 79. Let aj, a^, ag ■ ■ ■ be an infinite sequence of numbers. The symbol A = a^ + a^+ a^+ ■ ■■ (1 is called an infinite series. Let Ar, = a^ + a^-\ |-a„. (2 If lim A„ (3 n=oo is finite, we say the series 1) is convergent. If the limit 3) is infi- nite or does not exist, we say 1) is divergent. When 1) is conver- gent, the limit 3) is called the sum of the series. It is customary to represent a series and its sum by the same letter, when no con- fusion will arise. Whenever practicable we shall adopt the fol- lowing uniform notation. The terms of a series will be designated by small Roman letters,' the series and its sum will be denoted by the corresponding capital letter. The sum of the first n terms of a series as A will be denoted by A,,,. The infinite series formed by removing the first n terms, as for example, will be denoted by A„, and will be called the remainder after n terms. The series formed by replacing each term of a series by its nu- merical value is called the adjoint series. We shall designate it by replacing the Roman letters by the corresponding Greek or German letters. Thus the adjoint of 1) would be denoted by A= ai-|-a2 + a3+ •••= Adj ^ (5 ^^^^® a =\a\ "■n — I^Til" 77 78 SERIES If all the terms of of a series are > 0, it is identical with its adjoint. A sum of p consecutive terms as we denote by -4.„,p. Let T> , , . ^ ^ -D = fli, + tti, + a^, H , t] we shall call a positive term series ; similarly if its terms are all < 0, we call it a negative term series. If a„>0, n>m we shall say the series is essentially a pos- itive term series. Similarly if a„ < 0, n>m we call it an essen- tially negative term series. If A is an essentially positive term series and divergent, lim A„= -{-ao ; if it is an essentially negia,tive term series and di- vergent, lim A„= — CO. When lim J.„ = ± oo, we sometimes say ^ is ± oo. 80. 1. For A to converge, it is necessary and sufficient that e>0, m, |A„,p| m, p = l,2,-- (1 For the necessary and sufficient condition that lim An n=oD exists is r. , a a i e>0, m, \A^ — A„\m. (2 But ii V = n-\- p Thus 2) is identical with 1). PRELIMINARY DEFINITIONS AND THEOREMS 79 2. The two seriem A, A, converge and diverge simultaneously. When convergent, _ A = A, + A,. (3 For obviously if either series satisfies theorem 1, the other must, since the first terms of a series do not enter the relation 1); On the other hand, . _ j ■ j Letting p = 00 we get 3). 3. If A is convergent, ^„ = 0. ^°^ lim J„ = lim (A - .4 J = J. — lim A„ — A — A = 0. For A to converge it is necessary that a„ = 0. For in 1) take p = l; it becomes I a„^.i I < e n>m We cannot infer conversely because a„ = 0, therefore A is con- vergent. For as we shall see in 81, 2, 1 + 1 + H - is divergent, yet lim a„ = 0. 4. The positive term series A is convergent if A„ is limited. For then lim A^ exists by I, 109. 5. A series whose adjoint converges is convergent. For the adjoint A of A being convergent, €>0, m, |A„,p|TO, /)=1, 2, 3 ••• But A„,p = «„+! + «„+2 + ••• + a„+p> I ««+l + «n+2 + ••• + «n+p 1=1^ n, p r '^^^' \A„J some m', A„ — B„ > some term of A. Thus passing to the limit in 1), the theorem is proved. 7. Let A = a^ + a^+ •■■ The series B = ka^ + ka2+ •••, k^O, converges or diverges simultaneously with A. When convergent, B = kA. ^^^-^ B„ = kA„. We have now only to pass to the limit. From this we see that a negative or an essentially negative term series can be converted into a positive or an essentially positive term series by multiplying its terms by k = — 1. 8. Jff' A is simply convergent, the series B formed of its positive terms taken in the order they occur in A, and the series C formed of the negative terms, also taken in the order they occur in A, are both divergent. If B and C are convergent, so are B, F. Now A„ = B„^ + F„^, n = nj^ + n^. Hence A would be convergent, which is contrary to hypothesis. If only one of the series B, is convergent, the relation shows that A would be divergent, which is contrary to hypothesis. PRELIMINARY DEFINITIONS AND THEOREMS 81 9. The foUowing^heorem often affords a convenient means of estimating the remainder of an absolutely convergent series. Let A = a^ + a^+ ••■ he an absolutely convergent series. Let -B = ij + ij + ■■■ ^8 <^ positive term convergent series whose sum is known either exactly or approximately. Then *f | a„ | < J„, w > m For \ A \ ^ , I <-B„<5. Letting p = co gives the theorem. EXAMPLES 81. 1. The geometric series is defined by The geometric series is absolutely convergent when \g\<\ and di- vergent when |^|>1. When convergent, When^=?i=l, ^=l+g + g^+...+g-^ + -^!^. Hence 1 ^„ G- =_i » . "^re — -I 1 1-g 1-g When |5'|<1, lim^" = 0, and then lim (r„ 1-^ When 1^1 >1, lim^" is not 0, and hence by 80, 3, Q- is not conver- gent. 2. The series j^^ i , 1 , 1 , J_ . ... (3 82 SERIES is called the general harmonic series of exponent fi. When /* = 1, itbecomes j^= i + ^ + j + ^. + ... (4 the harmonic series. We show now that The general harmonic series is convergent when /* > 1 and is di- vergent for A'< 1. Let /i>l. Then — + - —4-— ! = -!-= • <1 i+i + i4-f n" n Thus 3) is divergent for /* < 1, if it is for /* = 1. But we saw, I, 141, that lim <^ = 00, hence J" is divergent. It is sometimes useful to know that lim-:^=l. (6 log n In fact, by I, 180, 1 lini^=lim^ Jn-Jn-^ ^lijj, « logW logW-log(M.-l) \og(_^!_\ = lim , ^ ,. =1. '»<'-;)" PRELIMINARY DEFINITIONS AND THEOREMS 83 Since * n> log n >> Z^^ • • ■ we have lim^=0 ; lim^ = oo , r>l. (7 n l^n Another useful relation is ^„=l + J + 5+- + ->log(w + l). (8 For log(l + w) — logm = log(l + — )< — . \ mj m Let m=l, 2--n. If we add the resulting inequalities we get 8). 3. Alternating Series. This important class of series is defined as follows. Let aj > ag > ag > • • • = 0. Then A= a^ — a^ + a^ — a^+ ■■■ (9 whose signs are alternately positive and negative, is such a series. The alternating series 9) is convergent and |A|<«n+l- (10 For let p>S. We have ^«,p = (-l)"!«n+i-««+2+ •••(-l)''^X+pi = (-l)»P. If p is even, P = («»+! - a»+a) + ■•■ + («n+p-i - osn+p)- If p is odd, ^ = («n+l - an+z) + ••• + (an+p-2 " «n+p-l)+ «n+p- Thus in both cases, i'>«„+i-«n+2>0. (11 Again, if p is even, -P = <*n+l — ('^n+2 " ''^n+s) — •■ • — (ftn+p-Z ~ <*n+p-l) ~ '''n+y * In I, 461, the symbol " lim " in the first relation should be replaced by Um. 84 SERIES If p is odd, P = '''n+l ~ ('''n+2 "■ ^n+i) — ••• — C«„+p_l — «n+p)- Thus in both cases, P < a„+i - (a„+2 - Sn+s) < «n+i- (12 From 11), 12) we have < a„+i - a„+2 < I ^„, p I < a„+i «- (a„+2 — a^+j). Hence passing to the limit p= cc, 0<|i:„| 0. Then A = X 1 ^■^ [(l+ai)-(l-(-a„_i) (l + ai)-(l + «J is telescopic. Thus °<''-° '- (! + ».) .'.(! + ..) <'• and A is convergent and < 1. 7'(l+ai).-(l+aO 1 «„ = (), 86 SERIES Example 3. is telescopic. '7(a; + w-l)(a; + w) = vl_i 1 ^ {.x + n — l x + x^O, -1, -2, ^,= --- + w 1 . 1 a; x+ n x 82. Dim's Series. Let A = a-^ + a^+ ■■■ be a (divergent positive term series. Then is divergent. For -Aj A^ Ag J) — '^'n+i I ... I ^m+p -"m+1 -'^m+p >-i («m+i+ ••• 4-a„+p) m, p = 1 -^m "I" -^171, p -^m+p Letting m remain fixed and p = , we have D^ > 1, since -^m+p = °o- Hence i) is divergent. Let Hence Let Then ^=1 + 1 + 1+ ••• Then^„ = w. D=\-'r^+\+ •••is divergent. i>=U 1 1 2(1 +i) 3(l+i + i) is divergent, and hence, a fortiori. ^ mi nA^ But Hence ^„_i>logw. ^^ fn\ c\Cf in. 9. Ino "Y w log n 2 log 2 3 log 3 is divergent, as Abel first showed. + PRELIMINARY DEFINITIONS AND THEOREMS 87 83. 1. AbeVs Sertks. An important class of series have the form B = ajij + «2*2 + "^3*3 + ••■ CI- As Abel first showed how the convergence of certain types of these series could be established, they may be appropriately called in his honor. The reasoning depends on the simple identity (^Abel's identity^, where as usual -4„^„ is the sum of the first m terms of the re- mainder series A„. From this identity we have at once the fol- lowing cases in which the series 1) converges. 2. Let the series A = a-^^ + a^+ ■■• and the series ^^\t„+i — t„\ converge. i^Let the t„ be limited.^ Then B = a-^t-^ + a^^ -|- ••• converges. For since A is convergent, there exists au m such that |^„^p|m, jo=l, 2, S--- Hence 3. Let the series A = a^+ a^+ ■■■ converge. Let t-^, t^, t^--- be a limited monotone sequence. Then B is convergent. This is a corollary of 2. 4. Let A = a.y + a^+ ■■■ besuchthat\A„\m. 5. Let\A„\t2>t^> ■■■ =0. Then B is convergent. This is a special case of 4. gg SERIES 6. As an application of 5 we see the alternating series S = ti — f 2 + <3 — • ■ • is convergent. For as the A series we may take J. = l— 1 + 1 — 1+ •■• as |-4„|<1. 84. Trigonometric Series. Series of this type are C = flj, + aj cos a; + a^ cos 2 a; + «3 cos 3 a; + ••• (1 *? = a^ sin a; + a^ sin 2 a; + flj sin 3 a; + ••• (2 As we see, they are special cases of Abel's series. Special cases of the series 1), 2) are r = J + cosa; + cos 2a; + cos 3a;+ ••• (3 2 = sina; + sin 2a; + sin 3a;+ ••• (4 It is easy to find the sums r„, S„ as follows. We have . , 2w— 1 2m+l 2 sni mx sin lx = cos x — cos -; x. Letting m = 1, 2, ••• n and adding, we get 2 sin ^ a; ■ 2„ = cos I a; — cos — - — x. (0 Keeping x fixed and letting w = oo, we see 2„ oscillates between fixed limits when x^Q, ± 2 tt, ■■• Thus 2 is divergent except when a; = 0, ±7r, ••• Similarly we find when x4=-2 mir, P _ sin(w-^)a; ,n 2 sin ^ a; Hence for such values r„ oscillates between fixed limits. For the values a; = 2 mir the equation 3) shows that r„ = + oo. From the theorems 4, 5 we have at once now If S I a^+i — a„ I converges and a„ = 0, and hence in particular if «! > dg _2 ■■• =0, the series 1) converges for every x, and 2) converges for x^2 mir. 1 f in 3) we replace a; by a: + tt, it goes over into A = ^ — cos a; + cos 2 a; — cos 3 a; + ••• (7 PRELIMINARY DEFINITIONS AND THEOREMS 89 Thus A„ oscillate* between fixed limits if x^±(2m — l^ tt, when n=^ cc. Thus If 2 1 a„^.i + a„ I converges and a„ = 0, and hence in particular if a^>a^> ••• =0, the series a„ — ajCOsa; + a^cos 2a;— a3COs32;+ ••• converges for a; ^ (2 m — 1) tt. 85. Power Series. An extremely important class of series are those of the type P = aQ + a-y(^x — a) + a^ (x — a)^ + a^(x — d)^ + ••• (1 called power series. Since P reduces to a^ if we set a: = a, we see that every power series converges for at least one point. On the other hand, there are power series which converge at but one point, e.g. a^ + V.(x-a~) + 2\(x-ay + n(x-ay+ - (2 For if a; =ifc a, lim n\ | a; — a | " = ao, and thus 2) is divergent. 1. If the power series P converges for x=b, it converges absolutely within -n ^ N ^ i II Jj)Xa) , X = I a — |. If P diverges for x=b, it diverges without D^Qa). Let us suppose first that P converges at b. Let a; be a point in D^, and set | a; — a | =f. Then the adjoint of P becomes for this = «0 + «lX • J + «2X2 .(ij + «3X3 . g ^ + ^"* lim«„X"=0, since series P is convergent for x = b. ^^'^°^ «„X"l,2, ... T^- n.<^(i.i.....e)<-5 and n is convergent. X If P diverges at a; = 6, it must diverge for all 6' such that I 5' — a I > X. For if not, P would converge at b by what we have just proved, and this contradicts the hypothesis. 90 SERIES 2. Thus we conclude that the set of points for which P con- verges form an interval Qa — p, a + p} about the point a, called the interval of convergence ; p is called its norm. We say P is developed about the point a. When a = 0, the series 1) takes on the simpler form ^^ ^ ^^ +a^+ - which for many purposes is just as general as 1). We shall therefore employ it to simplify our equations. We note that the geometric series is a simple case of a power series. 86. Cauchy's Theorem on the Interval of Convergence. The norm p of the interval of convergence of the power series, P =af^ + a-^x + a^x^+ •■■ is given by j - = limVa„ «„=«„. P We show n diverges if ^>p. For let P ^ Then by I, 338, l, there exist an infinity of indices tj, tj ••• for which Hence ^ a, and thus ..i ^ ^,.,im i since f /3 > 1. Hence ^ w n is divergent and therefore H. We show now that H converges if f < p. For let im. TESTS OF CONVERGENCE FOR POSITIVE TERM SERIES 91 Hence • ^ . Thus and n is convergent. Example 1. o o ^ri + 21 + 3!^"' ^''' ^^r.= ^^0 by I, 185, 4. Hence p = co and the series converges absolutely for every x. JExample 2. ^ ~6 1~3 "^5 ^^'^ V^„ = A=l by I, 185, 3. Hence jO= 1, and the series converges absolutely for | a; |< 1. Tests of Convergence for Positive Term Series 87. To determine whether a given positive term series ^z= flj 4- a^ + ••• is convergent or not, we may compare it with certain standard series whose convergence or divergence is known. Such com- parisons enable us also to establish criteria of convergence of great usefulness. We begin by noting the following theorem which sometimes proves useful. 1. Let A, B he two series which differ only hy a finite number of terms. Then they converge or diverge simultaneously. This follows at once from 80, 2. Hence if a series A whose convergence is under investigation has a certain property only 92 SERIES after the mth term, we may replace A by A^, which has this property from the start. 2. The fundamental theorem of comparison is the following : Let A=a^ + a2+ ■■■, j5 == Jj + ^2 + ■ ■ • ^^ **"" positive term series. Let r > denote a constant. If a„< rb„, A converges if B does and A < rB. If a„> rh^, A diverges if B does. For on the first hypothesis On the second hypothesis A„>rB,. The theorem follows on passing to the limit. 3. From 2 we have at once : Let A = a-^ + a^-\- •••, B = h-^^-\-h^+ ••• he two positive term series. Let r, s be positive constants. If r<^ 0. For U I ^ 1 which is thus < the wth term in the convergent geometric series Example S. ^ = T i log ^^ is convergent. ^""^ logfl + lUi-^ 0<^„<1. n^\ n j ti^ Thus A is comparable with the convergent series ^—^• 88. We proceed now to deduce various tests for convergence and divergence. One of the simplest is the following, obtained by comparison with the hyperhannonic series. Let A = a^-\- a^-\- ••• he a positive term series. It is convergent if lim a„W < CO ■ , /Lt > 1, and divergent if lim wa„ > 0. 94 SERIES For on the first hypothesis there exists, by I, 338, a constant 6r > such that «n<— w=l, 2, ■•• Thus each term of A is less than the corresponding term of the On the second hypothesis there exists a constant c such that convergent series -. ..^ ^ c a„>- w = l, 2, •.. n and each term of A is greater than the corresponding term of the divergent series c V - . ^-^ n Example 1. J[ = V , m>0. ^ log'" n Here n . , \ t aoo '^"■r, = p-^r- = + <»' by I, 463. log™ w Hence A is divergent. Example 2. ^ = T! — ; ■ ^-^ n log n Here 1 . „ w^n = :; = 0. logm Thus the theorem does not apply. The series is divergent by 82. Example S. i=2Z„ = 21og('l + ^+^') , r>l, where fi'is a, constant and | ^„ | < G-. From I, 413, we have, setting r = 1 + s, n\ n'J vfx n'J Hence nl„ = fj, , if /^ ^t 0, TESTS OF CONVERGENCE FOR POSITIVE TERM SERIES 95 and L is divergent. •If ^ > 0, i is an essentially positive term series. Hencei=4-oo- If /i< 0, i = — oo. Let 11 = 0. Then -j / q \ k-^l-^) 0<«„<1, which is comparable with the convergent series Thus Ii is convergent in this case. Example U. The harmonic series is divergent. For lim«a„ = l. Example 5. -i -^ = S -T\ — T- ^ arbitrary. ■^ w" logp n Here ^i-„ na„ = - — — = 00 , « < 1 logPji by I, 463, 1. Hence A is divergent for a < 1. Example 6. Here wa„=J- = l by I, 185, Ex. 3. Example 7. ^ ^ ^ /logC^ + 1) _ i\ ■^\ logw / Here, if /i > 0, wi+''a„ = «!+" logw log(l + l) W' log n log m since w*" > log n and ( 1 + - j =e. Hence A is divergent. log(l+l)"^cc, 96 SERIES 89 IfMemhert'i Test. The positive term series A=ai + a^-^ — converges if there exists a constant r < 1 for which ^i1, it diverges. Example 1. The Exponential Series. Let us find for what values of x the series X , x^ . a? E=\+~ + — + — +■■■ (1 1! 2! 3! ^ is convergent. Applying D'Alembert's test to its adjoint, we find ^ n — V. n\ yU-l n Thus jF converges absolutely for every x. Let us employ 80, 9 to estimate the remainder E^. Let a; > 0. The terms of E are all > 0. Since -n+p ^ z" / a: Y ■■■ n+p~ n\\n + \j '<^. m, \EA<\^ (3 TESTS OF CONVERGENCE FOK POSITIVE TERM SERIES 97 Example 2. The Lagarithmie Series. Let us find for what values of x the series r X x^ , a? a;* , is convergent. The adjoint gives a„ I w + 1 Thus L converges absolutely for any | a; | < 1, and diverges for \x\ >1. When x = l, L becomes which is simply convergent by 81, 4. When x= — 1, L becomes i + 2+i + i+- which is the divergent harmonic series. Examples. ^ = l+J- + i+... ■f 1^ 2" S** Kn + 1> As A is convergent when /a>1 and divergent if ai<1, we see that D'Alembert's test gives us no information when 1=1. It is, however, convergent for this case by 81, 2. Example 4. Y(l+2;)-(w + a;) Here *71 + T M+1 ^1 a„ n + 1 + « and D'Alembert's test does not apply. Example 5. A = Iwx"- Here '^+1V'.| = U1 98 SERIES Thus A converges for | a; |< 1 and diverges for | a: | > 1. For I a; I = 1 the test does not apply. For a; = 1 we know by 81, 2 that A is convergent for ^ < — 1, and is divergent for fi> —1. For x= — 1, ^ is divergent for fi^O, since a„ does not = 0. A is an alternating series for /* < 0, and is then convergent. 90. Cauchy's Radical Test. Let A = a-^-\- a^-\- ■••he a positive term series^ If there exists a constant r <\ such that "v/a„ < r M = 1, 2, ••• A is convergent. If, on the other hand, V^>1 A is divergent. For on the first hypothesis, so that each term of A is < the corresponding term in r + r^ + r^ +••• a convergent geometric series. On the second hypothesis, this geometric series is divergent and «„>?•"- Corollary. If lim Va„ = Z, and l<\, A is convergent. Jfl>l, A is divergent. Example 1. The series ^ 1 ^ log" n ^ ^ is convergent. For 1 .= 0. Example 2. log w is convergent. For n,— W" 1 .1^1 V a„ = - — -trr- = z — = - < 1. (w + 1)" /■, ,1Y e HJ Example S. In the elliptic functions we have to consider series of the type 0(i;) = 1 + 2 f j»' cos 2 TTwv 0<9<1. TESTS OF CONVERGENCE FOR POSITIVE TERM SERIES 99 This series converge^ absolutely if ? + 3* + 9«+- does. But here ■v^ = \/j"" = J" = 0. Thus 0(^v} converges absolutely for every v. Example 4. Let 0=6(- =00 n=2m + l, a\-\ =0 M = 2to. ^5 Thus tb e test gives us no information. 91. O'auchy^s Integral Test. Let (x) ie a positive monotone decreasing function in the interval (a, aj). The series * = <^(l)+<^(2) + <^(3)+- is convergent or divergent according as (f) (2;) dz is convergent or divergent. For in the interval (n, n + 1'), n>m> a, <^(m+l)<<^(a;)<.. 100 SERIES Hence ^n+i ('n + l)< I (n). Letting w = ?n, ?n+l, ••-mH-jD, and adding, we have Xm+p Passing to the limit jo = co, we get wfifish proves the theorem. Corollary. When is convergent ^™< Ccjxix. Example 1. We can establisfi Jit once the results of 81, 2. For, taking K^) = ^, is convergent or divergent according as /i>l, ojv /t 1; divergent if /li<; 1. We take here ^ and apply I, 637, 638. 8=1,2,- TESTS OF CONVERGENCE FOK POSITIVE TERM SERIES 101 92. 1. One way, as,already remarked, to determine whether a given positive term series A = a-^ + a2+ ■•■ is convergent or divergent is to compare it with some series whose convergence or divergence is known. We have found up to the present the following standard series S: The geometric series 1+^ + ^'+- (1 The general harmonic series The logarithmic series ^ + ^ + i-+- (2 1m 2" S** V_J__, (4 (5 ^ nl^nl^nl'^n We notice that none k.x these series could be used to determine by comparison the convergence or divergence of the series follow- ing it. In fact, let a„, 6„ denote respectively the wth terms in 1), 2). Then for5r0, -2^ = J- = ?. = 00 by I, 464, or using tlie infinitary notation of I, 461, Thus the terms of 2) converge to infinitely slower than the terms of 1), so that it is useless to compare 2) with 1) for conver- gence. Let ^f > 1. Then ^2±i = nf'g'' = 00, K This shows we cannot compare 2) with 1) for divergence. 102 SERIES Again, if a„, 6„ denote the wth terms of 2), 3) respectively, we have, if /t > 1, = 00 by I, 463, or IfM=l, or -2 = log m = 00, a«> 5«- Thus the convergence or divergence of 3) cannot be. found from 2) by comparison. In the same way we may proceed with the others. 2. These considerations lead us to introduce the following notions. Let A = aj + aj + •••, -B = 6j + ^2 + ■" ^^ positive term series. Instead of considering the behavior of a„/J„, let us gen- eralize and consider the ratios A„ : B„ for divergent and A„ : £„ for convergent series. These ratios obviously afford us a measure of the rate at which A„ and B„ approach their limit. If now A, B are divergent and . j^ we say A, B diverge equally fast ; if A^ < -B„, A diverges slower than B, and B diverges faster than A. From I, 180, we have : Let A, B he divergent and Hit. h \\xa^ = l. According as I is 0, ^0, oo, A diverges slower, equally fast, or faster than B. If A, B are convergent and we say A, B converge equally fast ; if A converges and B,B-^>d-^>...; if divergent, ^„>5„> (7„> ... Thus in both cases we are led to a sequence of functions of the type /j(n)>/2(n)>/3(^)> ... Thus to show the existence of a series O which converges (di- verges) slower than any series in 2, we have only to prove the theorem : 3. (Dm Bois Beymond.') In the interval (a, oo) let f^(ix)>f^(x)> ••• denote a set of positive increasing functions which =oo as a; = oo. Moreover, let ^ ^ -f -^ -p -^ 104 SERIES Then there exist positive increasing functions which = oc slower than anyf„. For as /i> /a there exists an aj>a. such that /i>/2 + l for X > flj. Since f^ >fz^ there exists an a^ > a-^ such that f^ >f^ + 2 for x>a2. And in general there exists an a„>a„_i such that /„ >/»+! + n iovx> a„. Let now ff(p) = fn(p) + w - 1 in (a„_i, a„) . Then g is an increasing unlimited function in (a, oo) which finally remains below any /^(a;) + m — 1, to arbitrary but fixed. T*^"« 0^L>\ W>TO, (1 A is convergent. If however. Q,(n)fd,+^n, or J log — ? ^ > M log ;,M = log I'^n. ajiii-^ ■•• (,_!« Hence ^ ; >l>, nr 1 a. < 'kZjW ••• Z,_iZ;m TESTS OF CONVERGENCE FOR POSITIVE TERM SERIES 105 Thus A is convergent. The rest of the theorem follows similarly. 2. For the positive term series A = a^ + a^-'r •■• to converge it is necessary that, for n = ao, lim a„ = 0, lim na„ = 0, lim naj-^n = 0, lim naj^nl^n =0, ••• We have already noted the first two. Suppose now that lim naJjU ••• l,n>0. Then by I, 338 there exists an m and a c> 0, such that naj-jfi ••■ l,n> , n > m, or ^ c nl jTi ■■■ l,n Hence A diverges. Example 1. /j _ V 1 ■^ n°- log^ n We saw, 88, Ex. 5, that A is divergent for a < 1. For a= 1, A is convergent for /3 > 1 and divergent if /3 ^ 1, according to 91, Ex. 2. If a > 1, let I , tt 11^ 1 Then if /3 > 0, . -, w° logp w w" and A is convergent since 5^ — is. If ^ < 0, let /3=-;8' , ;8'>0. Tli«° _ log^'w . J_ But log^' n < W' by I, 463, l ; and A is convergent since ^ -^ is. 106 Example 2. Here A = ^- SERIES •1 ,oi-— D log We 1 We"" _logw \n ^ _ a„w _ -logw+/^logw+^„ 51 \n /i-l + log w = fica by 81, 6). Hence A is convergent for /i>0 and divergent for /i<0. No test for )ii = 0. But for /i = 0, -( log _ _ a„nl^n _H„—l^ — l^ ZgW = -00, since l^n > l^n. Thus A is divergent for /t = 0. 94. A very general criterion is due to Kummer, viz. : Let A = a-y + a^+ ■■■ be a positive term series. Let k^ k^, ■■■ be a set of positive numbers chosen at pleasure. A is convergent, if for some constant k>0. K = K^-K+i>k w = l, 2, ... A is divergent if "■n+l ki k^ is divergent and For on the first hypothesis n = l,2,.- aK<^(*n-i«»-i-*»a„)- TESTS OF CONVERGENCE FOK POSITIVE TERM SERIES 107 Hence adding, • and A is convergent by 80, 4. On the second hypothesis, Hence -A diverges since i2 is divergent. 95. 1. From Kummer's test we may deduce D'Alembert's test at once. For take fC-t ^ rCn ^ • • • ^ J. ■ Then J. = aj + aj + ••• converges if i.e. if Similarly A diverges if ^±i >il. 2. To derive Raabe's test we take Then A converges if i.e. if K„ = n-^ - (w + 1)> A>0, an Similarly J. diverges if Va„+i / 108 SERIES 96. 1. Let A = a^ + a^+ ••• be a positive term series. Let ^-1-1 '■n+l -1 Then A converges if there exists an s such that \{n} >B>1 for some n>m; it diverges if ^•C*^) < 1 /<"■ ** > "*• We have already proved the theorem for Xq(w). Let us show how to prove it for \j(m). The other cases follow similarly. For the Kummer numbers A„ we take Then A converges if k„=n log n. an k„ = n log n ■ -^ (w + 1) log (n + 1) > A: > 0. a„ As n+1 = n(l + K„ = \,(«) - log(l + IJ - log (l + Vj = Xi(n)-log 1 + = \i(w)-(l + a) a>0. Thus A converges if Xi(w) >B>1 for n>m. In this way we see that A diverges if Xi(m) < 1, n>m. 2. Cahen's Test. For the positive term series to converge it is necessary that , , . , lim n TESTS OF CONVERGENCE FOR POSITIVE TERM SERIES 109 For if tliis upper iflnit is not + oo , n\n{-^-l]-li some m, and A is divergent by 1. Hxample. We note that Raabe's test does apply to the harmonic series ^ , , l + l+\+- (1 Here ^^"'^^ P„ = 0, and lim P^ = 0. Hence the series 1) is divergent. 97. Gauss' Test. Let A = «j + «2+ ■•■ he a positive term series such that , «„ _ w' + g,w''"-^+ •■■ + g^ where s, a-^ •■■ b^ ■■■ do not depend on n. Then A is convergent if flj — 6j > 1, and divergent ij aj — ij < 1 . Using the identity I, 91, 2), we have . , ai-b^ + -la^-b^+ -l Xo(n)=n(^-l) = f \a„+i / -1,-1 n+l / l+±\bi + n Thus limXQ(«)= flj — 6j. Hence, if a^ — b^>l, A is conver- gent; if flj — Jil and divergent if « 1 and diverges if « < 1. If a = 1, I n \i(w) = l^n \ \o(w) - 1 1 = -^j • /3„= 0, and A is divergent. EXAMPLES 99. The Binomial Series. Let us find for what values of z and /i the series £=l + ixx + fj. ■ n -1 x^ + fl ■ fJ,— 1 ■ fJL— 2 x^ + *n+1 1-2 " ■ l.'2.3 converges. If /«. is a positive integer, 5 is a polynomial of degree fi. For |t = 0, 5 = 1. We novir exclude these exceptional values of fi. Applying D'Alembert's test to its adjoint we find fi — n+l n Thus B converges absolutely for ja;] < 1 and diverges for | a;| > 1. Letx=l. Then ^ 1-2 1-2.3 Here D'Alembert's test applied to its adjoint gives ^n+1 «» |_ \fi-n + l\_^ = 1. As this gives us no information unless /i< — 1, let us apply Raabe's test. Here n[ — ^ — 1 )= ^^-^ — 1 for sufiSciently large n = 1 + M- TESTS OF CONVERGENCE FOR POSITIVE TERM SERIES 111 Thus B converges absolutely if /i > 0, and its adjoint diverges if /i< 0. Thus B does not converge absolutely for /ii< 0. But in this case wq note that the terms of B are alternately positive and negative. Also = 1 1 + M n so that the |a„| form a decreasing sequence from a certain term. Wq investigate now when a„ = 0. Now «„ = (-!) „ (-/^X-/^ + i)---(-/^ + ^-i) _ 1-2- •••« = (-l)» — 1. Thus B converges when /i > — 1 and diverges when ;it< — 1. Letx = -1. Then '^^ 1-2 1-2.3 If /i > 0, the terms of 5 finally have one sign, and J -^ -1)^1 + ,,. Hence B converges absolutely. If yu, < 0, let /t = — \. Then B becomes ^ X-X + 1 ^ X-X + 1 -x + s^ 1.2 Here 1-2-3 = l-\. /^_l) = _l^^l n Hence 5 diverges in this case. Summing up : The binomial series converges absolutely for \x\\. When x=lit converges for fi> — l and diverges for fi<—l; it converges absolutely only for fi>0. When x = — l, it converges absolutely for fi>0 and diverges for /i < 0. V 112 SERIES \x\ = \x\. (1 100. The Hypergeometric Series ^(«,^,,,.)=i+^.+ "-;+^-^-;+^ .^ 1-7 l-i!-7'7 + l «.a + l-« + 2-/3-/3+l-/3 + 2 , ■^ 1.2-3-7-7 + 1 -7 + 2 Let us find for what values of x this series converges. Passing to the adjoint series, we find (a + nY^ + n) (w+l)(7 + w) Thus F converges absolutely for | :«; | < 1 and diverges for | a; | > 1. Let x=l. The terms finally have one sign, and «,.+! ^ w^ + w(l + 7) + y «n+2 '>^^ + »»(« + /3) + a/3 Applying Gauss', test we find F converges when and only when « + /3 - 7 < 0. Let x= — 1. The terms finally alternate in sign. Let us find when a„ = 0. We have . I «^ (a + 1) •■• (« + w)(/3 + 1) •■• (/3 + n) ' "^'' 7 ■ (l + l)...(l + w)(7+l).-(7 + w)' /8^ Now a + 7n = m(lH m l + m = m(l-| — m. Thus «„+2 = n — 1 7 1 + /3 + m = m 1 + 1+^ m :m( 1+-^ V TO But by I, 91, 1), (i+iYi-f^) \ mj\ mj 1 + -=l-i+^ J_ mm? m 1 + _=l_l + l!l 7^ mm? m where o-„ = 1, t„ = 7^ as »w = 00. PRINGSHEIM'S THEORY 113 Hence « 1 V W\ w\ m wrjx m m?/ 1 -I- ')= + /3 - 7 - 1 _j_ O Hence n f i.gi...,i-i:iog(n- ?+'^-^-' +3.)=y;.=x.. and thus „ L = lim log I a„+2 1 = 2^ Z„. 1 Now for a„ to = it is necessary that iy„ = — oo. In 88, Ex. 3, we saw this takes place when and only when a + ;S — 7— 1<0. Let us find now when | a^^i | < | a„ |. Now 1) gives _1 I a + ^-7-1 ^ K Thus when a + /S — 7 — 1<0, | a„+2 1 < | ««+i I- Hence in this case F is an alternating series. We have thus the important theorem : The hyper geometric series converges absolutely when | a; | < 1 and diverges when | a: | > 1. When x = l, F converges only when a + /3 — 7 < ayid then absolutely. When a; = — 1, F converges only when a + /3 — 7 — 1<0, and absolutely if a + /8 — 7 < 0. Pringsheim' s Theory 101. 1. In the 35th volume of the Mathematische Annalen (1890) Pringsheim has developed a simple and uniform theory oi convergence which embraces as special cases all earlier criteria, and makes clear their interrelations. We wish to give a brief sketch of this theory here, referring the reader to his papers for more details. Let Mn denote a positive increasing function of n whose limit is + 00 for w =00 . Such functions are, for example, /it > 0, W , logf'w , l^n , Ijnl^n •■• l,_-^nli^n 114 SERIES A'!,, where A is any positive term divergent series. B„~'^ where B is any positive term convergent series. It will be convenient to denote in general a convergent positive term series by the symbol (7=Ci + = di + d^+ ... 2. The series is convergent, and conversely every positive term convergent series may he brought into this form. ^1 L_=Ji- and (7 is convergent. Let now conversely C= Cj+ Cj + ••• be a given convergent positive term series. Let Then ^ ^ 3. The series i> = i(ilf„+i-iK„) (2 is divergent, and conversely every positive term divergent series- may he hrought into this form. For i>n=2(itf„+i-itf„) PRINGSHEIM'S THEORY 115 Let now conweiselytD = di + d^ + ■•• be a given positive term divergent series. Let m — n 102. Having now obtained a general form of all convergent and divergent series, we now obtain another general form of a convergent or divergent series, but which converges slower than 1) or diverges slower than 101, 2). Let us consider first con- vergence. Let M'n < M„, then is convergent, and if M'„ is properly chosen, not only is each term of 1) greater than the corresponding term of 101, 1), but 1) will converge slower than 101, 1). For example, for M'^ let us take iHf^, < /i. < 1. Then denoting the resulting series by C = 2cJ„ we have Thus 0' converges slower than C. But the preceding also shows that C and ,_ ,. converge equally fast. In fact 2) states that Since M^ is any positive increasing function of n whose limit is oo, we may replace M„ in 3) by l^M^ so that is convergent and a fortiori ylM.,-lrM^ r=l,2, ... (4 is convergent. (3 116 SERIES Now by I, 413, for sufficiently large w, ,„.«..- log ^..-h(i- 4^) >*^. Replacing here 7tf„ by log i(f„, we get and in general ■^"n+l'l-'"n+l 'r-l-'"n+l Thus the series converges as is seen by comparing with 4). We are thus led to the theorem : The series wr i{/r iir li/r M^^, - itf„ 7M„,,l^M^^, - /,_iM'„,i?r''^nM r=l, 2, ■■■,fj.>0 form, an infinite set of convergent series; each series converging slower than any preceding it. The last statement follows from I, 463, i, 2. Corollary 1 {Abel). Let D= d-^ + d^+ ■■■ denote a positive term divergent series. Then ^ 7)1+" is convergent. Follows from 3), setting 7lf„+j = i>„. Corollary 2. If we take M„ = n we get the series 91, Ex. 2. Corollary 3. Being given a convergent positive term series C=c^ + c^+ ••• we can construct a series which converges slower than C. PRINGSHEIM'S THEORY 117 For by 101, 2 we nf&y bring Q to the form Then any of the series 7) converges slower than C. 103. 1. Let us consider now divergent series. Here our problem is simpler and we have at once the theorem : The series j^ i\/r wr 1 -'"n diverges slower than 2(if„^i-if„) = 2<. (2 That 1) is divergent is seen thus : Consider the product which obviously = 00. ^""^ Pn = (1 + <^l)(l + t?,) • • • (1 + C?„) = 1 + (cZj + •■• + C?„) + (^1^2 + '^Ws + ••■) + id^d^d^ + •••) + •■• + d-^d^ ■■■d„ 2 n ! Hence i)„= 00 and D is divergent. As ^ = -L = we see that 1) converges slower than 2). 2. Any given positive term series 2) = cZj + c?2 + ••• can be put in the form i). For taking il!fj>0 at pleasure, we determine M^, -^3 ••• by the relations tt/- ^^^=1+£Z„ n=l, 2.-. 118 SERIES Then M„+-^ > M„ and Moreover M„ = oo. For M But2)„ = oo. 3. The series >1 + D„ by I, 90, 1. 1 r = 0, 1, 2, form an infinite set of divergent series, each series divergent slower than any preceding it. IqM„ = M^- ^^°'- log M„,, - log M„ = log (l + ^-1^-^" ) This proves the theorem for r = 0. Hence as in 102 we find, replacing repeatedly M^ by log M^, Corollary 1. If we take ilfj, = w, we get the series 91, Ex. 2. Corollary 2 {Abel). Let D ■= d-^^ + d^-\- •■• be a divergent positive term series. Then , is divergent. We take here M^ = i>„. Corollary S. Being given a positive term divergent series D, we can construct a series which diverges slower than D. For by 101, 3 we may bring D to the form Then 1) diverges slower than D. PRINGSHEIM'S THEORY 119 104. In Ex. 3 of 1^454, we have seen that M„^i is not always ~ M„. In case it is we have 1. The series is convergent. Follows from 102, 3). 2. The series ^ e^n "^^ " is convergent if /i>0; it is divergent if fj,<0. For e'^^n > i ^^M^ -^Ml ti>0. Thus If /i<0 e>'«n ~ Ml ' ^rl+^ — ^n > nr _ lyr 3. If Mn+i ~ M„, we have I M —I M -■ ■^"+1"-^" -. ^n+^ — Mn For by 102, 5), 103, 3), 1 M —1 M -< -^ +1 ~ -^' MJ^M^-kM: M^^-,-M„ Now since i!f„+j~il!^, we have also obviously l^M„~l^M„+i m=l,2,-r. 105. Having obtained an unlimited set of series which converge or diverge more and more slowly, we show now how they may be employed to furnish tests of ever increasing strength. To ob- tain them we go back to the fundamental theorems of comparison of 87. In the first place, ii A = a^ + a^ + •■• is a given positive term series, it converges if ^0. (1 120 SERIES It diverges if ^>a. (2 In the second place, A converges if ^«±l_£«±J<0, (3 and diverges if „ ^ The tests 1), 2) involve only a single term of the given series and the comparison series, while the tests 3), 4) involve two terms. With Du Bois Reymond such tests we may call respec- tively tests of the first and second kinds. And in general any relation between p terms of the given series and p terms of a comparison series, ^n) ^n+\i ■■■ ^n+p-11 Or flf„, a„+j ••• (ln+p-\ which serves as a criterion of convergence or divergence may be called a test of the p"" kind. Let us return now to the tests 1), 2), 3), 4), and suppose we are testing A for convergence. If for a certain comparison series G — not always < G^ , n > m it might be due to the fact that c„ = too fast. We would then take another comparison series (7'= 2e{j which converges slower than C. As there always exist series which converge slower than any given positive term series, the test 1) must decide the con- vergence of -.4. if a proper comparison series is found. To find such series we employ series which converge slower and slower. Similar remarks apply to the other tests. We show now how these considerations lead us most naturally to a set of tests which contain as special cases those already given. 106. 1. Greneral Criterion of the First Kind. The positive term series A = a-^^+ a,^+ ••■ converges if PRINGSHEIM'S THEORY 121 It diverges if •. J!f. ^ „ ^ „ This follows at once from 105, 1), 2); and 101, 2; 103, l. 2. To get tests of greater power we have only to replace the series M^^,-M^ ^M^^,-M, '-- . X n+1 -^"n jf„+iifer„ ^ M^ .just employed in 1), 2) by the series of 102 and 103, 3 which con- verge (diverge) slower. We thus get from 1 : The positive term series A converges if hm ^yJ — "— a„ or lim — =-1 — ^ r i n+i , n « < oo . It diverges if y^^_^^ MJ,M^- IMn ^ >0 — M,^,-M, " Bonnefs Test. The positive term series A converges if lim nl-^n ■■• l^_-^niy'^n ■ a„ 0. It diverges if Ihn nl,n - l,n ■ a^>0. Follows from the preceding setting M„ = n. 3. The positive term series A converges or diverges according as e''^"ff, Tr^0, (3 <1 , /X<0. For in the first case and in the second case The theorem follows now by 104, 2. 4. The positive term series A converges if lim ^ >0 or lim --^^v-r. -r^-^ -" >o. M„,,-ilf„ >o _..,,.., ^MJ,M„-l,M^-a^ 01 ""^ 7 M 'r+l-'«n ^22 SERIES It diverges if Here r = 0, 1, 2, ••• and as before lf,M„ = M„. For taking the logarithm of both sides of 3) we have for con- vergence , M„+^ - M„ ^ = W -"■ As /A is an arbitrarily small but fixed positive number, A con- verges if lim q>0. Making use of 104, 3 we get the first part of the theorem. The rest follows similarly. Remark. If we take M„ = n we get Cauchy's radical test 90 and Bertram's tests 93. ^'''^ log- rr ^= log aJ1= -log 7a„>/.>0, it is necessary that jy— < „ ^ i Also if log r^ — ^ log + log — 1 it is necessary that log log- J , 107. In 94 we have given Kumraer's criterion for the conver- gence of a positive term series. The most remarkable feature about it is the fact that the constants Aj, k^ — which enter it are subject to no conditions whatever except that they shall be positive. On this account this test, which is of the second kind, has stood entirely apart from all other tests, until Pringsheim discovered its counterpart as a test of the first kind, viz. : PRINGSHEIM'S THEORY 123 Pringsheim's Critericm,. Let p^ p^'" ^^ ^ ®^* of positive numbers chosen at pleasure, and let P„=jOj+ ■•• +'p„. The positive term series A converges if logil=- lim "" > 0. (1 For A converges if lim ^ >0 , by 106, 4. (2 But ilfn+i — M„ = dn is the general term of the divergent, series Z> = c?i + cZ2+ ■•• Thus 2) may be written loff^ (3 log^ 0. M oreover A converges if £^>r>l, at is, if lim ^ > 0, a„ where as usual (7= t'j+ Cj + ••• is a convergent series. Hence A converges if c^ lim^>0. (4 But now the set of numbers Pi-, p^ ■••gives rise to a series P=p^+p^+ ■■■ which must be either convergent or divergent. Thus 3), 4) show that in either case 1) holds. 108. 1. Let us consider now still more briefly criteria of the second hind. Here the fundamental relations are 3), 4) of 105, which may be written : c„+^— c„ > for convergence ; (1 cZ^^j-^ — «?„:< for divergence. (2 124 SERIES Or in less general form: The positive term series A converges if lH]lfcn.l^^-O>0. (3 It diverges if ^ «n+l / Here as usual 0=c-^^+c^+ ••• is a convergent, and D=d-^^ + d^+ ••■ a divergent series. 2. Although w^e have already given one demonstration of Kummer's theorem we wish to show here its place in Pringsheim's general theory, and also to exhibit it under a more general form. Let us replace c„, e„+i in 1) by their values given in 101, 2. We get or since j^ ^ orby 103, 2 «„ „ a, ■n+l where i) = aj+tZ2+ "-is any divergent positive term series. Since any set of positive numbers k-^, k^, ■■■ gives rise to a series ^1 + ^2 + ■■■ ^vhich mast be either convergent or divergent, we see from 1) that 5) holds when we replace the cZ's by the k's. We have therefore: The positive term series A converges if there exists a set of positive numbers k^, k^ •■■ such that It diverges if a^ *„.i^^-/c„>0. (6 a„ d„^^^^-d„<0 where as usual d^ + d^+ ■■■ denotes a divergent series. ARITHMETIC OPERATIONS ON SERIES 125 Since the /fc's are enijfc-ely arbitrary positive numbers, tlie rela- tion 6) also gives A converges if ft as is seen by writing k - 1 reducing, and then dropping the accent. 3. From Kummer's theorem we may at once deduce a set of tests of increasing power, viz. : The positive term series A is convergent or divergent according as is > or is j< 0. For ^j, k^ •■■ we have used here the terms of the divergent series of 103, 3. Arithmetic Operations on Series 109. 1. Since an infinite series J. = ftj + ag + flg • • (1 is not a true sum but the limit of a sum ^=lim^„, we now inquire in how far the properties of polynomials hold for the infinite polynomial 1). The associative property is expressed in the theorem : Let -4 = a^ + a^ + •••he convergent. Let 6j = osj + ... 4- a^ , b^ = am+i+ ••• + c-m 1 ■■• Then the series B = h-^^ + h^+ ■•■is con- vergent and A = B. Moreover the number of terms which b„ em- braces may increase indefinitely with n. ^°^ B -A ^"^ lim A,„ =A by I, 103, 2. 126 SERIES This theorem relates to grouping the terms of A in parentheses. The following relate to removing them. 2. Let 5 = 6j + 62 + ••• ^« convergent and let Jj = aj+ ••• + a„, , h = «,«,+! + - + «m,> ••• If 1° A=a^ + a^+ ■■■ is convergent, A = B. 2° If the terms a„ > 0, A is convergent. 3° i)^ each TO„ — m„_i

0, m, -B„to. Then 5-^.«2„. On the third hypothesis we may set A = -Br +6^+1 where J|.+j denotes a part of the a-terms in h^^y Since h^+i con- tains at most p terms of A, J^+i = 0- ^^^°^ lim^.= lim£, , or A = B. Example 1. The series ^ = (1 - 1) + (1 - 1) + (1 - 1) + - is convergent. The series obtained by removing the parentheses ^=1-1 + 1-1+ - is divergent. Example 2. A = l — + - — + - — + •••; x^-1, -2,... l + x 2 2 + x S 2,+x B= y\(^ ^ WS ^ ^\n n + xJ ^n{n+x') As B is comparable with X — , it is convergent. Hence A is ^ n^ convergent by 3°. 110. 1. \je.t MS consiAev now \hB commutative property. Here Riemann has established the following remarkable theorem : ARITHMETIC OPERATIONS ON SERIES 127 The terms of a simplfg convergent series A = a-^ + a^^ ■■■ can be arranged to form a series S, for which lim S„ is any prescribed number, or ±00. ^^°^1^* B = b, + b,+ ... 0=0-^ + 02+ ••• be the series formed respectively of the positive and negative terms of A, the relative order of the terms in A being preserved. To fix the ideas let I he a. positive number ; the demonstration of the other cases is similar. Since 5„= +00, there exists an Wj such that £n„ > I (1 Let m^ be the least index for which 1) is true. Since C„= — 00, there exists an m^ such that B^,+ 0^<1. (2 Let m^ be the least index for which 2) is true. Continuing, we take just enough terms, say m^ terms of £, so that -8m, + O^ + 5„„ „, > I. Then just enough terms, say m^ terms of 0, so that ^my -^ (-'mi + -"mi, m, + ^mi. mj < ') etc. In this way we form the series S— -B„, + 0^^ -I- ^„„ „3 ■+■ ••• whose sum is I. For I a, I < e s > o- ; ^^^° r„ = mi+m^+ ••• + ?n„ 5 m. Hence | Z- >S'„ | < | a,J < e for n ><7. 2. Let J. = aj + aj + • • ^^ absolutely convergent. Let the terms of A be arranged in a different order, giving the series B. Then B is absolutely convergent and A — B. For we may take m so large that A„ < e. 128 SERIES We may now take n so large that A„ — B„ contains no term whose index is <. m. Thus the terms of A„ — J5„ taken with positive sign are a part of A„ and hence ^ - ^„ I < A„ < e n>m. Thus B is convergent and B = A. The same reasoning shows that B is convergent, hence B is absolutely convergent. 3. If A = a-^ + a^+ ■■■enjoys the commutative property, it it absolutely convergent. For if only simply convergent we could arrange its terms so as to have any desired sum. But this contradicts the hypothesis. , Addition and Subtraction 111. Let v4 = «i + a2 + "" 5 B = b^ + b2 + ■■■ be convergent. The series n^. 2. We owe the following theorem to Mertens. If A converges absolutely and B converges (not necessarily abso- lutely'), then C=a^b^+ (a^b^ + a^b^) + (ajJg + a^b^ + agJi) + • • • is convergent and C = A • B. We set C=Ci + c2 + e3+ ••• where Cj = a-fi^ ^n = «A + S^n-i + agi^-g + ••• + a„Si. Adding these equations gives a„ = ai5„ + a2-B„_i + ag^„_2 + •- + a„5i. ADDITION AND SUBTRACTION 131 But B„^B-£„ m=l, 2, .- X16IIC6 0„ = a,{B - 5„) + a^{B - 5„_i) + ■ .. + a„(£ - B,) = ^n-S — d„, where _ _ _ a« = «i^„ + «2^„-i + ■•• +a„By The theorem is proved when we show d„ = 0. To this end let us consider the two sets of remainders A , B, , .. •^». -^«i+i ' \+i 5 Bn,+n, n^ + n^ = n. Let * each one in the first set be | < | iHfj, and each in the second, set I < I M^. Then since \d^\< M^(a^ + ... + «„^)+ i!fi(«„^+i + ." + «„) Now for each e > there exists an Wj such that also a V, such that w, > v. ■■' 'I Ml Thus 1) shows that 3. When neither A nor B converges absolutely, the series may not even converge. The following example due to Cauchy illustrates this. VI V2 V3 V4 VI. V2 V3 V4 * The symbols | < |, | < | mean numerically <, numerically < . M, < 2A' K < e 2 Ml dn\ < e. 132 SERIES The series A being alternating is convergent by 81, 3. Its adjoint is divergent by 81, 2, since here /x = |^. Now VI VJ VViV2 V:^V1 VVl V3 V2 V2 V3 VT/ = Cj + Cg + (7^ + - viv^rn: V2Vjzr=^ v^rri vi By I' 95, n ^/7n{n— m) < -• A "^"'^^ 1 2 , ,^2(w-l) 'Vm(n — m) ** *'- Hence C is divergent since c„ does not = 0, as it must if C were convergent, by 80, 3. 4. In order to have the theorems on multiplication together, we state here one which we shall prove later. if all three series A, B, are convergent, then C= A- B. 113. We have seen, 109, 1, that we may group the terms of a convergent series A= aj + osg + "■ into a series 5 = ftj + Sg + •" each term J„ containing but a finite number of terms of A. It is easy to arrange the terms of A into a finite or even an infinite number of infinite series, B' , B", B'" ••• For example, let -S' = «! + flp+i + flap+i -I £('" = ap + a2p + «3P+ - Then every term of A lies in one of these p series B. To decom- pose A into an infinite number of series we may proceed thus : In B' put all terms a, whose index w is a prime number ; in B" put all terms whose index n is the product of two primes ; in TWO-WAY SERIES 133 5<'"' all terms whose intlex is the product of m primes. We ask now what is the relation between the original series A and the series B', B" - If A = a^ + a^+ •■• is absolutely convergent, we may break it up into a finite or infinite number of series B', B", B'", •■• Uaoh of these series converges absolutely and A = B' + B" +B"'+ ... ■ That each ^""^ converges absolutely was shown in 80, 6. Let us suppose first that there is only a finite number of these series, say p of them. Then A =-B; + B':^ +-+BjP^ n = n,+ ... + n,. As M=Qo, each Mj, Mj ■■•=="»• Hence passing to the limit n = 00 , the above relation gives A = B' + B" + •■• +5v, contains no term a„ of index < m, and m so large that A„v. Tivo-way Series 114. 1. Up to the present the terms of our infinite series have extended to infinity only one way. It is, however, convenient sometimes to consider series which extend both ways. They are of the type •••a_3 + a_2 + a_.^ + ao+ a^ + a^ + a^+- •■■ which may be written ao + «i + a2+ •••+a-i +a-2+ •■■ or S, ,-, 5;a„. (1 134 SERIES Such series we called two-way series. The series is convergent lim 5;a„ (2 »', s=oo n=-r is finite. If the limit 2) does not exist, A is divergent. The ex- tension of the other terms employed in one-way series to the present case are too obvious to need any comment. Sometimes w = is excluded in 1) ; the fact may be indicated by a dash, thus 2'a„. 2. Let m be an integer ; then while n ranges over 3,-2, -1, 0, 1, 2,3... v = n + m will range over the same 'set with the difference that v will be m units ahead or behind n according as m^O. This shows that 2a„ = Sa, n+m' 2a„ = 2a _ n=-ao n=— Similarly, 3. Example 1. ^ ^ |^„,+„„. z=\ -{■ 6^+" -|- g2i+4a I g3l+9a _i_ ... I Q—x+a _|_ g-2z+4a i g~Zx+^a ... This series is fundamental in the elliptic functions. Example 2. X TZ \x + n nj The sum of this series as we shall see is tt cot ttx. TWO-WAY SERIES 135 116. For a two-way*series A to converge^ it is necessary and sufficient that the series B formed with the terms with negative indices and the series C formed with the terms with non-negative indices be convergent. If A is convergent, A= B + 0. It is necessary. For A being convergent, \A-B,-0,\ some a- and r > some p. Hence adding, \C,-0,\ some p. Hence \B+0-CB,+ C,')\ 0. 136 SERIES ^^""^ «>0, v'^ = e-e»» = ifa<0 = 00 if a > 0; n=-n', w'>0 Vai, = e^^e""' = ifa<0 = 00 if a >0. The case a = is obvious. Thus the series defines a one-valued function of x when a < 0. As an exercise in manipulation let us prove two of its properties. 1° @(a;) is an even function. For 00 If we compare this series with 2) we see that the terms corre- sponding to n=m and n = — m have simply changed places, as the reader will see if he actually writes out a few terms of 2), 3). Of. 114, 2. 2° @(x + 2 ma) = e-'"(^+"""@(a:). w = ± 1, ± 2, ... For we can write 2) in the form @(a;)=e *" Se *" ■ (4 ^® (x+2ma)' „ (j+2(m4-w)a)' ®Cx + 2ma)=e *" 2e *" _2f „ (x-t-2t'a)' _. g-m(x+ma) _ g 4a ^e ^o which with 4) gives 3). CHAPTER IV MULTIPLE SERIES 116. Let a! = a;j, ••• a;„ be a point in m-way space 9?^- If the coordinates of x are all integers or zero, x is called a lattice point, and any set of lattice points a lattice system. If no coordinate of any point in a lattice system is negative, we call it a non-negative lattice system, etc. Let /(ajj ••• a;„) be defined over a lattice system t = tj,-..t„. The set |/('i"-tm)l i^ called an m-tuple sequence. It is customary to set /('i •■■'-) = «'.•-„• Then the sequence is represented by The terms j.^^^ ^ ^ jj^ ^ ^ ^^^ as tj ••• t„ converges to an ideal point have therefore been defined and some of their elementary properties given in the discussion of I, 314-328 ; 336-338. Let a; =.a;i ••■ a;„ y = y\---ym he two points in SR„. If Vi^^i--- Vm^^m we shall write more shortly y>x. If a; ranges over a set of points x' > x" > x'" ■■• we shall say that x is monotone decreasing. Similar terms apply as in I, 211. "°°^ f(y^-y^)>f(^i-^.) when y >x, we say /is a monotone increasing function. If f(.yi-yM)^, we say /is a monotone decreasing function. Similar terms apply as in I, 211. 137 138 MULTIPLE SERIES 117. A very important class of multiple sequences is connected with multiple series as we now show. Let ai,....„ be defined over a non-negative lattice system. The symbol 2a.,...« ti = 0, 1, •■•I'l , ■ ■•t„=0, 1, ■ ■!/„ (1 or 2ai,...i„ » or^^^...^„ denotes the sum of all the a's whose lattice points lie in the rec- tangular cell < a;i < i/j ■■■00 and A is diver- gent, the limit 2) is + oo. In this case we say A is infinite. Let us replace certain terms of A by zeros, the resulting series may be called the deleted series. If we delete A by replacing all the terms of the cell Ity^...^„ by zero, the resulting series is called the remainder and is denoted by A^^...^„ or by A^. Similarly if the cell J2„ contains the cell M^, the terms lying in B^ and not in M^ may be denoted by A^_ „. . The series obtained from A by replacing each term of A by its numerical value is called the adjoint series. In a similar manner most of the terms employed for simple series may be carried over to m-tuple series. In the series 2a,j....„the indices i all began with 0. There is no necessity for this; they may each begin with any integer at pleasure. 118. The Greometric Series. We have seen that 1 1-a 1 Hence l-b 1 = 1 + a + a^ + ■■■ |a| < 1, = l + b+i?+ - |6|<1. -=2a"'ft" (1 - a)(l - b) for all points a, b within the unit square. In general we see that is absolutely convergent for any point x within the unit cube < |a;J < 1 1=1, 2, —n, and j[ a = (l-Xi)(l-^2)-(l-^n)' 140 MULTIPLE SERIES 119. 1. It is important to show how any term of ^ = 2a,j....„ can be expressed by means of the -4v,...^„. Then -i^^ii-j-v^-i-i = -4r,.:.-^„-i-]i'„ — -^v,^j...v„_i-i, ^„_r (2 Let -^"l"! — •'m-2 ~ -^••i"! — "m-l -^>'i''2-""'ra-l-l' (3 Similarly ^•'l''^—''m-3 ~ -^"l"! — "^-2 ~ -^''1''2 — "m-J-l' (4 -^"l "m-^ ~ -^■'l''2--„_3"~ -^''l''!--„-3-l' (5 • Finally A, = -».,.. -i>.,..-l, (6 and «.,„,....„ =i).,-A,-l. (7 If now we replace the 2)'s by their values in terms of the A's, the relation 7) shows that a^, .•■..„ may be expressed linearly in terms of a number of A^^...^„ where each /i^ = v^ or v^ — 1. For m = 2 we find ^"l"! ^ -^•'l"! "^ -'^>'l-ll "2-1 ■'^"l. "2-1 "l-ll "2" C" 2. From 1 it follows that we may take any sequence {^.,... .„! to form a multiple series This fact has theoretic importance in studying the peculiarities that multiple series present. 120. We have now the following theorems analogous to 80. 1. For A to be convergent it is necessary and sufficient that e>0, p, |J.^,,|p. Hence passing to the limit p = x , limi),,...,^_^ 1. _(-ir! - + -1 2(a + l) Then by 119, 8) ,„ ,=1 + 1. a' a" 142 MULTIPLE SERIES lim J.r . = As A is convergent. But V I I 1 Till lira a, J =— ' iim «"!=-• r=» a* i=» a' That is when the point (r, ») converges to the ideal point (oo, 8), or to the ideal point (r, oo ), a„ does not = 0. 3. However, we do have the theorem : -^'* ^ = 2a...... a, >0 converge. Then for each e > there exists a \ such that a. ...j < e /or any t outside the rectangular cell R^. This follows at once from 120, l, since 122. 1. Letf^x-^ ••• x„) be monotone. Then ]im/(a;i ••• x^) = 1 x-^ < a^, ••• x„ < a„, a may be ideal. (1 x=a exists, finite or infinite. If f is limited, I is finite. If f is unlim- ited, Z = + 00 when f is monotone increasing, and 1= — cc whenf is monotone decreasing. For, let/ be limited. Let A = a^ < «2 < ••• = a. Then ,. <•,' s 7 hm/(aj = I is finite by I, 109. Let now 5 = ySj, /S^, •■• = a be any other sequence. ^^* Um/(;8„) = ; ih;7/(;8„) = f. ~zr - B Then there exists by I, 338 a partial sequence of B, say C=7i, 72 ••• such that also a partial sequence D=S^, B^ ••• such that lim/(S„) = l GENERAL THEORY 143 But for each «„ ther» exists a 7^ > «„; ^'""'^ /(7.„) >/(«.) and therefore Z > Z. (2 Similarly, for each d^ there exists an a,^^ > S„ ; and therefore 7 <- 7 /q Thus 2), 3) give ^^f^.^'^i, B Hence by I, 316, 2 the relation 1) holds. The rest of the theorem follows along the same lines. 2. As a corollary we have The positive term series A = Sa^ ... ^ is convergent if A^ ... ^ is limited. 123. 1. iei -4. = 2a.| ... .^ = Sa^ , B = '^h,^...,^ = 'Lh, he two non- negative term series. If they differ only hy a finite number of terms, they converge or diverge simultaneously. This follows at once from 120, 2. 2. Let A, B he two non-negative term series. Let r > denote a constant. If a^<.rh^, A converges if B is convergent and A < rB. If a^>rh^, A diverges if B is divergent. P'or on the first hypothesis and on the second /i ^ d Ax > riix . 3. Let A, B be two positive term series. Let r, s he positive constants. If r<~nimj"-mj-''l .'-2 •*-! converge at the point a = (flj, ••• a,), ^Aew if converges absolutely for all points x within the rectangular cell R whose center is the origin, and one of whose vertices is a; that is for | a;, | < | a. | , t=l, 2, ••• s. For since P converges at a, limc„,„,...afx...a™.= 0. Thus there exists an iHf such that each term af' I < M. Hence . . X » = mi^. 6. Absolutely convergent series are commutative. For let B be the series resulting from rearranging the given series A. Then any associate 58 of ^ is simply a rearrangement of an associate series 31 of A. But 31 = i8, hence A = B. 7. A simply convergent m-tuple series A can be rearranged, producing a divergent series. For let 31 be an associate of A. % is not absolutely convergent, since A is not. We can therefore rearrange 21, producing a series SB which is divergent. Thus for some SB limS8„ does not exist. Let 93' be the series formed of the positive, and 58" the series formed of the negative, terms of SB taken in order. 146 MULTIPLE SERIES Then either S8J, = + <» or SB!,' = — <»5 or both. To fix the ideas suppose the former. Then we can arrange the terms of SS to form a series S such that (5„ = + oo. Let now S be an associate series of 0. Then and thus lim (7„ = lim S„ = + oo. Hence C is divergent. 8. If the multiple series A is commutative, it is absolutely/ con- vergent. For if simply convergent, we can rearrange A so as to make the resulting series divergent, which contradicts the hypothesis. 9. In 121, 2 we exhibited a convergent series to show that a, .,..„ does not need to converge to if i^ ••• i„ converges to an ideal point some of whose coordinates are finite. As a counterpart we have the following : Let A be absolutely/ convergent. Then for each e > there exists a X, such that any finite set of terms B lying without R^ satisfy the relation \B\e. Hence, however large X is taken, there exist terms B='^n,p which do not satisfy 1). 10. We have seen that associated with any m-tuple series GENERAL THEORY 147 extended over a lattic^system SUi in $R„ is a simple series in 5Rj. We can generalize as follows. Let ^ = \il be associated with a lattice system Ti = \jl in 8I„ such that to each i corresponds a y and conversely. If t~ 9" we set a. .„ = «. .. Then A gives rise to an infinity of w-tuple series as }l —.In We say 5 is a conjugate n-tuple series. We have now the following : Let A he absolutely convergent. Then the series B is absolutely convergent and A = B. For let A\ B' be associate simple series of A, B. Then A, B' are absolutely convergent and hence A'=B'. But A = A', B = B'. Hence A = B, and B is absolutely convergent. 11. Let A = Sa^^... ,„ be absolutely convergent. Let B = Sa^^...^^, be any p-tuple series formed of a part or all the terms of A. Then B is absolutely convergent and l^l^AdjA For let A\ B' be associate simple series of A and B. Then B' converges absolutely and | -B' 1 ;5: Adj A. 125. 1. Let ^ = 2a, ...,„. (1 in the cell tj — l we may, and we have For the non-negative term series 1) to converge it is necessary and sufficient that the integral 3) converges. 2. iei /(a;, ••• a;„) > he a monotone decreasing function of X in H, the aggregate of points all of whose coordinates are non- negative. Let «, ..„ =/0, - O- The series j _ t,, -^ ^".] •••>m is convergent or divergent with J= f fdx^ ■•• dx^. J li For \&t R■^, R^, --hQ & sequence of rectangular cubes each R„ contained in i2„+i . Let R„^,= R,-R„ s>n. Then X, /* being taken at pleasure but > some v, there exist an I, m such that /^ But the integral on the right can be made small at pleasure if J is convergent on taking l> m > some n. Hence A is convergent if J" is. Similarly the other half of the theorem follows. Iterated Summation of Multiple Series 126. Consider the finite sum 2a.,...« tj = 0, 1, •••»! ••• i„ = 0, 1, •••«„. (1 One way to effect the summation is to keep all the indices but one fixed, say all but tj , obtaining the sum .,=(1 Then taking the sum of these sums when only tj is allowed to vary obtaining the sum „ „ •j=0 '1=0 ITERATED SUMMATION OF MULTIPLE SERIES 149 and so on arriving finally at '"n m, S-2«.,....„ (2 whose value is that of 1). We call this process iterated summa- tion. We could have taken the indices tj ••• t„ in any order instead of the one just employed; in each case we would have arrived at the same result, due to the commutative property of finite sums. Let us see how this applies to the infinite series, ^ = ^«.,-.™' ii--i,„=0, 1,...«). (3 The corresponding process of iterated summation would lead us to a series ^^£ £...|^ .4 which is an m-tuple iterated series. Now by definition 2l = lim 2 lim S ■ • • lim •-1 Cm— =*^ tm=^ 1*10-1==° 'm-l^*^ „,==c '>=0 = lim lim ••• lim A^^. ''m=« ■'m-l==° "1=== ■•I'm' ile A T . (5 (6 O Thus A is defined by a general limit while 21 is defined by an iterated limit. These two limits may be quite different. Again in 6) we have passed to the limit in a certain order. Changing this order in 6) would give us another iterated series of the type 4) with a sum which may be quite different. However in a large class of series the summation may be effected by iteration and this is one of the most important ways to evaluate 3). The relation between iterated summation and iterated integra- tion will at once occur to the reader. 127. 1. Before going farther let us note some peculiarities of iterated summation. For simplicity let us restrict ourselves to double series. Obviously similar anomalies will occur in wi-tuple series. 150 MULTIPLE SERIES ^^ A = Uqq + floi + Sa + ••• + «io + «ii + <^ii + - + - be a double series. The m'" row forms a series CO ^""' = «m,o + «">! + ••• = 2;a„„ ; and the w*" column, the series fn=0 ThpTl °° 00 CD ni=0 in=On=o are the series formed by summing by rows and columns, respec- tively. 2. A double series may converge although every row and every column is divergent. This is illustrated by the series considered in 121, 2. For A QO tX3 is convergent while 2a„, 2a,.j are divergent, since their terms are not evanescent. ~ '^ 3. A double series A may be divergent although the series R ob- tained by summing A by rows or the series C obtained by summing by columns is convergent. ^''"''le* A„=0 if r or 8 = if r, s > 0. r + s Obviously by I, 318, lim A„ does not exist and A - 2a„ is di- vergent. On the other hand, R = lim lim A„ = 0, (7= lim lim^„ = 1. Thus both R and C are convergent. ITERATED SUMMATION OF MULTIPLE SERIES 151 4. In the last example R and G converged but their sums were different. We now show : A double series may diverge although both R and G converge and have the same sum- For let A^^^ = if »• or s = '■^ ifr, s>0. r^ + s Then by I, 319, lim J.„ does not exist and A is divergent. On the other hand, n ^■ t a n ' R = lim lim J.„ = 0, T C O f^IOD C= lim lim A,, = 0. Then R and S both converge and have the same sum. 128. We consider now some of the cases in which iterated sum- mation is permissible. CO Let A = Sa,^ ... ,^ be convergent. Let I'j, tgi •" 'm ^^ c-'^'y permutation of the indices tj, tji "■ 'm- If <^^^ t^^ ^ ~ \-tuple series S 2 ••■ S a . ., . 2 3 m CO CO are convergent, J. = S "^S a,,...i„. This follows at once from I, 324. For simplicity the theorem is there stated only for two variables ; but obviously the demon- stration applies to any number of variables. 129. 1. Let f(x-^---x^ be a limited monotone function. Let the point a= («! ••■«„) be finite or infinite. When f is limited, all the s-tuple iterated limits Yixa ••• lim f (1 exist. When s = m, these limits equal lim/(a;i---a;„). (2 ac=a In these limits we suppose x may be called the first or principal polyant. Let us suppose now that the indices i in 1) run over one or more polyants. Let B^ be a rectangular cell, the coordinates of each of its vertices being each numerically < \. Let A^ denote the terms of A lying in R,,. Then I is the limit of J.^ for \ = cia, if for each e > there exists a X^ such that 154 MULTIPLE SERIES If lim A (3 A= exists, we say A is convergent, otherwise A is divergent. In a similar manner the other terms employed in multiple series may be extended to the present case. The rectangular cell jBa, which figures in the above definition may without loss of generality be replaced by the cube Moreover the condition necessary and sufficient for the exist- ence of the limit 3) is that I A - ^M I < 6 X, /i4 > Xo- 132. The properties of series lying in the principal polyant may be readily extended to series lying in several polyants. For the convenience of the reader we bring the following together, omitting the proof when it follows along the same lines as before. 1. For A to converge it is necessary and sufficient that lim A^ = 0. 2. A series whose adjoint converges is convergent. 3. Any deleted series £ of an absolutely convergent series A is absolutely convergent and \B\_Q be a monotone decreasing function of the distance of x from the origin. Let ., ITERATED SUMMATION OF MULTIPLE SERIES 155 Then . _ converges or diverges iviih the integration extended over all space containing terms of A. 133. 1. Let B, 0, D ••• denote the series formed of the terms of A lying in the different polyants. For A to converge it is sufficient although not necessary that B, 0, •■■ converge. When they do, A = B+C+I>+ - (1 For if B/^, C), •■• denote the terms of B, ■■• which lie in a rectangular cell Bx^> Passing to the limit we get 1). That A may converge when jB, C, ••• do not is shown by the following example. Let all the terms of ^= 1a,^...,^ vanish ex- cept those lying next to the coordinate axes. Let these have the value +1 if ij, i^--- i^>Q and let two a's lying on opposite sides of the coordinate planes have the same numerical value but opposite signs. Obviously, Ax = 0, hence A is convergent. On the other hand, every B., (7 ••■ is divergent. 2. Thus when B, ••• converge, the study of the given series A may be referred to series whose terms lie in a single polyant. But obviously the theory of such series is identical with that of the series lying in the first polyant. 3. The preceding property enables us at once to extend the theorems of 129, 130 to series lying in more than one polyant. The iterated series will now be made up, in general of two-way simple series. CHAPTER V SERIES OF FUNCTIONS 134. 1. Let I = (tj, tj ••• «p) i"un over an infinite lattice system g. Let the one-valued functions f^ ... .X^i •■■x^) = M^O = /. be defined over a domain 31, finite or infinite. If the ^-tuple series extended over the lattice sj'^stem 8 is convergent, it defines a one- valued function F(jx-^^--- x^) over 31. We propose to study the properties of this function with reference to continuity, differen- tiation and integration. 2. Here, as in so many parts of the theory of functions depend- ing on changing the order of an iterated limit, uniform convergence is fundamental. We shall therefore take this opportunity to develop some of its properties in an entirely general manner so that they will apply not only to infinite series, but to infinite products, multiple inte- grals, etc. 3. In accordance with the definition of I, 325 we say the series 1) is uniformly convergent in 31 when jP^ converges uniformly to its limit F. Or in other words when for each e>0 there exists a \ suchthat \F-F^X, for any x in 31. Here, as in 117, F^ denotes the terms of 1) lying in the rectangular cell ^^, etc. As an immediate consequence of this definition we have : Let 1) converge in 21. For it to converge uniformly in 21 it is necessary and sufficient that \Fi,\ is uniformly evanescent in 31, or in other words that for each e> 0, there exists a \ such that | J'^ | > e for any x in 31, and ii'>\. 163 GENERAL THEORY 157 135. 1. Let • lim /(a;i ■•• a;„, fj •■•*„) = (^(a;^ •••a;„) t = T in 21. Here 31, t may be finite or infinite. If there exists an 7/>0 such that /=<^ uniformly in F^(a), a finite or infinite, we shall say / converges uniformly at a ; if there exists no j? < 0, we say / does not converge uniformly at a. 2. Let now a range over 21. Let 93 denote the points of 21 at which no ?? exists or those points, they may lie in 21 or not, in whose vicinity the minimum oi t) is 0. Let D denote a cubical division of space of norm d. Let SQj^ denote as usual the cells of J) containing points of 93. Let (Eo denote the points of 21 not in S-". Then f =4> uniformly in S^) however small d is taken, but then fixed. The converse is obviously true. 3. Iff converges uniformly in 21, and if moreover it converges at a finite number of other points SQ, it converges uniformly in 21 + S. For iif=^ uniformly in 21, |/-<^| is limited in 21- For cf>=f(x,t)+e' \e'\ there exists a X such that I p I ^ „ /'I for any M, >_It^> Hx- When « = 1, these rectangular cells re- duce to intervals, and thus we have in particular I /„(a;i ••• a;„) I < e for any n > n'. When 8 > 1 we cannot infer from 1) that IA-'.(^i-^m)|< e , in 21, (2 for any i lying outside the above mentioned cell M^. A similar difference between simple and multiple series was mentioned in 121, 2. However if/ > in 21, the relation does hold. Cf. 121, 3. 136. 1. Let /{x-^ ■■•^mi ^1 •■• ^«) ^^ defined for each x in 21, and t inX. Let t ^ .^ n • or lim/=0(a;j...a;^) in 21, T finite or infinite. The convergence is uniform if for any x in 21 |/-^| < -f Oi ••• «„) t in Vs* (t), B fixed while lim 1^ = 0. For taking €>0 at pleasure there exists an 7;>0 such that I -f I < e , tin V,,* (t). But then ii S\<^ ■ for any t in Vs*{t} and any x in 21. Example. ,.„ sin a; sin V n . • of ^n ^ ^^rT^I^=' = ^' m2( = (0,co). Is the convergence uniform ? Let then w = 0, as y = — GENERAL THEORY 159 Then /-0 sin X cos u < 1 + x cot^ U sin a; sin^M X cos^ u _ I sin X cos u sin^ u I sin'''M + a; cos^m j < tan2M= 0. Hence the convergence is uniform in 21. 2. As a corollary we have Weierstrass' Test. For each point in 21, let \f^...,„\Q. Then |«^+i-B'"^^+ - +a„i2"lia^>_-- = 0. 138. 1. Let Ti K> X /■ \ be uniformly convergent in 21. Let A, B be two constants and ^ = %,-.(a;i-a;„) is uniformly convergent in 31. For then AF.,^< a.,^.,,. But F being uniformly convergent, 2. Let F=^f^...,ix,...x^~) f>0 converge uniformly in %. Then i=Slog(l+/0 is uniformly convergent in 21. Moreover if F is limited in 21, so is L. For /. > in 21, hence |/| - 0Cm=g^(t^ ... «„), y ... x^) lies in 21 as t = (t^ — t„) = T. If lim A = lim 1/(^1 ... g^, uni- formly in 21, GENERAL THEORY 163 For if / = ^ unif OKnly in 31, e>0, S>0 |/_01 S. Then \^nCx)\<.. 1 ^l+(« + l)S2 We now apply 136, 1. Example 4. Fix) = 1.x nCr^ + l)z^ - 1 GENERAL THEORY 165 Here /...{; n + 1 1 + nV 1 + (m + 1)V and F is telescopic. Hence p _ _x (n + V)x in ?I = (-^, iZ). 1 + 2;2 The convergence is not uniform at 2: = 0. For set a„ == — = — . Then w + 1 I F„(^a,^) I = i, does not = 0. It is, however, uniformly convergent in 21 except at 0. For if I a; I > S, l-P«(^)l = (n + \)x l+(n + 1)%2 (jn + 1) Jg 1 + (w + 1)2S2' < e for n > some m. 141. Let us suppose that the series F converges absolutely and uniformly in 31. Let us rearrange F, obtaining the series G. Since F is absolutely convergent, so is G- and F = Cr. We can- not, however, state that Gr is uniformly convergent in SI, as Bocher has shown. Fxample. -, F= i— l^jl- 1 + X - X + x^ - z^ + x^ - 3^ + ■■■ \. X ^^'^ F.,^ = 0. Hence F'\s uniformly convergent in St = (0, 1). Let -, a= t^:^\l- I + X + x"^ - X + a^ + x* - x"^ + ■■■ I. x Then a,,^,= '^^=^]{l-l) + (x + x'^-x) + Cx^ + x*-x'') + -: X + (a;2" + a;2''-i - 2;") i = L^-?fz"+iH \- x"^" \ = z\l - x'^) . 166 SERIES OF FUNCTIONS Let 1 a„ = 1 - -. n = -1 — as«=oo. e\ ej Hence Gr does not converge uniformly at a; = 1. 142. 1. Let f = uniformly in a finite set of aggregates 2lj, I 2l25 ••• Sly Thenf converges uniformly in their union (21^, ••• 31^,). For by definition €>0, S. >0,|/-<^| 0. Then 1) holds if we replace S, by S. 2. The preceding theorem may not be true when the number of aggregates Slj, Slj ••• is infinite. For consider as an example jF' = 2(1 - x)3f', which converges uniformly in 21 = (0, 1) except at x= 1. Let '21. = f*-^, -^) « = 1, 2, Then F' is uniformly convergent in each 21,, but is not in their union, which is 21. 3. Letf= , g = -\fr uniformly in 21. Then /± 5' = <^ ± V' uniformly. If ^, yjr remain limited in 21, fg = yfr uniformly. (1 If moreover | '•I' | > some positive number in 21, - = — uniformly. (2 g ^|r The demonstration follows along the lines of I, 49, 50, 61. GENERAL THEORY 167 4. To show that 1), 2) may be false if ^, y]r are not limited. Let -. f=g = ±+t, 2l=(0M), T = 0. X Then rt, X the rest being as before. Then . But setting a; = <, ■(x^ ••• x^), uniformly. The demonstration is entirely analogous to that of I, 292. limM.(a;i"-a;„, there exists a fixed o- > 0, such that I F(u) - F(v^ I < 6 M in V„iv) , d in aj. But as u^ = v^ uniformly there exists a fixed S > such that i Wc — ■"i I < ^' ) a; in 31 , tin F8*(t). Thus if e' is sufficiently small, m=(mj, ••• Mj,) lies in V„(v) when a; is in 21 and t in Fj*(t). lim/(2;i - 2:„, t^ - tj = , i|r be limited in 21, and some positive number. Then /" = <^* uniformly in 21. (1 For GENERAL THEORY 169 But by 2), log^=\og(f} uniformly in 21; and by 142, 3 g log f =yjr log (f), uniformly in 31. Hence 2) gives 1) by 1. 145. 1. The definition of uniform convergence may be given a slightly different form which is sometimes useful. The function is a function of two sets of variables x and t, one ranging in an 9?^ the other in an 3i„. Let us set now w = (x-^ — x„, t^ — *„) and consider w as a point in m+p way space. As X ranges over 31 and ( over Vs*{t^, let w range over Ss- Then ,. „ , uniformly in 21 when and only when €>0, S>0 |/-<^| uniformly in 21 when and only when e>0, 8>0 |/— ^|0 ; X>0. (1 '^^^^ (j>Cx} = lim/(2;, w) = , in 21 = (0, oo). Let us investigate whether the convergence is uniform at the point X in 31. First let a; > 0. If < a < a; < J, we have 170 SERIES OF FUNCTIONS As the term on the right = as w = oo , we seef= uniformly in (a, 6). When, however, a = 0, or 6 = oo , this reasoning does not hold. In this case we set _ ^^ which gives ^ y^ . ^ x^ — • As the point (x, m) ranges over 3E defined by x>0 , w>l, the point (t, n) ranges over a field 2; defined by * > 1 , w > 1, and the correspondence between 36 and X is uniform. Here The relation 2) shows that when a;>0, ^ = oo as»=^oo; also when x= 0, t = l for any n. Thus the convergence at a; = is uniform when ^ ^ fi The convergence is not uniform at a; = when 3) is not satisfied. For take < a: = — - , m=l, 2, ... For these values of re ih which does not = as w = oo . 146. 1. {Moore, Osgood.') Let lim/(a;i ■■■ a;„, t^ ■■■ tj = (^x^ ■■■ x„) uniformly in 21. Let a he a limiting point of 21 and lim/(a:i - a;„, t^ -. t„) = V^( = lira <^(x^ •••«„) , ^ = lim -^{t^ - t„) x=a t=T exist and are equal. Sere a, t are finite or infinite. GENERAL THEORY 171 We first show eMsts. To this end we show that 6>0 , S>0 , I <^(a;') - ^(a;") I < e a;', a;" in Fj*(a). (1 Now since /(a;, f) converges uniformly, there exists an i7>0 such that for any a;', a;" in 21 (x') = /(a;', + e' t in F,*(t) (2 4>(x")=f(x'\t) + e". |e'|,|e"|<|. (3 On the other hand, since /= -^ there exists a S > such that f(x') I < I a; in 31 , t in F;*(t). (6 Since /= i^, we have I f(x, - -f (0 I < I « "1 T'^«'*(«) . * fixed in F;* (t). (7 o Since =*, I I < I a; in F6"*(a). (8 o Thus 7), 8) hold simultaneously for S < S', S". Hence |^(<)_^| ^ and L = ^l.= Xi = - Hence limF(x)=L Also iZlim/„ = 0; hence Elim F(x}=0. a!=0 Example 2. 1 n! \ X J converges uniformly in any interval finite or infinite, excluding x=0, where F is not defined. For I f„ I < — , and I -4- > - 'nl i+S.^,- Hence lim F{x') = e. x=0 GENERAL THEORY I73 Example S. • 1 for x^Q Here while Thus here 1 + 3^ ■■ for x= 0. lim F(x) = 1, Slim /„(2;)= 20=0. a!=0 liin2/„(a;)=^Slim/„(:c), But J' does not converge uniformly at a; = 0. On the other hand, it does converge uniformly at a; = ± 00. ^°^ ]imJ'(x)=0 , lim/„(x)=0, ^^'^ lira S/„(:r) = Slim/„(a;), as the theorem requires. Example 4. p, \_'S^ f**^^ (n + V)x- 21 2^2 e which converges about x^O but not uniformly. However, n,^ 2/„(a.) = Slim /„(:.) = 0. Thus the uniform convergence is not a necessary condition. 147. 1. Let lim /(iCj ■ • • a;„, ij ••• ^„) = <^(a;j ••• a;^) uniformly at x — a. Let f(x, f) be continuous at x= a for each t in F^6*(t). Then (a + /i) = lim /(a, t) = (/>(«). Q.E.D. /l=0 t=r 174 SERIES OF FUNCTIONS A direct proof may be given as follows : fix, = ^{x) + e' I e' I < €, a; in Vi^a) Cx')-(ix")=f{x', f)-fix", 0+e". But \fix",t)-fix',t)\ are continuous. There is, however, an impor- tant case noted by Dini. Let f(xi ••• x„, fj ••• <„) be a function of two sets of variables such that X ranges over 31, and t over a set having t as limiting point, T finite or ideal. Let lim/(a;, 0= <^(a') i" 31. Then we can set j?^ ^n j.r ^ , i r *\ 176 SERIES OF FUNCTIONS Suppose now | •^(a;, *') I < i '^(^i I ^oi" ^^J *' i° t^ie rectangu- lar cell one of whose vertices is t and whose center is t. We say then that the convergence of / to ^ is steady or monotone at x. If for each x in 21, there exists a rectangular cell such that the above inequality holds, we say the convergence is monotone or steady in 21. The modification in this definition for the case that t is an ideal point is obvious. See I, 314, 315. 2. We may now state Din€s theorem. Let f(x-^ ■•■ x^, be continuous functions of x in 21. Then f converges uniformly to (x) + ^jr(ix,t). We may take t' so near t that | ■^(a;, <') | < -• Let x' be a point in VriCx}. Then f{x',t') = (x')+ylrCx',t'). As /is continuous in x, \f(x',t'}-f(x,t'-)\ + ylr. Since /= 4> uniformly |^| <• ^ for any ( in some V*(t) and for any x in 21. Thus I /^ /»' I — \ff-f4>\ to be limited we may suppose that /(a;, t) is limited in 21 for each t near t. 2. As corollary we have Let limf(xi---x^, t^---t„) = = flim/. 3. From 1, 2, we have at once: Let -F= 2/. ....,(^i*"^m) ^^ uniformly convergent in the limited field 21- Let each /,...., be limited and integrable in 2[. Then F is integrable and rp_ v Cf •'t* 178 SERIES OF FUNCTIONS If the f^^...^, are not integrable, we have Example. p_'^ ^ does not converge uniformly at a; = 0. Cf . 140, Ex. 3. Here #„ = 1 and -p. ^ (1 for x^O, 1 + na;^ for X for a; = 0. Hence CjFdx = 1, fFjx=i-r-'^ + na^ _ 3^ _ arctg Vw _^ ^ Thus we can integrate F termwise although F does not converge uniformly in (0, 1). 151. That uniform convergence of the series ^(^)=/l(^)+/2(^)+- (1 with integrable terms, in the interval 31 = (a < 6) is a sufficient condition for the validity of the relation Xb fb /»6 Fdx = ^ f^dx + j^f^dx + is well illustrated graphically, as Osgood has shown.* Since 1) converges uniformly in 21 by hypothesis, we have F„{x) = F(x) - F^ix-) (2 and \F„ix)\<€ n>m (3 for any x in 21. * Bulletin Amer. Math. Soc. (2), vol. 3, p. 59. GENERAL THEORY 179 In the figure, the graph of F(x) is drawn heavy. On either side of it are drawn the curves F—e, F+e giving the shaded band which we call the e-hand. From 2), 3) we see that the graph of each J'„, n>m lies in the e-band. The figure thus shows at once that £Fdx and dx can differ at most by the area of the e-band, i.e. by at most '2ecZa; = 2e(J-a). 152. 1. Let us consider a case where the convergence is not uniform, as nx _(n—V)x n^) = X g(m-lte2 = 0. Here Fx^:> = If we plot the curves «/ = FX^O, we observe that they flatten out more and more as m = ao, and approach the a;-axis except near the origin, where they have peaks which increase indefinitely in height. The curves FnQc), n>m, and m suf- ficiently large, lie within an e-band about their limit F(x~) in any inter- val which does not in- clude the origin. If the area of the region under the peaks could be made small at pleasure for m sufficiently large, we could obviously integrate termwise. But this area is here 180 SERIES OF FUNCTIONS Jo " 2^0 dx I e'"'] 2L e'^'X 2V e^'J as w = 00 . Thus we cannot integrate the F series termwise. 2. As another example in which the convergence is not uniform let us consider Here F =• The convergence of F is uniform in 21 = (0, 1) except at a; = 0. The peaks of the curves F„(x^ all have the height e"i. Obviously the area of the region under the peaks can be made small at pleasure if m is taken sufficiently large. Thus in this case we can obviously integrate termwise, although the convergence is not uniform in 21. We may verify this analytically. For rF„dx= r^dx=- Jo Jo e'^" n 3. Finally let us consider 1 + nx . we" ^(■^■) ^ ^ j (n + lYx _ n^x ^ ^ ^ 1 1 +(w + 1)3^2 1 + w3a;2 as w = 00 , = 0. Here ^.(^) = 1 + n^x^ The convergence is not uniform at a; = 0. The peaks of -F„(a;) are at the points x = n~\ at which points F„=^Vn. GENERAL THEORY 181 Their height thus %icreases indefinitely with n. But at the same time tliey become so slender that the area under them = 0. In fact CFJx)dx = f ^ c? log (1 + wV) ^ii}^^^"- "^1 loff (1 + wV) . Q 2 n "In We can therefore integrate term wise in (0 < a). 153. 1. Let Urn 0-(x, «j ••• «„) =g{x) in 21 = (a, a + S), t finite or infinite. Let each Cr'^i^x^ f) he continuous in 21 ; also let Gr'x(,x, <) converge to a limit uniformly in % as t = t. Then Urn Gr'Xx,t)=g'(x) in % (1 and g' (jc) is continuous. For by 150, 2, G'^dx = I lim Gr'xdx. By I, 538, ^, Gt'^dx = a(x, - a^a, t). Also by hypothesis, ^^^ ^ ^ ^^^ ^^ _ g_^^^ ^^ j ^ ^^^^ _ ^^^^_ Hence ^(a;)-Ka) = f'lim a',(x, t)dx. (2 But by 147, l, the integrand is continuous in 21. Hence by I, 537, the derivative of the right side of 2) is this in- tegrand. Differentiating 2), we get 1). 2. Let F(x)= '2/,^..., X^} converge in 2l = (a, a -f-S). Let each f/Cx') be continuous, also let be uniformly convergent in 21. Then F'(x) = l.f'Xx\ in%\. This is a corollary of 1. Hence Let 182 SERIES OF FUNCTIONS 3. The more general case that the terms /.j...,, are functions of several variables xy- x„ follows readily from 2. 154. Example. Here p — _ '"'''^ a function whose uniform convergence was studied, 145, 3. We saw F(x^=0 for any a; >0. J"(a:)=0 x>0. aix:>=ifL(x-). (1 Then G^„(,)=^,(,)._^ + ^!^^!^. hence F'(x')='S.Mx\ (2 and we may differentiate the series termwise. If a; = 0, and «= 1, \>0; (r„(0)= — m^ = — oo as m = qo. In this case 2) does not hold, and we cannot differentiate the series termwise. For a; = 0, and a > 1, G^„(0) = 0, and now 2) holds ; we may therefore differentiate the series termwise. But if we look at the uniform convergence of the series 1), we see this takes place only when t ^ « — 1 \ 155. 1. ( Porter.) Let ^^ ^ ^ FCx)=-Lf,^ (X) converge in 21 = (a, 6). For every x in 'SSL let |/i'(a:)| 0, S>0, D = For < e, I Aa; I < S. ^F _ ^ f„(x + Ax) - Ux) ^f,... Ax Ax ~ ^■'''^^^^ where |^„ lies in V^^x). Thii„, we may take S so small that "^^"^ |-»I-jrx + — 00 '^2 (a') = 2 S^*"""*'' cos (2 n + 1) Tra; = 2qi cos Tra; + 2 g^ cos Sirx + •■• i^j (a;) = 1+2 2 J"' cos 2 WTra; = l + 2q cos 2 Tra; + 2 9* cos 4 7ra;+ ••• ,Vg (a;) = 1 + 2 i ( - l)"g"' cos 2 WTra; 1 = 1 — 2g'Cos 27ra;+ 2^ cos 47ra:— ••• Let us take i , ^ i Then these series converge uniformly at every point x. For let us consider as an example iVj. The series T=\q\ + \q\*+\q\'+- is convergent since the ratio of two successive terms is 2 =02«+l. q"' ^ and this = 0. Now each term in 'Vj is numerically <|j|("+i)»<|g|»^ and hence < the corresponding term in T. Thus &i (a;) is a continuous function of x for every x by 147, 2. The same is true of the other >» 's. These functions were discovered by Abel, and were used by him to express the elliptic functions. Let us consider now their derivatives. Making use of 155, l it is easy to show that we may differentiate these series termwise. Then A{ (a;) = 2 Tri ( - 1)»(2 n + 1} q^'^+i'^ cos (2 w + 1) vx = 2 TT (gri cos Tra; — 3 g* cos B-n-x + •••). GENERAL THEORY - 186 •H (^) = - 2 ttI (2 « + 1) ^c+i)= sin (2 w + 1) ttx = — 2 TT (g* sin TTz + Sj^sin 3 tts; + .••). CO ''3 (a;) = — 4 7r2 nj"' sin 2 uttx = — 4 TT (g- sin 2 Tra; + 2g^sin 4 7r2; + ...). GO ''0 (a;) = — 4 ttS ( - l)"^^"' siri 2 WTra; 1 = + 47r(g'sin 2 7ra; — 2 g* sin 4 tts; + •••). To show the uniform convergence of these series, let us con- sider the first and compare it with S=i + S\q\ + 5\q\'^+7\qf+ - The ratio of two successive terms of this series is 2^+3 |?ri)' _ 2n + 3 |2„,i 2n+ 1 |g|"' 2w + 1 '-" ' which = 0. Thus *S' is convergent. Tlie rest follows now as before. 156. 1. Let jj^ Gr(a + h, t^ - O- G-(a. t-^ - t„) ^ g(a + h)-ff(a) t=T h h uniformly for < | A | < ?;, t finite or infinite. Gl{a, t) exist for each t near t. Then g'(a') exists and limG^.;(a,0 = /(«)• This is a corollary of 146, l. Here a(a + h, t)-a(a,t) h takes the place of /(a;, f). 2. From 1 we have as corollary : (Dint), Let jp(^) ^ 2/., ... .» 186 SERIES OF FUNCTIONS converge for each x in 21 which has x = a as a proper limiting point. Letfl^a) exist for each t = (ti, — £„). Let ^ h converge uniformly with respect to h. Then FXa)=^fi...S'^). CHAPTER VI POWER SERIES 157. On account of their importance in analysis we shall devote a separate chapter to power series. We have seen that without loss of generality we may employ the series , , ■2 , ' /-i instead of the formally more general one a^ + ai(a;— a)+ ^2(3; — a)^+ •■• We have seen that if 1) converges for a; = c it converges abso- lutely and uniformly in (—7, 7) where < 7 < | c |. Finally, we saw that if c is an end point of its interval of convergence, it is unilaterally continuous at this point. The series 1) is, of course, a continuous function of x at any point within its interval of convergence. 158. 1. Let PQc) = «o + *i^ + V^ + ■■■ converge in the interval 21 = ( — a, a) which may not be complete. The series P„ = 1 . 2 • ••• wa„ + 2 • 3 • ••• (n + l)a„+i2; + - obtained by differentiating each term of P n times is absolutely and uniformly convergent m 93 = (— A ^)) /3< «i «»i^ ^=P„(2;), in 58. dx" For since P converges absolutely for x= 0, u„^'' ttiq, a small at pleasure, ^''^^^ \Q^\ = \rP^dx<-^<^ on taking a sufficiently small. 159. 1. Let us show how the theorems in 2 may be used to obtain the developments of some of the elementary functions in power series. The Logarithmic Series. We have , =l+x + x'^+x^+ ■■■ I —X for any a; in 31 = (— 1*, 1*). Thus J I ^ =r— log(l — x)= i dx+ i xdx+-- 1 — x ^ ^0 *^ Hence , 2 ™3 1 log(l-a;)=-J2;4-| + J+- I ; a; in 31. This gives also log(l-|-2:) = a;-^ + f -••• ; x in 21. (1 GENERAL THEORY ' 189 The series 1) is alsc^valid for x = 1. For the series is conver- gent for x=l, and log (1 + x) is continuous at x = l. We now apply 147, 6. For a; = 1, we get 2. The Development of arosin x. We have by the Binomial Series 1 1 2^ 1 -3 .^ l-S-S .^ Vl-a;2 2 -ZA 2.4.6 for a; in 21 = ( - 1*, 1*) . Thus — = arcsiu x= x + - — -x^-\ '- — -x^+--- (2 ° VF^ 2-3 2.4.5 ^ It is also valid for a; = 1 . For the series on the right is conver- gent for x=l. We can thus reason as in 1. For a; = 1 we get 2 2.3 2.4.5 2.4.6.7 3. The Arctan Series. We have ~ — =l-a^H-a;*-a;6 4-... l+a;2 for a; in 31 = (- 1*, 1*). Thus I = arctan a; = I dx—\ x^dx+--- 1 -|- a;2 »/o i/o valid in 21. The series 3) is valid for a; = 1 for the same reason as in 2. For a:=lweget 7r_^_ll_l, 4~ 3 5 7 4. The Development of e^. We have seen that converges for any x. Differentiating, we get 190 POWER SERIES Hence j^'^a-) = ^ix) (a) for any x. Let us consider now the function We have e^W - He" E'-E „ f (x) = = = U by (a). Thus by I, 400, /(a;) is a constant. For x = 0, fQc) = 1 . Hence ^ ^ x ^ x^ ^ s? , '=^ + r! + 2! + 3! + - valid for any x. 5. Development of cos x, sin x. The series 2 ^4 ^e converges for every x. Hence, differentiating, (II — X vr X" ^ - i + 31 6! + ri" — — 1 -I-— — -I- ... Hence adding, ^^ 0" = 0. (b) Let us consider now the function f(x)= Csin z + C cos X. Then ^,^^^ = Ccos a; + 0" sin x- O' sin a; + C" cos a; = {C+ C'^cosx = by (b). Thus /(a;) is a constant. But C= 1, (7' = 0, for a; = 0, hence /(2=)=0, or (7sina;+ C" cos a;= 0. (c) In a similar manner we may show that or 51(3;)= C cos a;— 0" sin a; = 1. (d) GENERAL THEORY 191 If we multiply (c) liy sin x and (d) by cos x and add, we get C= cos X. Similarly we get 0" = — sin x. Thus finally valid for any x. X a? , 7^ 160. 1. Let P = a„2;'"+ a^+ia;""*! + ••• , a^^O, converge in some interval 21 about the origin. Then there exists an interval S3 < 21 iw which P does not vanish except at x = Q. P = ^(am + Clm+l^ + • • •) = x'"Q. Obviously Q converges in 21. It is thus continuous at x=0. Since Q^O a,t x = it does not vanish in some interval 53 about a; = by I, 351. In analogy to polynomials, we say P has a zero or root of order m at the origin. 2. Let P = aQ + a^x + a^aP + ••• vanish at the points h-^,h^, •■■ = 0. Then all the coefficients a„ = 0. The points b„ are supposed to be different from each other and from 0. For by hypothesis P(bn) =0. But P being continuous at a; = 0, P(0) = limP(J„). Hence P(0)=0, Hence and thus n P = xPy Thus Pj vanishes also at the points 6„. "We can therefore reason on Pj as on P and thus asj = 0. In this way we may continue. 3. If P = ao + «ia^+- Q = h^^ + b^x+- 192 POWER SERIES he equal for the points of an infinite sequence B whose limit is a; = 0, then a„ =b„, n = 0, 1, 2 ••• For P — Q vanishes at the points B. H^^'^^ a„-&„ = , ri = 0, 1,2... 4. Obviously if the twro series P, Q are equal for all a; in a little interval about the origin, the coefficients of like powers are equal; that is ^^^j^ ^ w = 0, 1, 2... 161. 1. Let y = a^ + a^x + a^i^+... (1 converge in an interval 21. As x ranges over SI, let y range over an interval S3. Let z = h^ + h^y + l^y-^+- (2 converge in 33- Then z may be considered as a function of x de- fined in 21. We seek to develop z in a power series in x. To this end let us raise 1) to the 2°, 3°, 4° ••• powers ; we get series „ , , 2 , y = «2o + "-21^ + 'hz^ + ••• / = «30 + «3l2: + «32*''' + •• • (3 which converge absolutely within 21. We note that a„„ is a polynomial. «m.„=-f'm,n(«0' «l"-«n) in Uq. ■■• a„ with coefficients which are positive integers. If we put 3) in 2), we get a double series + ^2*20 + *2«21^ + *2«22^^ + ' " ' (* + ^s^so + h'^zi^ + \<^vi^ + • •• + If we sum by rows, we get a series whose sum is evidently z, since each row of i) is a term of z. Summing by columns we get a series we denote by C= c^-V c-^x-\-c^+ ••• (5 GENERAL THEORY 193 Ci = ^iftj + ^ga^i -f ijagi + • • • (6 We may now state the following theorem, which is a solution of our problem. Let the adjoint y-series, '? = «j + ail+«2P + - (7 converge for | = ^q to the value rj = t]q. Let the adjoint z series converge for 7} = i;g. Then the z series 2) can be developed into a power series in x, viz. the series 5), which is valid for | x1 < f^. For in the first place, the series 8) converges for v^Vo- We show now that the positive term series S) = (/3o + /3i«o) + /3i«i^ + /3i«2P + - + /32«20 + /32"21? + ^2«a2? + ••• + converges for f < |q. We observe that J) differs from Adj D, at most by its first term. To show the convergence of 33 we have, raising 7) to successive powers, r,^ = A,, + A,,^ + A,,^^+- We note that A^„ is the same function F^^„ of «„, Kj, ••■ a„ as a„„ is of «„, ••• a„, i.e. As the coefficients of jF„_„ are positive integers, •*m, » = I *m, n I — -^m, n • (9 194 POWER SERIES Putting these values of ij, r^, t;^ • ■ • in 8), we get + Summing by rows we get a convergent series whose sum is f or 8). But this series converges for f 1 converge for some x > 0. If the series z = \+h^y + \y^+ ... converges for some y > 0, it can be developed in a power series z = Cg+ CiX + c^a? H — convergent for some x > 0. For we may take | = ] a; | > so small that has a value which falls within the interval of convergence of 4. Another corollary of 1 is the following : y= a„ + ajX+ a^ + •.• GENERAL THEORY 195 converge in 21= (— A^A). Then y can be developed in a power series about any point c of % y=Co + c^(x-c} + c^{x-cy^+ ••• which is valid in an interval 58 whose center is c and lying within 31. 162. 1. As an application of the theorem 161, i let us take 1! 2! 3! « = Z._£l4.^_ ... ■^ 1! 3! 5! As the reader already knows, 3 = 6* , ^ = sin a;, hence z considered as a function of x is s = e"" ". We have + *a;2+ 1 ,.4 6 ^ + ix'+ +7rV**+" -La^ + +.. (1 + Tk*'+ + Summing by columns, we get g^ gsinx^ 1 ^. 2.4. i,2^_ 1 3.4 __i_2,B_ ^i_ a^ ... 2. As a second application let us consider the power series 2 = -PCy) = «o + «i2/ + «2i/'^ + ••• convergent in the interval 21 = ( — ^, -B). Let a; be a point in 21. Let us take 1; > so small that y = x + h lies within 21 for all \h\ + («2 + *2 V + •• • -P- $ = («0-^o)+(«l-*l> + («2-*2>'^+ ■•• P- Q = a^h^ + (ajJ„ + a^h^x + {a^b^ + a^b^ + o-oh)^^ + ■" These are valid within 31, and the first two in 21. 2. Let us now consider the division of P by R. Since ^=P.l B R the problem of dividing P by 22 is reduced to that of finding the reciprocal of a power series. -^^* P = aQ + aiX+ a^+ ■■■ , a^^^Q converge absolutely in R=(^— R, R). Let Q = ajX + a^3^+ ■■• be numerically < | a^ | m SB = ( — >*, »*) r < R. GENERAL THEORY 197 Then l/P can he developed in a power series -p = Co + <'i^ + V^+ ••■ valid in 93. The first coefficient c^ = — . *^°' i = _J_ = l._L_ «o t «0 «0 «0 for all X in SB. We have now only to apply 161, l. 3. Suppose p ^ ^^^^ ^ ^^^^^^,, ^ .. . ^^ ^ ()_ To reduce this case to the former, we remark that where r> Then j. ^ 1 J. But l/Q has been treated in 2. 164. 1. Although the reasoning in 161 affords us a method of detei-mining the coefficients in the development of the quotient of two power series, there is a more expeditious method applicable also to many other problems, called the method of undetermined coefficients. It rests on the hypothesis that/(a;) can be developed in a power series in a certain interval about some point, let us say the origin. Having assured ourselves on this head, we set /(a;) = flo + <^i^ + 'h^ + ••• where the a's are undetermined coefficients. We seek enough relations between the a's to determine as many of them as we need. The spirit of the method will be readily grasped by the aid of the following examples. Let us first prove the following theorem, which will sometimes shorten our labor. 198 POWER SERIES ^' "^ f{x) = aQ^-a^x^a^^-V ■■•; -R0 in any interval S3 = (— ^ + S, ^— Sj,S>0, it follows that i^^i^i ij^jg Thus by 163, 2, tan x can be developed in a power series about the origin valid in 58- We thus set tan X = a,x + ao^ + afX? 4- • •• (2 GENERAL THEORY 199 since tana; is an odd^f unction. From 1), 2) we have, clearing fractions, ^-fT + fT-- = ^^'^^ + «3-' + -)(^-f? + S--) \' 2:^4! 6!/ ^V' 2! ^4! 6 ! ^ 8 V ^ Comparing coefficients on each side of this equation gives 1 .•. Uo = -. ^ 3 ^ a, , a, 1 2 . _i^ + fa_^=_l. .-.a -.11. ' 2! 4! 6! 7! ^ 315 a _^ + ^_^ + ih.= 1 • - 62 flj = 1. «1 ^~2! = 1 3! Thus valid in 2!4! 6!8! 9! ^ 2835' t^ux = X + ^x^ + j\:x^ + ^\\x'' + ^J^x^ + ... (3 TT* TT* 2^ ' 2 Example 2. j., ^ 1 / (•''^) = cosec X = sin X 1 ""v 3!+5T ; Since Q^i_^j^ X we see that 1 1 < 1 when X is in 93 = (-7r + S, tt-S), S>0. Thus xf(x) = l/F can be developed in a power series in 33. As f(x) is an odd function, xf(x) is even, hence its development contains only even powers of x. Thus we have ^/(*) = «o + '*2^ + ^4** H 200 POWER SERIES Hence Comparing like coefficients gives 1. an n _ 1 «o=l- ^3! ■ ^ 6 Thus 3! 5! 7! " 3-7! sin x « 6 360 3 • 7 ! valid in (— tt*, tt*). 166. Let -F(x)=/i(a;)+/2(2;)+ - wlaere /• ^ n , , 2 , -i o /„(a;) = a^ + a„i2; + a^^x^ + •■• w = 1, 2 ••• Let the adjoint series «nO + «nl? + «n2P + "• converge for | = i? and have „ as sums for this value of ^. Let <|) = 0i + as sum. Thus the double series 1) converges absolutely for | a; | < ^, by 123, 2. Thus the series 1) may be summed b)'- columns by 130, l and has F(x') as sum, since 1) has F as sum on summing by rows. 167. Example. Fix) = J ^^4^ -J— = S/„(a;) a>l. S n\ l + a^x ^ ■' This series we have seen converges in 21 = (0, 6), b positive and arbitrarily large. Since it is impossible to develop the /„(a;) in a power series about the origin which will have a common interval of convergence, let us develop J' in a power series about a;Q>0. We have 1 + a^x 1 + a"a;o -, a^jx — x^) 1 + a% ^ aHx-x^^ ^ a^"(a:-a:o)^ __ l + a.%P l + a% ' (l + a%)2 = ^„o + A,^ix- Xo) + A^^(x - a:o)2 + . . . where . (-l)''a"' "•" (l + a%)«+i 202 POWER SERIES Thus J' give rise to the double series + A[o + A',,(x - Xo) + ^iaCa; - x^y + + where a, _ (-!)" a nl The adjoint series to/„(«) is, setting ^=\x — Xq\, This is convergent if ^"^ <1 or if f , ^-. have 6q^ 0, and let it converge for t=t^. If we set it goes over into a series of the form u = X — a^ — a^3? — ••• (2 which converges for z = l. Without loss of generality we may suppose that the original series 1) has the form 2) and converges for x=l. We shall therefore take the given series to be 2). By I, 437, 2 the equation 2) defines uniquely a function x oi u which is continuous about the point m = 0, and takes on the value x= 0, for M = 0. We show that this function x may be developed in a power series in u, valid in some interval about w = 0. To this end let us set X = u + c^ + CgM^ + ••• (3 and try to determine the coefficient c, so that 3) satisfies 2) formally. Raising 3) to successive powers, we get a;2 = m2 + 2 c^u? + {,e^ + 2 c^u^ + (2 c^ + 2 c^c^u^ + •• ■ 2^ = m3 + 3 CjM* + (3 c} + 3 c^vJ' + ••• (4 a;* = M*+t c.-^K' + ••• Putting these in 2) it becomes M = M + (Cj — «2)**^ + ('^s "~ 2 a^c^ — a^v? + (c^ - a^ic^ + 2 (jg) - 3 a^c^ - a^)u^ 4- (Cg -2a^Xc^^- c^n^} - 3 agfc/ + Cg) - 4 a^c^ - a^'yu^ (5 + Equating coefficients of like powers of u on both sides of this equation gives _ ^2 — ''^2 Cj = 2 a^c^ + flg "i = a2(<'2^ + "^ ''3) + ^ «3«2 + «4 (6 Cg = 2 a^(c^ + CjCg) + 3 agCcj^ + Cg) + 4 a^Cj + a^ . 204 POWER SERIES This method enables us thus to determine the coefficient c in 3) such that this series when put in 2) formally satisfies this relation. We shall call the series 3) where the coefficients c have the values given in 6), the inverse series belonging to 2). Suppose now the inverse series 8) converges for some w^^fcO ; can we say it satisfies 2) for values of u near the origin ? The answer is, Yes. For by 161, 3, we may sum by columns the double series which results by replacing in the right side of 2) y*» />«« /*«0 . _ . by their values in 3), 4). But when we do this, the right side of 2) goes over into the right side of 5), all of whose coefficients = by 6) except the first. We have therefore only to show that the inverse series con- verges for some m ^ 0. To show this we make use of the fact that 2) converges for x=\. Then a„ = 0, and hence I a„ I < some a w = 2, 3, ••• (7 On the other hand, the relations 6) show that c„=/„(a2, ftg, •••«„) (8 is a polynomial with integral positive coefficients. In 8) let us replace aj , osg ■ • ■ by «, getting 7n=/n(«> «i ••■ «)•• (9 Obviously I ''r. I < 7n- (10 Let us now replace all the a's in 2) by a ; we get the geometric U=x — a7? — a3? — ca^—--. (11 = x--^. (12 1 — x The inverse series belonging to 11) is a; = M+ 72'jt2 4- From this we conclude at once : If 1°, /(z) aw(i its derivatives of every order are continuous in 31 = (a, J), and 2° lim i2„ = lim ~ fHa +0K) = O , n = aD, (4 w! a =/(«) + ^/(«) + 1^/"(«) + • ■ • (5 The above theorem is called Taylor s theorem; and the equa- tion 6) is the development of f{x) in the interval 31 by Taylor's series. Another form of 5} is f (X-) =f(a} + ^^ff^f(a) + (^^fXa) + ... (6 When the point a is the origin, that is, when a= 0, 5) or 6) gives 2 /(^) =/(0) + xfXO^ + |y/"(0) + - (7 This is called Maolaurins development and the right side of 7) Maclaurin's series. It is of course only a special case of Taylor's development. 2. Let us note the content of Taylor's Theorem. It says : If 1° /(a;) can be developed in this form in the interval 9l=(a< J); 2° if f{x) and all its derivatives are known at the point x = a; then the value of / and all its derivatives are known at every point X within 21. The remarkable feature about this result is that the 2° condi- tion requires a knowledge of the values of /(a;) in an interval (a, a + S) as small as we please. Since the values that a func- tion of a real variable takes on in a part of its interval as (a < c), have no effect on the values that /(a;) may have in the rest of the interval (c < J), the condition 1° must impose a condition on /(x) which obtains throughout the whole interval 3t. 170. Let f(x) he developable in a power series about the point a, viz. let f{x~) = ao -I- a.^(x -a) + a^(x - a)2 + ... (1 Then m)c„\ a„=/-l^ n=0, 1, ... (2 n ' i.e. the above series is Taylor'' s series. 208 POWER SERIES For differentiating 1) « times, we get Setting here a;= a, we get 2). The above theorem says that if f(x) can be developed in a power series about x=a, this series can be no other than Taylor's series. 171. 1. Let f'-'^\x) exist and he numerically legs than some con- stant M for all a 0, we have valid for all x and a > 0. 172. 1. To develop (1 + a;)'' and log (1 + a;) we need another expression of the remainder i2„ due to Cauchy. We shall con- duct our work so as to lead to a very general form for i2„. From 169, l we have K =f(x) -fia-) -ix- a-)f'(:a-) ^''~ 'l^ "" '/'"-"(«)• n— 11 TAYLOR'S DEVELOPMENT 209 We introduce the auiflliary function defined over (a, J). g(f) =/(0 +f(t-)(x-t) + ... +/<"-i'(0 ^^^^=^. (1 n — \\ Then , ^ ^/ n and g («) = /(a) + /' (a) (.r - a) + • • • + /"""(a) '^'^ ~ ^^^"'' ' Hence n ^ \ / n /'o If we differentiate 1), we find the terms cancel in pairs, leaving y(Q = (^^il!lV<'"(0. (3 n— \-\ We apply now Cauchy's theorem, I, 448, introducing another arbitrary auxiliary function Gr(x') which satisfies the conditions of that theorem. Then 9<:x-)-g(a-) ^IS^ a 1. When x=l, T converges only when ft > — 1 ; when a; = — 1, r converges only when /u.^0. We wish to know when (l + xy = l + ^x + ^'^~'^ x^->r -. (1 The cases when T diverges are to be thrown out at once. Con- sider in succession the cases that T converges. We have to investigate when lim i2„ = 0. Case 1°. 0<|a;|— 1. We employ here Lagrange's form of the remainder, which gives \ -l ■ ■■• n ^"- 1 . 2 . ... « ' Consider W„. Since n increases without limit, fi-n becomes and remains negative. As ^ > lim TF„ = 0. For U„, we use I, 143. This shows at once that lim ?7'„ = 0. Hence limi2„=0 and 1) is valid in this case, i.e. for a; = 1, /^ > — 1- 212 POWER SERIES Case S. a; = — 1, /i ^ 0. We use here for /i > the Schlomilch- Roche form of the remainder 172, 5). We set a = 0, A = — 1 and get Rn = i- 1)" '^^"?)"'^ M-/*~i — M-w + i-(i- ey- n—llfj, _ , _ iv„ jtt-l . fi-2- ■■■ fi-n+1 ^ ' 1.2--W-1 Applying I, 143, we see that lim B,^ = 0. Hence 1) is valid here if /t > 0. When ^ = equation 1) is evidently true, since both sides reduce to 1. Summing up, we have the theorem : The development of (1 + x^ in Taylor's series is valid when I a; I < 1 for all fi. When x = + 1 it is necessary that /i > — 1 ; when x= — l it is necessary that /i>0. 2. We note the following formulas obtained from 1), setting x=l and — 1. 1 1-2 1-2.3 '^ 11-2 1.2.3 '^^ 174. 1. We develop now log (1 + a;). The corresponding Taylor's series is f -. 1j._ _4._ 1 2 8 We saw, 89, Ex. 2, that T converges when and only when |a;| < 1 or x = 1. Let < a; < 1. We use Lagrange's remainder, which gives here ^ ^ (- ly-'^x" " n^l + dxy Thus . |A|<-. n Hence lim i2„ = 0. TAYLOR'S DEVELOPMENT 213 Let — 1 < a; < 0. We use here Cauchy's remainder, which gives, setting a; = — ^, 00 but otherwise arbitrary, which has derivatives in 21 of every order, viz. : 216 POWER SERIES The Taylor's series about the origin for F(x) is T(x) = J ^ ^('^)(0) ; \ ! = 1 for \ = 0, 3)^! A=0 and by 2) Hence -yw(O) \! S «! e' n^)=S^^=^^=2(-iy«,. (3 ^=0 e"* As iA>0 and lim «a = 0, (^-'-^) =4.68 ••■. T(3>) does not equal F(x) throughout any interval 21 = (0, 6), however small J > is taken. We show 1° that if F{x) = T{x) throughout 21, this relation is true in S8 = (0, 2J«). In fact let < 2;o < 6. By 161, 4 we can develop T about a;o, getting a relation T(ix-) = lC,ix-x,y (1 valid for all x sufficiently near a;j. On the other hand, we saw in 167 that F(x) = lBXx-x,y (2 is also valid for 0 is arbitrarily large. To prove our theorem we have now only to show F^T for some one a;>0. Since we have h , Fix) > -i — L_ = a^x-) . 1+x 1 + ax ^ "^ On the other hand ^«=;-C?-?)-( )-• Hence T(x)- take x = a~K For this value of X ai + l Observe that G considered as a function of a is an increasing function. For /^ , -j^n 2 j Hence F>Tior x>a-i. 178. Criticism 2. It is commonly thought if /(a;) and its derivatives of every order are continuous in an interval 21, that tlien the corresponding Taylor's series is convergent in SI. That this is not always so is shown by the following example, due to Pringsheim. It is easy to see that converges for every x>_^i ^^nd has derivatives of every order for these values of a;, viz. : 218 POWER SERIES Taylor's series about the origin is 2a=o T=^l(i- 1)X«'"'' + e-'V' The series T is divergent for a; > 0, as is easily seen. 179. Criticism 3. It is commonly thought if /(a;) and all its derivatives vanish for a certain value of x, say for x = a, that then/ (a;) vanishes identically. One reasons thus: The development oif(x) about a;= a is /(:.)=/(a) + ^/'(a)H-i^=j^/"(a)+ - As /and all its derivatives vanish at a, this gives /(a;) = + • (a; - a) + • (a; - a)2 + ••• = whatever x is. There are two tacit assumptions which invalidate this conclusion. First, one assumes because / and all its derivatives exist and are finite at x = a, that therefore f(x) can be developed in Taylor's series. An example to the contrary is Cauchy's function 0(x). We have seen that 0(x') and all its derivatives are at a; = 0, yet Oix) is not identically ; in fact C vanishes only once, viz. at a; = 0. Secondly, suppose /(a:) were developable in Taylor's series in a certain interval 31 = (a— A, a + A). Then / is indeed through- out 21, but we cannot infer that it is therefore outside 21. In fact, from Dirichlet's definition of a function, the values that/ has in 21 nowise interferes with our giving / any other values we please outside of 21. 180. 1. Criticism 4. Suppose /(a;) can be developed in Taylor's series at a, so that fix)=fia-) + ^^^/(a) + (^^/"(a) + .■■ = T for 2l=(a<6). TAYLOR'S DEVELOPMENT 219 Since Taylor's serie^y is a power series, it converges not only in 21, but also within 33= (2 a — 6, a). It is commonly supposed that /(a;) = T also in SB. A moment's reflection shows such an assumption is unjustified without further conditions on /(»). 2. Hxample. We construct a function by the method considered in I, 333, viz. f(x) = lim a+^)"cos^+l + sina; _ ^ Then f{x) = cos x, in 21 = (0, 1) = 1 + sin X, within 93 = (0, - 1 ). We have therefore as a development in Taylor's series valid in 21, ^2 -r* ffi ^^^)=^-f!+f!-fl + - = ^- It is obviously not valid within 93, although T converges in 93. 3. We have given in 1) an arithmetical expression for / (x). Our example would have been just as conclusive if we had said : Let /C^') = cos X in 21, and = 1 + sin x within 93. 181. 1. Criticism 5. The following error is sometimes made. Suppose Taylor's development /(:.)=/(«) +^^^/'(a)+ ^£^/"(a) + ... (1 valid in 21 = (a < S) . It may happen that T is convergent in a larger interval S8 = (a<5). One must not therefore suppose that 1) is also valid in 93- 2. Example. ^^^ fix')=e^ in 21 = (a, J), '^"^^ =e^+sin (a;-J) in 93 = (J, 5). Then Taylor's development is valid for 21. The series T converging for every x converges in 93 but 1) is not valid for 93. 220 POWER SERIES 182. Let f(x) have finite derivatives of every order in 3l=(a<6). lu order that/(a;) can be developed in the Taylor's series 72 /(a;)=/(a + A)=/(a)+A/'(a) + |^/"(a)+ ••. (1 valid in the interval 21 we saw that it is necessary and sufficient *^^* limi2„=0. But Rn is not only a function of the independent variable A, but of the unknown variable 6 which lies within the interval (0, 1) and is a function of n and h. Pringsheim has shown how the above condition may be replaced by the following one in which 6 is an independent variable. For the relation 1) to he valid for all h such that 0<_h < H, it is necessary and sufficient that Cauchy' s form of the remainder i2n(A, 0) = (1 - ^>";'^" /""(a + eh-), n — 1 : the h and 6 being independent variables, converge uniformly to zero for the rectangle D whose points (h, 0) satisfy 0 an wi such that for every point (A, 6) of D. Let us fix A ; then | ^„ | < e no matter how 6 varies with n. 2° It is necessary. Let h^ be an arbitrary but fixed number in 2l = (0, if*). We have only to show that, from the existence of 1), for h<_\, it follows that i2„(A, ^) = uniformly in the rectangle 2>, defined by 0(«) +... n — 1'. But obviously G(^a, ^) is continuous in Aq, and evidently all its terms are also continuous there. Therefore by 149, 3, *"~^ G[(n-])(^«) ^ uniformly in A„ . (5 n-1! But if we show that |/<">(a + a)|(a)+^/"'+2'(a)+ ... (7 and from 4) (?(»-"(«) = |/<">(a) I + « /<'-!'(«) ■|^|/"'^^'(«) + ••• (8 The comparison of 7), 8) proves 6). Circular and Hyperbolic Functions 183. 1. We have defined the circular functions as the length of certain lines ; from this definition their elementary properties may be deduced as is shown in trigonometry. From this geometric definition we have obtained an arithmeti- cal expression for these functions. In particular ^"^" = rT-37 + 5!-7-:-^- ^^ /y-2 /y4 /y£ cos.= l-- + ---+... (2 valid for every x. As an interesting aud instructive exercise in the use of series we propose now to develop some of the properties of these func- tions purely from their definition as infinite series. Let us call these series respectively S and C. Let us also define tan x = , sec x = , etc. cos X cos X 2. To begin, we observe that both S and C converge absolutely for. every a;, as we have seen. They therefore define continuous one-valued functions for every x. Let us designate them by the usual symbols gj^^^ ^ ^.^sa;. We could just as well denote them by any other symbols, as {x) , fCx). 3. Since s=Q , C=l iovx = 0, we have • n n n i sin = , cos = 1. CIRCULAR AND HYPERBOLIC FUNCTIONS 223 4. Since S involve* only odd powers of x, and only even powers, sin X is an odd, cos a; is an even function. 5. Since *S' and C are power series which converge for every x, they have derivatives of every order. In particular dx 2! 4! 6! dC _ _X 3? _7^ X' _ __o Hi" i sT 5T 77 ■" ~ Hence d sin X d cos x ■ ^n — - — = cos X , — - — = — sin X. (_d dx dx 6. To get the addition theorem, let an index as x, y attached to S, C indicate the variable which occurs in the series. Then f ^r^ XTJ \ f ^u (XytJ ecu \ 'S'xC; = a: - (^- + — j -f- (^- + gj2! + 1T4T j _fx' I ^ «/^ , x^ ^ L ^y^ \ VTI 5!2T 3!4! 6!y ■" V7 ! 5 1 2 : 3 ! 4 ! ^ 1 ! 6 ly Adding, SA +O^S, = x + y-^^[x^+ {^^x-^y + (^^xy-^ 4 y^ + 5! _x±y^_(x + yy (x±by_ _ ~ 1! 3! 5! — "H-V Thus for every x, y sin (x + y') = sin 2; cos y 4- cos x sin y. In the same way we find the addition formula for cos a;. 224 POWER SERIES 7. We can get now the important relation sin^ X + cos^ x = \ (4 directly from the addition theorem. Let us, however, find it by- aid of the series. We have 1 V3! 3!; V5! 3!3! 5!y ._:^fl + ll + lJ_ + J_U... \7! 3!5! 5!3! 7!/ V2! 2!; 1,4! 2!2! 4.7 V6! 4!2!2!'*'4! 6iy V8 ! 6 ! 2 ! 4 ! 4 ! 6 ! 2 ! 8 ly H6nc6 6! Now by I, 96, / \ / \ / \ 1 -©+(?)-(?)+ •■■-»■ '^''"^ *S2 + C2 = sin2 a; + cos2 a; = 1. 8. In 2 we saw sin a;, cos a; were continuous for x; 4) shows that they are. limited and indeed that they lie between ± 1. For the left side of 4) is the sum of two positive numbers and thus neither can be greater than the right side. 9. Let us study the graph of sin a;, cos a:, which we shall call 2 and r, respectively. Since sin a; = 0, -~^ = cos a; = 1, for a; = 0, 2 cuts the a;-axis at under an angle of 45 degrees. CIRCULAR AND HYPERBOLIC FUNCTIONS 225 Similarly we see «/ »= 1 for a; = 0. T crosses the y-axis there and is parallel to the a;-axis. '"" s.Ji-Jt\^^(i-jL\^... \ 2.3/ 5!V 6. 7 J and each parenthesis is positive for < a;^ < 6, sina;>0 for 00 for <2;< V2 = 1.414 ••• ^^°^^ (7=l_^ + ^_^fl_^V— fl "^ 214! 6!V 7.87 10 !V 11-12, cosa;<0 for a: =2. Since D^ cos a; = — sin a; and sin a; > for 0 for x = V2, but < for a; = 2, cos x vanishes once and only once in (V2, 2). This root, uniquely determined, of cos x we denote by - • As a first approximation, we have V2<|<2. From 4) we have sin^ ^= 1- As we saw sina;>0 for a;< V6, we have sin- = + l. Thus sin x increases constantly from to 1 while cos x decreases from 1 to in the interval (0, ^ j= /j. We thus know how sin x, cos X behave in /j. From the addition theorem sin( — -|-a;)= sin ^ cos a: + cos ^ sin a; = cos a;. cosf--|-a;j= cos- cos X — sin ^ sin a; = — sin x. 226 POWER SERIES Knowing how sin a;, cos a; march in /j, these formulae tell us how they march in ^ = f ^, tt j • From the addition theorem, sin QjT + x) = — sin x, cos (ir + x)= — cos x. Knowing how sin x, cos x march in (0, tt), these formulae inform us about their march in (0, 2 tt). The addition theorem now gives sin (a; + 2 tt) = sin x, cos (a; + 2 tt) = cos x. Thus the functions sin x, cos x are periodic and have 2 tt as period. The graph of sin x cos x for negative x is obtained now by recalling that sin x is odd and cos x is even. 10. As a first approximation of tt we found V2 0, sinh a; > since each term in 3) is > 0. The relation 4) shows that cosh x is positive for every x. If a/ > a; > 0, sinh a;' > sinh a;, since each term in 3) is greater for a/ than for x. The same may be seen from 5). THE HYPERGEOMETRIC FUNCTION 229 Evidently from 3), 4^ lim sinh a; = + ao , lini cosh a; = + oo . At x= 0, cosh X has a minimum, and sinh x cuts the a;-axis at 45°. For a; > 0, cosh x > sinh x since The two curves approach each other asymptotically as x= +oo . For the difference of their ordinates is e~' which = as a; = + oo . The addition theorem is easily obtained from that of e". In fact sinh x cosh y = e^ — e"^ ev+e'" 2 2 = |(e^+''+ e^~» — 6-^+" — fi-^-"). imi ar y ^^^^ ^ ^^^^^ ^ _ ^^g^+j, _ gi-i, ^ g-i+j, _ g-x-yy^ Hence sinha;coshy + cosh2;sinhy = ^Qe"^" — e-<^+!'>) = sinh (a; + y). Similarly we find cosh (a; + «/) = cosh a: cosh «/ + sinh a; sinh y. In the same way we may show that cosh^ X — sinh^ x= 1. ITie Hypergeometric Function 185. This function, although known to Wallis, Euler, and the earlier mathematicians, was first studied in detail by Gauss. It may be defined by the following power series in x: F(ia,^,y; a;) = 1 + -— ^a: + ; , 7 ^ I . ry 1.2-7-7+1 a-«+l-« + 2-/3-;8+l./3 + 2 ^ ^ 1.2-8 ■yy+l-y + 2 The numbers «, /8, 7 are called parameters. We observe that a, /8 enter symmetrically, also when a = 1, yS = 7 it reduces to the geometric series. Finally let us note that 7 cannot be zero or a negative integer, for then all the denominators after a certain term = 0. 230 POWER SERIES The convergence of the series F was discussed in 100. The main result obtained there is that F converges absolutely for all I a; I < 1, whatever values the parameters have, excepting of course 7 a negative integer or zero. 186. For special values of the parameters, F reduces to ele- mentary functions in the following cases : 1. If a or ;8 is a negative integer —n,F\s a polynomial of degree n. 2. J-a, 1, 2; -a;)=hog(l + 2;). (1 For TT^iio \ 1 X , x^ a? , F(l, 1, 2, _2;)=1 -- + ---+ ... Also o log(l+a;)=a;(l-| + |-...). The relation 1) is now obvious. Similarly we have J'(l, 1, 2; a;) = ^log(l-a;). X l'ai,f,.^)=f^log[±|. 8. Fi-a,^,^; x-y^l-'^^t + '^-f^:,?- ... =il-xy. 4. 2;J'(^, ^, |, a;''') = arcsin x. 5. ^F{\, 1, |, — a;2) = arctan x. 6. lim F(a, 1, 1, ?^ = e-. (2 \ a/ 1 1 « 1-2.1.2 W a.«+l-a + 2 1.2.3/ For + • 1.2.3 1.2.3 ©" -^-h<^-%-{'--X'-'^t'.^- ■ ^^ THE HYPERGEOMETRIC FUNCTION 231 Let < a < ^. 'Phen is convergent since its argument is numerically < 1. Comparing 3), 4) we see each term of 3) is numerically < the corresponding term of 4) for any \x\ yS. Thus the series 3) considered as a function of a is uniformly convergent in the interval (/3 + oo ) by 136, 2 ; and hereby x may liave any value in (— (r, (r). Applying now 146, 4 to 3) and letting «= +qo, we see 3) goes over into 2). 7. lim xF(a, a,%; - ■^\ = sin x. (5 For Let a; = (? > and a = G^. Then is convergent by 185. We may now reason as in 6. 8. Similarly we may show : + lim ^1 «e, a, _ ; a=+oo \ Z X" , 4^2; = «•'«- lim ^( a, «, -, -T— ^ )= sinh x. a=+oo \ 2 4 «v lim F\ a, a, -, — ■„ ) = cosh x. ' 2 4W 187. Contiguous Functions. Consider two F functions J'(«, /8, 7; a;) , F(a',0',y'; x^. If a differs from a' by unity, these two functions are said to be contiguous. The same holds for /8, and also for 7. Thus to F(^uj3yx^ correspond 6 contiguous functions, -F(a ± 1, /3 ± 1, 7 ± 1 ; a;)- 232 POWER SERIES Between F and two of its contiguous functions exists a linear relation. As the number of such pairs of contiguous functions is 6-5 1-2 = 15, there are 15 such linear relations. Let us find one of them. ^® ^^* „ ^ a -\-l ■ a+ 1 ■■■■ a + n-1- ^ ■ fi + \-- + 71-2 1 . 2 • ••• w • 7 • 7+ 1 ■ ••• 7 + w— 1 Then the coeiificient of a;" in Fi^a^'yx) is in F(a + 1, /3, 7, x) it is (« + «)(/3 + w- !)om 3), 5) let us eliminate F(a, /3+ 1, 7, 2;), getting (a _ 1 _ (^ _ /3 _ l)a;)i^(a, /3, 7, x) + (7 - a)J'(« - 1, /3, 7, a;) + (l-7)(l-a;)J'(«, /3, 7-l,^)=0. (6 In 1) let us replace os by a — 1 and 7 by 7 + 1 ; we get (7 -a+l~)F(ia- 1, A 7 + i, a;) + (a- !)!'(«, /3, 7 + 1, a;) -7l'(«-l, ;8, 7, a;)=0. (a). In 6) let us replace 7 by 7 + 1 ; we get («_l_(^_^)a;)J'(«,/3, 7+1, a:) + (7 + l-a)J'(«-l,/3, 7 + l,a;) -7(l-a:)^(«,A7,a;) = 0. (b) Subtracting (b) from (a), eliminates F(^a — 1, /8, 7 + 1, x) and gives 7(1 - x)F(a^yx') - yF(a - 1, yS, 7, .-c) + iy-/3)xF{a,/3,y+l,x-)=0. (7 From 6), 7) we can eliminate F{a— 1, /3, 7, a:), getting 7?7-l+(« + /3+l-2 7)a;(^(«, ;8, 7, a;) + (7 - a) (7 - /3)a;J'(a, /S, 7 + 1, x) + 7Cl-7)(l-a=)^(«./3, 7-1. ^)=0. (8 In this manner we may proceed, getting the remaining seven. 188. Conjugate Functions. From the relations between con- tiguous functions we see that a linear relation exists between any three functions FCa, A 7, a;) FCa', I3\ 7', ^) ^(«". /8", 7", ^^ whose corresponding parameters differ only by integers. Such functions are called conjugate. 234 POWER SERIES For let p, q, r be any three integers. Consider the functions F^a^'ix'), FQa + 1, /3, 7, x) - F(a +p, /8, 7, x), FQa +p, yS + 1, 7» a;), -F(« +p, ;8 + 2, 7, a;) ••■ Z(a+J», fi+q, 7, a:), J^(«+;?,/3+y,7+1.2:),^(a+p,/3+j,7 + 2,a;)---^(«+j»,/3+9,7+r,a;). We have p + q + r + 1 functions, and any 3 consecutive ones are contiguous. There are thus p+ q + r— 1 linear relations between them. We can thus by elimination get a linear relation between any three of these functions. 189. Derivatives. We have ;;^i 1 ■ 2 • •••M-7- 7+1 • •••7 + w— 1 ^ -g- a-a+1- ■■■a + n. 0.^+1- ...0 + n ^„ ^ 1 • 2 . ••• n + 1 ■ 7 ■ 7 + 1 ■ ••• 7 + n n=0 «/3V « + l....tt+W;g + l-"-/3+W •w4-l'7 + l-'-'7 + w X" ==^^(«+l, y3 + l, 7 + 1, a;). 7 Hence F" («, A 7, :r) = ^ ^' (a + 1, ;8 + 1, 7 + 1, x^) 7 ^ «.«+l.^.^ + l .7.7 + 1 and so on for the higher derivatives. We see they are conjugate functions. 190. Differential Equation for F. Since jF, F' , F" are conju- gate functions, a linear relation exists between them. It is found to be a;(a;- 1)jP" + \{a+ ^ + 1')x- r^]F' + a^F=0. (1 To prove the relation let us find the coefiBcient of 3^ on the left side of 1). We set p _ a.«+l....« + w-1-j8-/g+l----/3 + w-l 1 . 2 . ■•• w • 7 • 7 + 1 - ••• 7 + w— 1 THE HYPERGEOMETRIC FUNCTION 235 The coefficient of a;*in a?F" is n(»i-l)P„, in — xP" it is 7 + w in <^a + ^ + V)xF' it is w(a+/3 + l)P„, in — 7^' it is -" ^i+^ ^"' in a^F it is «y8P„. Adding all these gives the coefficient of a;" in the left side of 1). We find it is 0. 191. Expression of F^a^yx") as an Integral. We show that for | a; | < 1, B<'^,y-^')-F(a^yx)= Cuf'-^ (1 - uy-^-^ (1 - xuy du (1 where B(^p, q) is the Beta function of I, 692, B{p, q)= I mp-1 (1 — m)«-i du. For by the Binomial Theorem (l-xu)-''=^l+-xu+—-~xht? + iT- 2 • 3 "" for I a;M I < 1. Hence J'= f V-i (1 - m)t-P-^ (1 - xuydu Jo = r M^-^ (1 - w)^-^-^<^M + ^ J^ V (1 - m)V-P-1 «•« + ! . ^ rV+i(l - m)v-p-1c?m + ••• 1-2 -'o = 5 (y8, 7 - /3) + "a^-S (yS + 1, 7 - ^) + ^^^^^^(/3 + 2,7-/3)+- (2 236 POWER SERIES Now from I, 692, 10) Hence 7+1 7-7+1 etc. Putting these values in 2) we get 1). 192. Value of F (a, A 7. a;) for x=l. We saw that the F series converges absolutely for a; = 1 if « + ;8 — 7 < 0. The value of F when a; = 1 is particularly in- teresting. As it is now a function of a, ^, y only, we may denote it by -F(«, y3, 7). The relation between this function and the F function may be established, as Gauss showed, by means of 187, 8) ■V17 • 7i7- 1 +(« + /3+ 1 - 27)a;| ^(a;87a:) + (7 - «) (7 - l3')xF(a, /3,y+l, x} + 7(l-7)(l-a;)^(«,^,7-l,a;) = 0. (1 Assuming that „ + ^ _ ^ < q, (2 we see that the first and second terms are convergent for a; = 1 ; but we cannot say this in general for the third, as it is necessary for this that a + /8 — (7 — 1) < 0. We can, however, show that ilim(l-x)^(a, /3, 7- 1, a;) = 0, (3 x=l supposing 2) to hold. For if | a; | < 1, J'(a, y8, 7 - 1> a;) == «(, + a^x + a^x^ + ••■ (4 Now by 100, this series also converges for a; = — 1. Thus lim a„ = 0. ' (5 From 4) we have (1 - x')F(a, ^,y - l,z) = aQ + (a^ - aQ)x + (a^ -aj)3?-\ Let the series on the right be denoted by Q-(x). As (r„+i(l) = a„, we see Cr(V) is a convergent series, by 5), whose sum is 0. But then by 147, 6, (7 (x) is continuous at x = 1. Hence ^^ j.^^ (y{x}=aO) = 0, THE HYPERGEOMETRIC FUNCTION 237 and this establishes 3^. Thus passing to the limit x= 1 in 1) gives 7 (« + /S - 7) ^(«, A 7) + (7 - «) (7 - ^)F(ia, y8, 7 + 1) = 0, FCa, A 7) = ^'y-"K7-^) j,(„, ^, ^ + 1). 7(7 -«-^) Replacing 7 by 7 + 1, this gives ^(«, A 7 + 1) = C7 + l-«K7 + l-g ^(„, ^, ^ + 2), (7 + 1) (7 + 1 - « - ^) etc. Thus in general ^(«, yS, 7) = (7-«)(7 + ^-«) — C7+w-l-«)-(7-"(7+w-l-ja) 7(7+l)—(7+w-l)(7-«-/8)C7-«-/S+l)---(7-«-/3 + «-l) •J'(a,/3,7 + M). Gauss sets now U(n x-) = ^^^ '^ ' ^ (a;+l)(a:+2)...(:c + n)' Hence the above relation becomes n (n, 7 — a — 1) II (w, 7 — yS — 1) ^^^^ lim J-Ca, /3, 7 + «) = 1. (7 n=co For the series j'(«,A7)=i+f^+ '^-;t'-^-ft' +- (8 1.7 l-z-7-7 + 1 converges absolutely when 2) holds. Hence ■^ 1. (? 1-2- a-a + 1 ^ is convergent. Now each term in 8) is numerically < the corre- sponding term in 9) for any y >G-. Hence 8) converges uni- formly about the point 7 = + 00. We may therefore apply 146, 4. As each term of 8) has the limit as 7 =+ 00, the relation 7) is established. 238 POWER SERIES We shall show in the next chapter that lim n (w, X) n=co exists for all x different from a negative integer. Gauss denotes it by n (a;) ; as we shall see, r(a;) = n(a;- 1) , for a; > 0. Letting m = co, 6) gives n (7 — a — 1) n (7 — /8 — 1) We must of course suppose that 7, 7 — «, 7 - A 7 — a - ^, are not negative integers or zero, as otherwise the corresponding n or ^ function are not defined. Bessel Functions 193. 1. The infinite series j-„(a;)=a;»i(-i)'— — ^1— — »i=o,i, 2... (1 ,=0 2"+^« ! (w + s) ! converges for every x. For the ratio of two successive terms of the adjoint series is r^p 22(8 +l)(n + «+•!) which = as « = 00 for any given x. The series 1) thus define functions of x which are everywhere continuous. They are called Bessel functions of order w = 0, 1, 2.-. In particular we have "^ 2 • 2 22 . 42 22 . 42 . 62 ^ JJx) = ^-J^Jr + •■• (3 ^^ ^ 2 22 . 4 22 . 42 . 6 22 . 42 . 62 . 8 ^ Since 1) is a power series, we may differentiate it termwise and BESSEL FUNCTIONS 239 2. The following llfiear relation exists between three consecutive Bessel functions : 2 n "^n+i(a;) = — J„(x) - Jr^iQc) w > 0. (5 For J a;"-i g. . j^. a~^'+"-i "-^ 2"-i(m-1)!'^^A ^ 2»+2'-is!(7i-l + s)!' ^ «> ™2s+n— 1 J" = — S<'— 1^' — f1 "+^ »=i^ ^ 2«+2.-i(g_i)!(^+s)! '^' Hence = ^""' lir IV '^"''""M 1 1 ) 2"-i(w-l)! 1^ ^ 2''+2'-Ms!(^w-l + s)! (s-l)!(/i+s)! 1 /j.n— 1 <» ™2s+n— 1 * +„2(-l)'- ^ 2"-i(w - 1) ! 1 2"+2'-i8 ! (w + «) ! w. _2_w a; 3. We show next that 2JI(a;) = J-„_i(a;)-J-„^i(a:) w>0. (8 For subtracting 7) from 6) gives T -T - ^""^ ir l^' ^'"^""^ n+2s "-1 "■'I ~ 2»-i(w - 1) ! "*" 1 '^ ^ 2«+2.-i ■ s ! (w + s) ! ^ " ._ ^., (:w + 2s)a:^+"-i ^<. ^ 2»+2»-is!(w + s)! From 8) we get, on replacing J^+i by its value as given by 5) : JXx-) = - ^ J-„(:c) + J-„_i(a;), « > 0. (9 X From 5) we also get Jl{x)='^Jjix)-J^^^{x) n>0. (10 4. The Bessel function J"™ satisfies the following linear homo- geneous differential equation of the 2° order : ^: + -^^+fl-^Vn = 0. (11 X \ X J 240 POWER SERIES This may be shown by direct differentiation of 1) or more sim- ply thus : Differentiating 9) gives ' n — Equation 10) gives j':='-^j.--Ji+JL-v (12 Replacing here J"„_i by its value as given by 9), we get Putting this in 12) gives 11). 5. e'^ = iMV„(a;) (13 —00 for any a;, and for m ^ 0. For e 2 = e' e 2 K — I 1 4. 5?^ J- ^^ 4. ^''^ 4- ""1 2 22.2I23-3: 2m 22.2Im2 Now for any x and for any m =^ 0, the series in the braces are absolutely convergent. Their product may therefore be written in the form l._±, (xY J_ _ N V 22'*"\2y2!2! '") + Kl - 2W+ 3721(1)' -••■) + + «"( ) (-!)»«-»( ) + = J^ix-) + uJ^ix) + v?J^(x) + ••• J.jx-) , J-/a:) BESSEL FUNCTIONS 241 194. 1. Uxpressiornpf J„(x) as an Integral. 2»V'7r p/ ^w+ l \Jo cos M= 2(— 1)' ^ ' (2s)! Hence <» / _ i ^. cos (x cos ^) = S ^^ Y 2^ cos^ d> C2s)! and thus cos (a; cos (^) sin^" ^ = ^ ^^ — y *^' '^o^^' ^ s™^" ^• As this series converges uniformly in (0, tt) for any value of x, we may integrate termwise, getting I cds (x cos -, ) -^ We shall show in 225, 6, that r^2s^\ l-3-5--2s-l ^ V 2 y 2* Thus the last series above V 2 yA(2s)! 2'(w + «)! Thus - j cos(a;cos<^)8in''" 2»v^r^2^^J. 4'22'+''g!(« + s)! "^ ^ CHAPTER VII INFINITE PRODUCTS 195. 1. Let 5a.,...ijJ be an infinite sequence of numbers, the indices t = (ti"-t,) ranging over a lattice system 8 in s-way space. The symbol p ^ ^^ ^ ^ ^^ ^^ c ' ' s is called an infinite product. The numbers a^ are its f acton. Let P^ denote the product of all the factors in the rectangular cell ^"- ^^ liraP^ (2 "(»• is finite or definitely infinite, we call it the value of P. It is customary to represent a product and its value by the same letter when no ambiguity will arise. When the limit 2) is finite and :^ or when one of the factors = 0, we say P is convergent, otherwise P is divergent. We shall denote by P^ the product obtained by setting all the factors a^ = 1, whose indices i lie in the cell R^. We call this the co-product of P^. The products most often occurring in practice are of the type oo P = a^-a^-a^- ••• = Tla„. (3 The factor P^ is here replaced by and the co-product P^ by Pm = "^ro+l ■ ''m+2 ■ <^m+3 ' ' ' ' Another type is +„ P=na„. . (4 The products 3), 4) are simple, the product 1) is s-tuple. The products 3), 4) maly be called one-way and two-way simple products when necessary to distinguish them. 242 GENEKAL THEORY • 243 2. Let p==i.i.|.|.|.... Obviously the product P = 0, as n Hence P = 0, although no factor is zero. Such products are called zero products. Now we saw in I, 77 that the product of a finite number of factors cannot vanish unless one of its factors vanishes. For this reason zero products hold an exceptional posi- tion and will not be considered in this work. We therefore have classed them among the divergent products. In the following theorems relative to convergence, we shall suppose, for simplicity, that there are no zero factors. 196. 1. For P= Ilai, ..., to converge it is necessary that each P^ is convergent. If one of these P^ converges, P is donvergent and P = P • P The proof is obvious. 2. If the simple product P = a^- a^- a^---is convergent, its fac- tors finally remain positive. For, when P is convergent, | P„ | > some positive number, for n > some m. If now the factors after a„ were not all positive, P„ and P^ could have opposite signs v>n, however large n is taken. Thus P„ has no limit. 197. 1. To investigate the convergence or divergence of an infinite product P = Ila.^...,, when a. > 0, it is often convenient to consider the series L = l\og a ..,. = '21^...,., C called the associate logarithmic series. Its iiuportance in this con- nection is due to the following theorem : The infinite product P with positive factors and the infinite series L converge or diverge simultaneously. When convergent, P = e^, L = log P. For ldgP^ = i^, (1 P^ = e^M. (2 244 " INFINITE PRODUCTS If P is convergent, P^ converges to a finite limit ^ 0. Hence L^ is convergent by 1). If L^ is convergent, P^ converges to a finite limit =!tO by 2). 2. Example 1. is convergent for every x. For, however large | a; | is taken and then fixed, we can take m so large that „ l+5>0 n>m. n Instead of P we may therefore consider P„. \ nj n n^ Hence L^=Xm^x^--„ m+l n' which is convergent. The product P occurs in the expression of sin x as an infinite product. Let us now consider the product m+l But by I, 413 Q=Xl{l + ^e'n w = ± 1, ±2, The associate logarithmic series i is a two-way simple series. We may break it into two parts L', L", the first extended over positive n, the second over negative n. We may now reason on these as we did on the series 3), and conclude that Q converges for every x. 3. Example 2. / i\i x^ 1+5 n is convergent for any x different from 0, - 1, - 2, - 3, - GENERAL THEORY 245 For let p be taken«Bo large that | a; | < p. We show that the co-product . ...^ (1+-) p+i 1 + - n converges for this x. The corresponding logarithmic series is 00. i = 2 L logf 1 + 1) - logf 1 + -") I = tf?-log(l + --)l-.Sfl-log(l + l)l. p+1 [n \ nJ } p+i in \ nJ . As each of the series on the right converges, so does L. Hence G converges for this value of x. 198. 1. When the associate logarithmic series Zr=2]oga.,...., , a, > is convergent, ^.^ ^^^ ^^^^^ ^ q, by 121, l, |il=°o and therefore ,. i hm a =1. For this reason it is often convenient to write the factors a.j....^ of an infinite product P in the form 1 + J........ When P is written in the form ^ „ ,^ , p = n(i + 6.,...o, we shall say it is written in its normal form. The series 2J,..., = 26. we shall call the associate normal series of P- 2. The infinite product P=U(l+a,^...J , a. >-l, and its associate normal series A = 2a,, ....,, converge or diverge simultaneously. 246 INFINITE PRODUCTS For P and 7- v 1 /-1 . s i = S log (1 + a J converge or diverge simultaneously by 197. But A and L con- verge or diverge simultaneously by 123, 4. 3. If the simple product P = a^- a^- a^--- is convergent, a„i 1. For by 196, 2 the factors a„ finally become > 0, say for n > m. Hence by 197, 1 the series 2 log a„ a„ > is convergent. Hence log a„ = 0. .-. a„ = 1. 199. Let ^A, < R^^ < ••• \\\=oo bea sequence of rectangular cells. Then if P is convergent. For P is a telescopic series and V Pk+1 = P^ + S(A„+i - -Pa„)- 200. 1. Let P=n(l + «.,... J. We call ^=n(l + a,, ) , a. = | a. | the adjoint of P, and write ^ = Adj P. 2. P converges, if its adjoint is convergent. We show that € > 0, \, \P^-P,\<€ At, V > X. Since ^ is convergent, is also convergent by 199. Hence 0<«|5,-^^ by 196, 2. Then ^=eSioga„ by 197, 1, = e^ioB"'!-'. by 124, 3, = P by 197, 1. 3. If the associate simple product Q is absolutely convergent, so is p. For let 73 TT/-1 , s P = n(l4-S.....), Q=U(l+aJ. Since Q is absolutely convergent, n(l + a„) , «„=|a„| is convergent. Hence 11(1 + a.^.... ) is convergent by 2. 4. Let P = n(l + a.j...,^") he absolutely convergent. Then each associate simple product Q= 11(1 + a„) is absolutely convergent and P=Q. For since P is absolutely convergent, converges by 200, 4. But then by 124, 6 is convergent. Hence Q is absolutely convergent. 5. IfP— Haij....^ is absolutely convergent, the factors a,...i >0 if they lie outside of some rectangular cell R^. For since P converges absolutely, any one of its simple associ- ate products ^=na„ converges. But thena„>0 for w>?m, by 198, 3. Thus a.j..., > if t lies outside of some R^. 6. From 5 it follows that in demonstrations regarding abso- lutely convergent products, we may take all the factors > 0, without loss of generality. GENERAL THEORY 249 For • P=P^.P^; and all the factors of P^ are > 0, if ft is sufficiently large. This we shall feel at liberty to do, without further remark. 7. A=U(l + a^^...J a, >0 """^ i=Slog(l + a.,...,) converge or diverge simultaneously. For if A is convergent, is convergent by 200, 4. But then L is convergent by 123, 4. The converse follows similarly. 202. 1. As in 124, lo we may form from a given wi-tuple as infinite number of conjugate w-tuple products Jl. ■'n where a^ = bj if i a.ndj are corresponding lattice points in the two systems. We have now : 2. If A is absolutely convergent, so is B, and A = B. For by 201, 6, without loss of generality, we may take all the factors > 0. Th en A 21og«j . = e ' = B. ■■Jn 3. Let A = na,.. "' 'm be an absolutely convergent m-tuple product. Le\ J B = m,^. -K be any p-tuple product formed of a part of or all the factors of A. Then B is absolutely convergent. 250 INFINITE PRODUCTS For 2 log a. is convergent. Hence 2 log yS^. is. Arithmetical Operations 203. Absolutely convergent products are commutative, and con- versely. be absolutely convergent. Then its associate simple product 2l=na„ is absolutely convergent and J. = 21, by 201, 4. Let us now re- arrange the factors of A, getting the product B. To it corre- sponds a simple associate series 33 and 5 = 33. But 21 = 33 since 21 is absolutely convergent. Hence A = B. Conversely, let A be commutative. Then all the factors «i,....^ finally become > 0. For if not, let i^i < iij < ••• = °° 0- be a sequence of rectangular cells such that any point of 9i„ lies in some cell. We may arrange the factors a, such that the partial products corresponding to 1), ■"1 ' -^2 ' ■"■& '" have opposite signs alternately. Then A is not convergent, which ' is a contradiction. We may therefore assume all the a's > 0. Then . sioga^.... A = e " remains unaltered however the factors on the left are rearranged. Hence .^ n 2 log a,, ....^ is commutative and therefore absolutely convergent by 124, 8. Hence the associate simple series 2l = 21oga„ = 21og(l-|-6„) is absolutely convergent by 124, 5. Hence 2/3„ is convergent and therefore A is absolutely convergent. ARITHMETICAL OPERATIONS 251 204. 1, Let •^ TT he absolutely convergent. Then the s-tuple iterated product ^ = nn...na, , is absolutely convergent and A = B where tj ••• ti is a permutation of tj, tj ••• tj. For by 202, 3 all the products of the type Ha. . Hffl , are absolutely convergent, and by I, 324 n = nn. •■a-l'a ^a-1 ^« Similarly the products of the type n 'j— l''8-2^a— B are absolutely convergent and hence n= n n n. In this way we continue till we reach A and B. 2. We may obviously generalize 1 as follows : ^'* A = na.... be absolutely convergent. Let us establish a 1 to 1 correspondence betieeen the lattice system 8 over which t = (tj • • • tj) ranges, and the lattice system 2Ji over which 3 '^\3\\)\^ '"^Zl^TS '"JrlJri '"Jrp) ranges. Then the p-tuple iterated product 5 = n • n • ••■ Ha,. ,. ... 1 2 r " " " is absolutely convergent, and A = B. 252 INFINITE PRODUCTS 3. An important special case of 2 is the following : -^^* A = Ua„ , w=l, 2, - converge absolutely. Let us throw the a„ into the rectangular array '^l ' *22 ■ ■ ■ Then B, = Ua,„ , B, = na^„... converge absolutely, and 4. The convergent infinite product P = (l + ai)(H-a2)- is associative. For let ^ ^ • mi Ciu. — Afi • Ji^. Now by hypothesis A^ = A, B^^ = B us fi = co. He'^^e O.^A.B. To show that O is absolutely convergent when A, B are, let us write a,= 1 + a, , b,= 1 + b, and set | Oi | = a^ , | b. | = ^,. Since A, B converge absolutely, 21og(l + «0 , Slog(l + A) are convergent. Hence 2 llog (1 + «0 + log (1 + A) S = 2 log (1 + a.) (1 + A) is absolutely convergent. Hence is absolutely convergent by 201, 7. 206. Example. The following infinite products occur in the theory of elliptic functions : §1 = 0(1 + 92") (?2 = n (1 + J^n-l) W=l, 2, ... 0 Je uniformly convergent and limited. Then J I, I, ig uniformly convergent in 2J. For Now L^^L uniformly . Hence by 144, 1, F is uniformly con- vergent. 208. If the adjoint of ii uniformly convergent in 21 (^finite or infinite^, F is uniformly convergent. For if the adjoint product, ^=n(i + 0,...j, is uniformly convergent, we have for any x m UNIFORM CONVERGENCE 255 But as already notidfed in 200, 2, 1) \P,-P^\<\%-'^.\. Hence F is uniformly convergent. 209. The product is uniformly convergent in the limited or unlimited domain 21, if <& = 2<^.,...,.(a;i-x„) , 0 (1 converge in 21= (a, a + 8). Then logJ'=i = 21og/.. (2 If we can differentiate this series termwise in 21 we have £log^=SA (8 Thus to each infinite product 1) of this kind corresponds an infi- nite series 3). Conditions for termwise differentiation of the series 2) are given in 153, 155, 156. Other conditions will be given in Chapter XVI. 2. Example. Let us consider the infinite product eix') = 2 ^g sin -n-xflO- - 2 ^" cos 2 ,r2; + q*") (1 1 which occurs in the elliptic functions. Let us set 1 — M„= 1 - 2 5^ cos 2 Tra; + ^". Then | m„ | <2 | y |2»+ | j|<". Thus if I y I < 1, the product 1) is absolutely convergent for any x. It is uniformly convergent for any x and for \q\ sin' rir ir'^r'^ 4 Tr^r*' n nr and hence for an wij so large that - — - < 1, we have, m-t log But the series sin^- sm' n -" some v the first term on the right of 8) is < e/3, and thus 7) holds. 260 INFINITE PRODUCTS 2. In algebra we learn that every polynomial Uq + UjX + a^ + • • • + a„2;" can be written as a product «n(a;-ai)(a;-a2) ••• {x-a^), where Wj, "^ • • • are its roots. Now ''"" = T!-3! + 5!-- ^^ is the limit of a polynomial, viz. the first n terms of 9). It is natural to ask, Can we not express sin x as the limit of a product which vanishes at the zeros of sin x ? That this can be done we have just shown in 1. 3. If we set x = 7r/2 in 5), it gives, ,_7r /. INtt (2r-l)(2r + l) 2 V 4W 2 2r-2r Hence ^^ 2r-2r _ 2 • 2 . 4 • 4 • 6 • 6 ••• ,.q 2 (2r-l)(2r + l) 1 . 3 • 3 • 6 • 5 ■ 7 .••' ^ a formula due to Wallis. 4. From 5) we can get another expression for sin x, viz. : sin a; = a;n fl - — V** r=±l, ±2, ■•• (11 \ rirj For the right side is convergent by 197, 2. If now we group the factors in pairs, we have (i_^)/^Ci+^y^=i-^. This shows that the products in 5) and 11) are equal. 5. From 5) or 11) we have sina; = limP„(2;) = lim2;ri'^^i-?^ (12 n=oo s=-n Stt where the dash indicates that « = is excluded. THE CIRCULAR FUNCTIONS 261 214. We now show that cosa; = nfl — V (1 iV (2n-l)27rV '^ To this end we use the relation sin 2 a; = 2 sin x cos x. Hence 1 2 a; \ r/W cos a; = K • 7— 2 ^n(^i ^) Ufl i^Yl- ^^ ) rM \ (2w)27rVV (2m-l)27rV (\ _ -^A \ (2 m- 1) VV from which 1) is immediate. From 1) we have, as in 213, 4, cosa;=nfl- ,„ ^^, V(2n-i). M = 0, ±1, ±2, ... (2 V (2w— l)7r>/ 215. From the expression of sin a;, cos a; as infinite products, their periodicity is readily shown. Thus from 213, 12) sina; = ]im Pn(x)- n=ao P„(a;) a; — WTT Hence ^^^ p^^^ + ^) = - lim P„(a;), °^ sin (a; + tt) = — sin a;. ^*^°^® sin(a: + 2 tt) = sin a; and thus sin x admits the period 2 tt. 216. 1 . Infinite Series for tan x, cosec a;, e . = i log (l - -^^^^^^\ (4 valid for any x such that cos a; ^ 0. If we differentiate 3), 4) we get cotr.= i+2j— ^, (5 l' X^— 8 V^ l-2-J"^^ valid as in 3), 4). Remark. The relations 5), 6) exhibit cot a;, tana; as a series of rational functions whose poles are precisely the poles of the given functions. They are analogous to the representation in algebra of a fraction as the sum of partial fractions. 2. To get developments of sec x, cosec x, we observe that cosec X = tan J a; + cot x. Hence coseca;=2|; 1^ a:^ +^-^|^V^ = l+V i^ 2T ^ a; 7.(2 8 - 1) V - a;2 ^ sV _ 3^2 X "Y « V — a;^ valid for x4=± sir. THE CIRCULAR FUNCTIONS 263 3. To get sec «, we observe that cosec i — — x\= sec x. Now cosec Hence X 1 {Sir — X siT + x) ■^-x ' |«7r-- + a: Stt + 1 - a; Let us regroup the terms of S, forming the series if-^ f+4 W- x+4 ^^ |^„-2'J = -.^ 1 =0, -TT — X 2 we see that T is convergent and = ^S". Thus sec a; valid for all x such that cos x=^0. 217. As an exercise let us show the periodicity of cot x from 216, 5). We have n 1 cot X = lim F^Cx} = lim V x =^ stt. n=« -^ a; + Stt ^ X + Sw No^ !'„(. + ^) = J'„(.) + a;+(w+l)7r a; — wtt Letting w = oo we see that lim F„(x + 7r) = lim F^(x) and hence cot (^x+Tr')= cot a;. 264 INFINITE PRODUCTS 218. Development of log sin x, tan x, etc., in power series. From 216, 1) sm X x^ \ If we give to its limiting value 1 as a; = 0, the relation 1) X holds for I a; I < TT. Now for 1 a; I < TT 1 (^ ^ \ 2^ ^ 1 a;* Thus 1 sin X x^ , 1 x^ , 1 afi , X it'' Z TT* O 77° 22,r2 22*7r* 326,r« x^ 1 x^ 1 1^ 32,r2 2 3*7r* 3 36,r6 provided we sum this double series by rows. But since the series is a positive term series, we may sum by columns, by 129, 2. Doing this we get X TT^ 77* 77° where ^ ^ ^ ^ " 1" 2" 3» 4" '" The relation 2) is valid for | 2; | < 77. In a similar manner we find -logcosx=G,^+ia,^ + ia,^+... (3 77 77 77 valid for | a; | < — • Here 1" 3" 5" THE CIRCULAR FUNCTIONS 265 The terms of (r„ ar« a part of 5"„. Obviously On 1 These coefficients put in 3) give -lQgcosa;=(22-l)ir2^+l(2*-l)^44+K2«-l)^6^+-(4 valid for I a; I < ^- If we differentiate 4) and 2), we get tan a;= 2(22- 1)524+ 2(2*- l)^,^ + 2(2«-l)^54+ -(5 valid for | a; | < — ; Li cotx=\-2H,^-2H,^^-2Hfi-^- ... (6 a; 71^ TT* tt" valid for < | a; | < tt. Comparing 5) with the development of tan x given 165, 3) gives 2 12^02^S?2^ ft ft 9 1 1 9! ^4-r4+2'*'^3"*'^ ■■■ ~90"30' "IT"^' TT ^ ^,1^1 + 1+... =^ 1 .2i^=«. 6 16^26 36 945 42 6! ^6! 8 18 "^28 38 9450 30 81 ^8! Let us set 2^"-^^" (2«)! Then 5) gives ^2" = ^n ..^ , -Szn-r (8 tan.= ^^q-'^ -"^-+^^^Tr^^-^+^^^^^-^^-^^ valid for \x\<~- The coefficients 5^, -Bg ••• are called Ber- nouillian numbers. From 7) we see 266 INFINITE PRODUCTS From 6), 8) we get cotan x-^=- t tI^^^-i^^'""' (10 valid for < | a; | < tt. 219. Mecursion formula for the Bernouillian Numbers. If we set f(x) = tan x, we have by Taylor's development f(x) = xf'iO) + a^-^^ + x^-^ + ... where y(2»-i)(0) ^ 2(2^"- 1)E,^ ^ 2'"(2'" - 1) „ ,. (2w-l)! TT^" C2w)! '""^ ^ Now by I, 408, /C2n-i)(0)_ ^2 n- iy2„-3)(0)C^2 « - iy(2„-5,(0)_ •.•=(- l)"-i.(2 From 1), 2) we get 22n-l(22» - 1) p /'2W - IN 2'''-3(-22n-2 _ 1^ H'v T''"?-2"'' ^-'---<-^-- <■' We have already found B^, B^, B^, B^ ; it is now easy to find successively : ^9 = ^6 -Bn = MTT ^3 = 1 A5=¥^/ Thus to calculate 5j, we have from 3) 29(2^0 - 1) D 9-8 2^(28 - 1) J_ 9-8-7-6 2^(28 - 1) J_ 5 * 1-2 4 "30 1.2. 3. 4 3 42 _9,^^7 23(2^-l).jL ,_ 1^^ 1.2.3 2 30 ^ ^6 Thus ^« = 512.^23 !1 - 9 + 168 - 2016 + 9792| 5 . 7936 5 512 . 1023 66 THE B AND T FUNCTIONS 267 !fhe B and T Functions 220. In Volume I we defined the B and F functions by means of integrals: Xoo which converge only when m, t; > 0. Under this condition we saw We propose to show that r(M) can be developed in the infinite product / jv„ M 1 j^ M This product converges, as we saw, 197, 3, for any m ^ 0, — 1, — 2, ••• From 201, 7 and 207 it is obvious that G converges abso- lutely and uniformly at any point u different from these singular points. Thus the expression 4) has a wider domain of definition than that of 2). Since 6r = T, as we said, for m> 0, we shall ex- tend the definition of the F function in accordance with 4), for negative u. It frequently happens that a function /(a;) can be represented hj different analytic expressions whose domains of convergence are different. For example, we saw 218, 9), that tan x can be de- veloped in a power series tan X = '• ^ '- B^x -\ ^— — '-B^3? ^ valid for | a; | < ^ . On the other hand, X _^ ,^ __ n IT 5l ■" sin a; tan X = 5 -J = I _ ar' a:* _ cos a; 2l"''4] 268 INFINITE PRODUCTS and „ ~ tanx = 22 , by 216, 6) are analytic expressions valid for every x for which the function tan X is defined. 221. 1. Before showing that Gr and F have the same values for M > 0, let us develop some of the properties of the product 6r given in 220, 4). In the first place, we have, by 210: The function G-{n) is continuous, except at the points m = 0, — 1, -2,... Since the factors of 4) are all positive for m > 0, we see that (?(m) is positive for w > 0. 2. In the vicinity of the point x = — m, m = 0, 1, ••• x + m where ff(u) is continuous near this point, and does not vanish at this point. For f -1 X „ (1 + i) a(u) = - — "^HQuy 1 + ^ m where M is the infinite product Cr with one factor left out. As we may reason on H as we did on Q-, we see H converges at the point x = — m. Hence H^ at this point. But H also converges uni- formly about this point ; hence H is continuous about it. '''• (? = liml- ^■^^■■ 0, a" = e" '"^ ". Hence «log(l+^) G = -Ii- No-W- and 1 1 M 270 INFINITE PRODUCTS are convergent. Hence „ Tie '^ " »-' — 7 — r^ Mn(i + -V» \ nj from which 1 ) follows at once, using 223. 225. Further Properties of Gr. 1. a(u+i)=ua(iu). (1 Let us use the product "^^ u (M + 1) ...(„ + „_!) employed in 222. Then P„(«+1) = ?^!^^zl(^. (2 u +n As = M as n = CX3 we get 1) from 2) at once on passing to the limit. 2. G^(m+w)=m(m+1) •••(« + %- !)(?(«). (3 This follows from 1) by repeated applications. 3. G^(w)=l-2- ...w-l=(w-l)! (4 where w is a positive integer. 4. a(u-)a(l-u-) = -^ — (5 sin iru ^°^ ff (1 - w) = - uGC- m) by 1, , by 224, 1). nfi-^V n. Hence -. (?(m)G^(1-M)=; U(l + '^y-^.(l-'^]e K-S)' THE B AND T FUNCTIONS 271 We now use 213, ^. Let us note that by virtue of 1, 2 the value of 6r is known for all M > 0, when it is known in the interval (0, 1). By virtue of 5) Cr is known for m < when its value is known for m > 0. Moreover the relation 5) shows the value of G- is known in Q, 1) when its value is known in (0, ^}. As a result of this we see Gr is known when its values in the interval (0, |^) are known ; or indeed in any interval of length l. Gauss has given a table of log €r(u') for l when w > 0, by 221. 6. ^(2n_^l). 1.3.5^.2n-l .^ ^, where w. is a positive integer. Si.il„rf, c(^y^a(^-i^}eu,. aC^y-^.'-^-l i 226. Expressions for log (r(M), and its Derivatives. From 224, 1) we have for m > 0, Differentiating, we get That this step is permissible follows from 155, 1. 272 INFINITE PRODUCTS We may write 2) i'=-c+yf- — ^}- (3 "7 U u + n—l) ^ That the relations 2), 3) hold for any M=?fc 0, — 1, — 2 ■•• follows by reasoning similar to that employed in 216. In general we have i. = (_iy(._l)!5__A__ , r>l. (4 In particular, i'(l) =-0. (5 i^Cl) = (- lX(r- 1) ! 2;^= (-l/Cr- 1)! 5;. 227. Development of log (3'(m) in a Power Series. If Taylor's development is valid about the point u=l, we have log acu} = i(w) = ici) + ^ i'ci) + (^^^'i"(i) + ... ; or using 226, 5), and setting m = 1 + a;, log (7(1+ x) = - Ci + I; ^Ili>^ 2r„a;». (1 71=2 ** We show now Mis relation is valid for — J < a; < 1, by proving that si converges to 0, as « = 00 , For, if 0-llogl±|^^og-^-ii(^,„.,-l)£^ (2 valid in 21. This series converges rapidly for 00. We have for 0<_u<_l, r(M + n)= je-^x''+''-'^dx * Math. Annalen, vol. 31, p. 456. 274 INFINITE PRODUCTS Now for any x in the interval (0, n), x'^Kn'^ , x''>_xrC'-'^ since m > and m — 1 < 0. Also for any x in the interval (w, oo ) Hence i"-ir e-^a;"c?a:4-w"J e-'x''-'^dx 1 ^ — r + • • • + - — -— , for any m n+1 (w + 1) ••• (w + m) -^ mn'" m m (w + 1) ... (w + TO) ~ A _^ 1\ __ /j _^ m\^/j_|_m\'»' \ wy \ nj \ n) Let us take ^ w > TO^ or — < — . n m Then m m i) Yi + i^ Since wj may be taken large at pleasure, lim v„= ca and hence ,. hm q„ = 0. Thus „. ^ - hm ^ '^ 7" / = 1 , 0 0. "^^ r(M + w)=M(M + l) ... (M + w-l)r(M), we have t^^ , ^ r(u-) = T(u+n) m(w+1) ••• (m + m— 1) Hence using 1), ^ (n-l)\n- . r(^ + n) m(m4-1) ••• (M + n- 1) ' (m-1)!w"' Letting m= oo, we get T(u)= G(u} for any m>0, making use of 1) and 222, 1). 2. Having extended the definition of r(M) to negative values of M, we may now take the relation as a definition of the B function. This definition will be in accordance with 220, 1) for m, v > 0, and will define B for negative M, V when the right side of 2) has a value. CHAPTER VIII AGGREGATES Uquivaie^ce 229. 1. Up to the present the aggregates we have dealt with have been point aggregates. We now consider aggregates in general. Any collection of well-determined objects, distinguish- able one from another, and thought of as a whole, may be called an aggregate or set. Thus the class of prime numbers, the class of integrable func- tions, the inhabitants of the United States, are aggregates. Some of the definitions given for point aggregates apply obvi- ously to aggregates in general, and we shall therefore not repeat them here, as it is only necessary to replace the term point by object or element. As in point sets, 21 = shall mean that 21 embraces no elements. Let 21, S be two aggregates such that each element a of 21 is associated with some one element b of SS, and conversely. We say that 21 is equivalent to SS and write 21 ~«. We also say 21 and SQ are in one to one correspondence or are in uniform correspondence. To indicate that a is associated with b in this correspondence we write a '~ J. 2. if 21 ~ S3 awi « ~ S, then 21 ~ S- For let a ~ 6, 6 ~ c. Then we can set 21, S in uniform corre- spondence by setting a -^ e. B. Let 2l = S-i-S-f-S)4- - A = B + C+D+ ••• // » ~ 5, (5 ~ C\ •••, then 21 ~ ^. 276 EQUIVALENCE 277 For we can associate the elements of 21 with those of A by keeping precisely the correspondence which exists between the elements of © and B, of S and (7, etc. Example 1. SI = 1, 2, 3, ••• If we set a„~ M, 31 and S will stand in 1, 1 correspondence. Example 2. 31 = 1, 2, 3, 4, ••• « = 2, 4, 6, 8, - If we set w of St in correspondence with 2 w of 48, SI and SQ will be in uniform correspondence. We note that S is a part of 31 ; we have thus this result : An infinite aggregate may be put in uniform correspondence with a partial aggregate of itself. This is obviously impossible if 31 is finite. Example S. 31 = 1, 2, 3, 4, ••• SB = 101, 102, 103, 104, ... If we set n ~ 10", we establish a uniform correspondence be- tween 31 and 48. We note again that 31 — 48 although 31 > ©. Example 4- Let S= l^\, where, using the triadic system, ?=-li?2ls-^ f„ = 0,2 denote the Cantor set of I, 272. Let us associate with f the point X ^ • XjX^X^ ■•• (1 where a;„ = when ^„ = 0, and =1 when f „ = 2 and read 1) in the dyadic system. Then {x] is the interval (0, 1). Thus we have established a uniform correspondence between £ and the points of a unit interval. In passing let us note that if f < |' and x, a:' are the correspond- ing points in \x\, then x b , or a < 0. (1 This property has not yet been established for transfinite car- dinal numbers. There is in fact a fourth alternative relative to 31, S, besides the three involved in 1). For until the contrary has been shown, there is the possibility that : No part of 21 is ~ SS, and no part of © is ~ 31. The reader should thus guard against expressly or tacitly assuming that one of the three relations 1) must hold for any two cardinal numbers. 3. We note here another difference. If 21, -8 are finite with- out common element, Card (31 -I- «)> Card 21. (2 Let now 21 denote the positive even and © the positive odd numbers. Obviously Card (2[ -I- «) = Card 31 = Card SS and the relation 2) does not hold for these transfinite numbers. 4. We have, however, the following : Let 31 > «, then Card 2[ > Card 33. For obviously ^ is ~ to a part of 21, viz. © itself. 5. This may be generalized as follows : ^^* 3l = S + e-h©+ •■• A=B+0+D+ - If Card SB < Card B , Card (5 < Card (7, etc., ^^^^ Card 31 < Card A. For from Card 39 < Card B follows that we can associate in 1, 1 correspondence the elements of 53 with a part or whole of B. The same is true for g, Q; 35, i> ; ••• Thus we can associate the elements of 31 with a part or the whole of A. 280 AGGREGATES Enumerable Sets 232. 1. An aggregate which is equivalent to the system of .positive integers 3^ or to a part of ^ is enumerable. Thus all finite aggregates are enumerable. The cardinal num- ber attached to an infinite enumerable set is Kq, aleph zero. At times we shall also denote this cardinal by e, so that 2. Every infinite aggregate 21 contains an infinite enumerable set S. For let aj be an element of 21 and 2l = ai-|-2li, Then 2li is infinite ; let a^ be one of its elements and 2li = «3+2l2. Then 2I2 is infinite, etc. Then ™ is a part of 21 and forms an infinite enumerable set. 3. From this follows that Xq is the least transfinite cardinal number. 233. The rational numbers are enumerable. For any rational number may be written m ,-, r = - (1 n where, as usual, m is relatively prime to n. The equation \m\ + \n\=p (2 admits but a finite number of solutions for each value of p = 2, 3,4, ... Each solution m, w of 2), these numbers being relatively prime, gives a rational number 1). Thus we get, e.g. p=2 , ±1. P = 2> , ±2, ±^. ^0 = 4 , ±3, ±^. ^^ = 5 , ±4, ±1 , ±1 , ±|. ENUMERABLE SETS 281 Let us now arrange tliese solutions in a sequence, putting those corresponding to p'= q before those corresponding to p = q + l. We get which is obviously enumerable. 234. Let the indices tj, tji ••■ 'p range over enumerable 8ets. Then IS enumerable. For the equation ^1 + ^2+ - +i'p = n, where the vs are positive integers, admits but a finite number of solutions for each n=p, p + 1, p + 2, p + S--- Thus the elements of m ti > may be arranged in a sequence by giving to n successively the values p, p + 1, ■■■ and putting the elements by^ ... „^ corresponding to w = ^ + 1 after those correspond- ing to w = 5'. Thus the set .95 is enumerable. Consider now 31. Since each index („, ranges over an enumerable set, each value of i^ as i'^ is associated with some positive integer as m' and conversely. We may now establish a 1, 1 correspondence between 21 and SB by setting b,n[mi - m'p ~ "^ij'j - ly Hence 21 is enumerable. 235. 1. An enumerable set of enumerable aggregates form an enumerable aggregate. For let 21, S, S ■•• be the original aggregates. Since they fprm an enumerable set, they can be arranged in the order 2li , 2I2 ' 3I3 ' - (1 But each 21^ is enumerable ; therefore its elements can be arranged in the order 282 AGGREGATES Thus the a-elements in 1) form a set \a„J m, n,= 1, 2, ••• • which is enumerable by 234. 2. The real algebraic numbers form an enumerable set. For each algebraic number is a root of a uniquely determined irreducible equation of the form a;» + aia;"-! + ■■■' + a„=0, the a's being rational numbers. Thus the totality of real algebraic numbers may be represented by where the index n runs over the positive integers and a-y- a„ range over the rational numbers. 3. Let 31, S be two enumerable sets. Then Card 31 = Card « = «„. Card (31 +«)=«„. And in general if Slj , 3I2 • ■ • are an enumerable set of enumerable aggregates, ^ard (Slj , Sl^ , • • • ) = Xo- This follows from 1. 236. Every isolated aggregate 31, limited or not, forms an enumer- able set. For let us divide 9i„ into cubes of side 1. Obviously these form an enumerable set (7j, C^--. About each point a of 31 in any (7„ as center we describe a cube of side a, so small that it contains no other point of 31. This is possible since 31 is isolated. There are but a finite number of these cubes in (7„ of side 0- = -, v= 1, 2, 3, ••• V for each v. Hence, by 235, 1, 21 is enumerable. 237. 1. Every aggregate of the first species 21, limited or not, is enumerable. For let 21 be of order n. Then 21 = 21. + 3i; ENUMERABLE SETS 283 where 21^ denotes the isolated points of 21, and 21^ the proper limit- ing points' of 21. Similarly, , 2i;' = 2i;v + 2i;" Thus, 2i = 2i. + 2i;,. + 2i';. + -+2r < 2I<">. Thus 21 being the sum of w + 1 enumerable sets, is enumerable. 2. If 21' is enumerable, so is 21. For as in 1, 91 = 21, + 2i; and Wp 2I2, we would infer from 1) that also 3l~3li. (2 As 31^ ~ 53 by hypothesis, the truth of the theorem follows at once from 2). To establish 2) we proceed thus. In the correspondence 1), let 2(3 be that part of 2I2 which ~ 2li in 21. In the correspondence 2I1 ~ 2(3, let 9l4 be that part of 2I3 which ~ % in 2li. Continuing in this way, we get the indefinite sequence 21 > 2li > 2l2 > 2I3 > - such that 2I~2l2~2l4~- Let now 2l = 2ti4-(5i , ^^ = ^^ + (1, , - S) = i>t<(2l, 2li,2l2"0 >0. Then 91 =, 2;) + (^^ + g^ + gg + g^ + ... (3 and similarly 91^ = S) + S^ + S3 + S, + ©5 + - We note that we can also write 21i = S)+g3 + (J2 + S6 + S4+ - (4 Now from the manner in which the sets 2I3, 21^ ••• were obtained, it follows that Si-Sj , (53-65- (5 Thus the sets in 4) correspond uniformly to the sets directly above them in 3), and this establishes 1). ENUMERABLE SETS 285 2. In connection %ith the foregoing proof, which is due to Bernstein, tlie reader must guard against the following error.- It does not in general follow from '^'"^ . Si ~ S3 which is the first relation in 5). Example. Let 31 = (1, 2, 3, 4, •••). 2li=(2, 3, 4, 5...) , 2[2=(3, 4, 5, 6-.) 2l3=(5, 6, 7, 8-). T^^" Si = l 63= (3, 4). Now 31, 9lj, 2I2 , 9I3 are all enumerable sets ; hence 3l~3l2 , 3li~3l3. But obviously Sj is not equivalent to S3, since a set containing only one element cannot be put in 1 to 1 correspondence with a set consisting of two elements. 240. 1. J/"3l>S>e, a/iiSl-g, t^ew 3l~«. For by hypothesis a part of 53, viz. S, is ~ 31. But a part of 31 is ~59, viz. Sd itself. We apply now 239. * 2. Let a he any cardinal number. If a < Card S3 < a, *^^" «=Card«. For let Card 31 = a. Tlien from a < Cards it follows that 31 ~ a part or the whole of SB ; while from Card « < a it follows that S is ~ a part or the whole of 31. 3. Any part 53 of an enumerable set 31 is enumerable. For if 53 is finite, it is enumerable. If infinite, Card53>«o- On the other hand Card 53 < Card 31 = Ko- 286 AGGREGATES 4. Two infinite enumerable sets are equivalent. For both are equivalent to 3^, the set of positive integers. 241. 1. Let S le any enumerable set in 31 ; 8e( 21 = (g + S3. If S3 is infinite, 2I~93. For -93 being infinite, contains an infinite enumerable set jj. Let S = g + ®. Then 3I=(g + g + ®, S3 = g + ®. Bute + g~g- Hence 21 ~ S3. 2. We may state 1 thus : Card (31 -g)= Card 31 provided 21 — @ is infinite. 3. From 1 follows at once the theorem : Let 31 be any infinite set and (5 an enumerable set. Then Card (3t + g) = Card 21. Some ■ Space Transforviatitms 242. 1. Let y be a transformation of space such that to each point X corresponds a single point Xj,, and conversely. Moreover, let x, y be any two points of space. After the trans- formation they go over into Xj., y^. If Dist (a;, y') = Dist (a;y , y^) we call ya displacement. If the displacement is defined by aij = a;j -|- aj , • • ■ a5„, = x,^ -f- a^ it is called a translation. If the displacement is such that all the points of a line in space remain unchanged by T, it is called a rotation whose axis is the fixed line. THE CARDINAL c 287 If 9? denotes the «riginal space, and 9tj, the transformed space after displacement, we have, obviously, «/i = tei , ••• y„^te,„ , <> 0. (1 Then when x ranges over the m-way space •£, y ranges over an w-way space 9). If we set a; ~ «/ as defined by 1), ^^®° Dist (0, y-) = t Dist (0, X). We call 1) a transformation of similitude. If i > 1, a figure in space is dilated ; if t < 1, it is contracted. 3. Let Q be any point in space. About it as center, let us de- scribe a sphere yiS of radius ^. Let P be any other point. On the join of P, Q let us take a point P' such that Dist (P', (?) = ^ L>ist (P, (?) Then P' is called the inverse of P wi^A respect to S. This trans- formation of space is called inversion. Q is the center of inversion. Obviously points without *S' go over into points within, and con- versely. As P = 00 , P' = Q. The correspondence between the old and new spaces is uniform, except there is no point corresponding to Q. Tlie Cardinal c 243. 1. All or any part of space ® may he put in uniform cor- respondence with a point set lying in a given cube O. For let @i denote the points within and on a unit sphere S about the origin, while ©„ denotes the other points of space. By an in- version we can transform ©, into a figure ©,- lying in S. By a transformation of similitude we can contract ©^, ©,- as much as we choose, getting @;, @y. We may now displace .these figures so as to bring them within in such a way as to have no points in common, the contraction being made sufficiently great. The 288 AGGREGATES correspondence between © and the resulting aggregate is obviously uniform since all the transformations employed are. As a result of this and 240, i we see that the aggregate of all real numbers is ~ to those lying in the interval (0, 1) ; for example, the aggregate of all points of 5R„ is ~ to the points in a unit cube, or a unit sphere, etc. 244. 1 . The points lying in the unit interval 21 = (0*, 1*) are not enumerahle. For if they were, they could be arranged in a sequence a^, a^, flj ••• (1 Let us express the as as decimals in the normal form. Then Consider the decimal b = ■ JjSj^g ... also written in the normal form, where Then 6 lies in 31 and is yet different from any number in 1). 2. We have (0*, 1*)~ (0, 1) , by 241, 3, ~(a, &) , by 243, where a, h are finite or infinite. Thus the cardinal number of any interval, finite or infinite, with or without its end points is the same. We denote it by c and call it the cardinal number of the recti- linear continuum, or of the real number sygtem 9?. Since 31 contains the rational number system R, we have c>«o- 3. The cardinal number of the irrational or of the transcendental numbers in any interval 21 is also c. For the non-irrational numbers in 21 are the rational which are enumerable ; and the non-transcendental numbers in 21 are the algebraic which are also enumerable. THE CARDINAL c 289 4. The cardinal nvmber of the Cantor set (^of I, 272 is c. For each point a of S has the representation in the triadic system _ „ •' a = • a^a^a^ •■• , a = 0, 2. But if we read these numbers in the dyadic system, replacing each a„ = 2 by the value 1, we get all the points in the interval (0, 1). As there is a uniform correspondence between these two sets of points, the theorem is established. 245. An enumerable set 21 is not perfect, and conversely a perfect set is not enumerable. For suppose the enumerable set 21 = a^, flj ■■• 0- were perfect. In D*(^a-^) lies an infinite partial set 2li of 21, since by hypothesis 21 is perfect. Let a^^ be the point of lowest index in 2li . Let us take r^ < r-y such that D^^(^a^^ lies in D*(a^. In I)*(an,) lies an infinite partial set 2t2 of 2li. Let a„3 be the point of lowest index in 2I21 etc. Consider now the sequence It converges to a point a by I, 127, 2. But a lies in 21, since this is perfect. Thus a is some point of 1), say a = a,. But this leads to a contradiction. For a, lies in every D*J^amJ ; on the other hand, no point in this domain has an index as low as m„ which = 00, as w = 00. Thus 2t cannot be perfect. Conversely, suppose the perfect set 91 were enumerable. This is impossible, for we have just seen that when 2t is enumerable it cannot be perfect. 246. Let 21 be the union of an enumerable set of aggregates 2l„ each having the cardinal number c. Then Card 21 = c. For let ^„ denote the elements of 2l„ not in 2li,2l2 ••• 2l„_, . ^^^"^ 2l = 2li + «2 + «3+- Let S„ denote the interval (n— 1, w*). Then the cardinal number of gi+ S2+ ••• is c 290 AGGREGATES ^^^ Card «„< Card g„. " Card 21 < c , by 231, 5. (1 On the other hand, , „, ^ „ , „, Card 21 > Card Slj = c. (2 From 1), 2) we have the theorem, by 240, 2. 247. 1. As already stated, the complex x= (^x^, x.^, ••• x„) de- notes a point in w-way space. Let x^, x^, ■■■ denote an infinite enumerable set. We may also say that the complex x= (ajj, x^, ••• in inf.) denotes a point in oo -way space 9J„. 2. Let 21 denote a point set in 9i„, n finite or infinite. Then Card 21 < c. (1 For let us first consider the unit cube (5 whose coordinates x^ range over SS = (0*, 1*). 'Let 35 denote the diagonal of g. Then c = Card © < Card g. (2 On the other hand we show Card S < c For let us express each coordinate x^ as a decimal in normal form. Then x^ = • 0,21^2,^7,3'^u ' ' ' Let us now form the number obtained by reading the above table diagonally. Let ^ denote the set of y's so obtained as the a;'s range over their values. Then For the point y, for example, in which aj„= 0, w= 1, 2, ••• lies in 53 but not in 2) as otherwise a;j = 0. Let us now set x~ y. Then e ~ 2) and hence ^.^^^ g < ^_ ^3 From 2), 3) we have Card g = c. THE CARDINAL C 291 Let us now comple^p S by adding its faces, obtaining the set G. By a transformation of similitude T we can bring C^ within S. ^^"^^^ Card S > Card C. On the other hand, S is a part of (7, hence Card S < Card C. Thus Card (7 = c. The rest of the theorem follows now easily. 248. iei i? = l/i denote the aggregate of one-valued continuous functions over a unit cube E in 9?„. ^^^'^ Cardg = c. Let C denote the rational points of S, i.e. the points all of whose coordinates are rational. Then any / is known when its values over C are known. For if a is an irrational point of S, we can approach it over a sequence of rational points aj, asg ••• == «. But / being continuous, /(a) = lim/(a„), and / is known at a. On the other hand, being enumerable, we can arrange its points in a sequence ^_ Let now 5R„ be a space of an infinite enumerable number of dimensions, and let y= (y^, y^ ••■) denote any one of its points. Let / have the value rj^ at Cj, the value t)^ at c^ and so on for the points of O. Then the complex tjj, tjji •" completely deter- mines / in S. But this complex also determines the point j; = (t/j, 772 •••) in 9t„. ^® ^°^ associate / with -q. Thus Card S< Card 5R = c. But obviously Card ^ >(-, for among the elements of % there is an/ which takes on any given value in the interval (0, 1), at a given point of S. 249. There exist aggregates whose cardinal number is greater than any given cardinal number. Let Sb=\b\ be an aggregate whose cardinal number b is given. Let a be a symbol so related to SS that it has arbitrarily either the value 1 or 2 corresponding to each h of SS- Let 21 denote the 292 AGGREGATES aggregate formed of all possible a's of this kind, and let a be its cardinal number. Let /3 be an arbitrary element of -33. Let us associate with /8 that a which has the value 1 for J = /3 and the value 2 for all other 6's. This establishes a correspondence between 93 and a part of 21. Hence a>6. Suppose a=h. Then there exists a correspondence which associates with each h some one a and conversely. This is impossible. For call aj that element of 21 which is associated with b. Then flj has the value 1 or 2 for each yS of S. There exists, however, in 21 an element a' which for each ;8 of 23 has just the other determination than the one a^ has. But a' is by hypothesis associated with some element of 93, say that a' = ay . Then for h = 6', a' must have that one of the two values 1, 2 which %' has. But it has not, hence the contradiction. 250. The aggregate of limited integrable functions § defined over 21 = (0, 1) has a cardinal number f > c. For let f(x) = in 91 except at the points S of the discrete Cantor set of I, 272, and 229, Ex. 4. At each point of 6 let / have the value 1 or 2 at pleasure. The aggregate ® formed of all possible such functions has a cardinal number > c, as the reasoning of 249 shows. But each / is continuous except in S, which is discrete. Hence / is integrable. But ^ > ®. Hence f>c. Arithmetic Operations with Cardinals 251. Addition of Cardinals. Let 21, 93 be two aggregates with- out common element, whose cardinal numbers are a, b. We define the sum of a and b to be Card(2l, 93)=a-|-l). ARITHMETIC OPERATIONS WITH CARDINALS 293 We have now the flowing obvious relations : K(j + n = Sq , n a positive integer. (1 So+«o+ ••• +>5o = Ko ' n terms. (2 So + So+ ••• = Ko 1 an infinite enumerable set of terms. (3 If the cardinal numbers of 21, SS, G are a, 6, c, then * a+(b + c) = (o + b) + c, The first relation states that addition is associative, the second that it is commutative. 252. Multiplication. 1. Let3l=5a|, ^ = {5| have the cardinal numbers a, b. The union of all the pairs (a, 6) forms a set called the product of 91 and SS. It is denoted by 31 • S. We agree that (a, 5) shall be the same as (6, a). Then We define the product of a and b to be Card ?l • « = Card 93 • 21 = a • b = b • a. 2. TFe Aave obviously the following formal relations as in finite cardinal numbers : ^^^ ■ c) = (a • b)c, a ■b = h • a, a(h + c) = ab + ac, which express respectively the associative, commutative, and dis- tripulative properties of cardinal numbers. Uxample 1. Let 31 = {ai, SB = ]b\ denote the points on two indefinite right lines. Then 21. 58= {(a, 6)!. If we take a, b to be the coordinates of a point in a plane di^, then 21.58 = 5R2- * The reader should note that here, as in the immediately following articles, c is simply the cardinal number of S which is any set, like 31, SB ••• 294 AGGREGATES Example 2. Let 21 = {aj denote the family of circles x^+y^=aK (1 Let S = i6J denote a set of segments of length 6. We can interpret (a, 6) to be the points on a cylinder whose base is 1) and whose height is h. Then 31 • iB is the aggregate of these cylinders. (1 253. 1. Sf) — M • Sq ' or ne = e. For let 9'J = (ai, aj, • - an), e = (ei,e2 •• • in inf.) Then 9fi.g = (ai, ej) , («1> «2) ■ . (aiieg)-" (^2' «l) , (aj, fig) 1 1 (^2' «3) ••• The cardinal number of the set on the left is w^^Q, while the cardinal number of the set on the right is Sg • 2. ec = c. (2 For let S = \c] denote the points on a right line, and S = (1, 2, 3,...). Then ee={(w, e)j may be regarded as the points on a right line l„. Obviously, Card \U=c. Hence ec = Card @g = c. 254. Exponents. Before defining this notion let us recall a problem in the theory of combinations, treated in elementary algebra. Suppose that there are 7 compartments ^1' ^2' "■ ^Y' and that we have k classes of objects ARITHMETIC OPERATIONS WITH CARDINALS 295 Let us place an Object from any one of these classes in Cj, an object from any one of these classes in Cj ••• and so on, for each compartment. The result is a certain distribution of the objects from these k classes IC, among the 7 compartments 0. The number of distributions of objects from h classes among 7 compartments is k^. For in (7j we may put an object from any one of the k classes. Thus Oi may be filled in k ways. Similarly O^ may be filled in k ways. Thus the compartments C^, 0^ may be filled in P ways. Similarly C^, O^, C^ may be filled in h^ ways, etc. 255. 1. The totality of distributions of objects from k classes JST among the 7 compartments Cform an aggregate which may be denoted by j^u We call it the distribution of K over O. The number of distri- bution of this kind may be called the cardinal number of the set, and we have then Card K'^ = k"* 2. What we have here set forth for finite Caud ^may be ex- tended to any aggregates, 21 = \al, S3 = \b\ whose cardinal num- bers we call a, 6. Thus the totality of distributions of the a's among the 6's, or the distribution of % over S3i is denoted by and its cardinal number is taken to be the definition of the symbol ""'• Thus, Card . as = a*. 256. Example 1. Let x" -t- a^x"-^ -I- ■•• 4- a„ = (1 have rational number coefficients. Each coefficient < a, can range over the enumerable set of elements in the rational number system B = \r\, whose cardinal number is K,,. The n coefficients form a set 21 = (at, •••«„) = fa}- To the totality of equations 1) corresponds a distribution of the /s among the a's, or the set whose cardinal number is 296 AGGREGATES ^® Card R^=^„ = e. we have the relation : Kg = Ko ' or e" = e for any integer n. On the other hand, the equations 1) may be associated with the complex («17 ••• «»)' and the totality of equations 1) is associated with ^"* Ka,, a,)\ = ]a,\ . \a,\, {(aj, a^, a^yl = i(«i' 'h)l ' l^sl ' etc. ^^''''^ (i=\a,\.\a^\...la„\. Card (^ = t ■ i. • ■■• t , n times as factor. ^^* Card g = Card R^, since each of these sets is associated uniformly with the equations ^' e" = e • e ■ •■• c , w times as factor. 257. Example 2. Any point x in m-way space 9i„ depends on m coordinates aij, x^, ••• x„, each of which may range over the set of real numbers 9t, whose cardinal number is c. The m coordi- nates Xy ••• x„ form a finite set Thus to $K„ = \x\ corresponds the distribution of the numbers in 5R, among the m elements of 3£, or the set whose cardinal number is cm ^^ Card $R^ = c we have c*" = c for any integer m. (1 As in Example 1 we show C" = c • c ■ ••• c , w times as factor. ARITHMETIC OPERATIONS WITH CARDINALS 297 258. • a*+^ = a*-a'. (1 To prove this we have only to show that 21®+^ and 31® • 2t^ can be put in 1-1 correspondence. But this is obvious. For the set on the left is the totality of all the distributions of the elements of 31 among the sets formed of SS and S. On the other hand, the set on the right is formed of a combination of a distri- bution of the elements of 31 among the Sgi and among the S. But such a distribution may be regarded as the distribution first con- sidered. 259. {aiy = a^< (1 We have only to show that we can put in 1-1 correspondence the elements of (31®)^ and 31®'®. (2 Let 31= \a\, S8= \bl, S= {c|. We note tha.t 21© is a union of distributions of the a's among the J's, and that the left side of 2) is formed of the distributions of these sets among the e's. These are obviously associated uniformly with the distributions of the a's among the elements of 93 • S. 260. 1. c" = (to^)" = «i"= = m^ = c (1 where m, n are positive integers. For each number in the interval S = (0, 1*) can be represented in normal form once and once only by • a^a^a^ . • • in the wi-adic system, (2 where the < a, < to. [I, 145] . Now the set of numbers 2) is the distribution of a)?=(0, 1, 2, • ••m — 1) over @= (aj, a^, «3"-)' "^"^ whose cardinal number is On the other hand, the cardinal number S is c. 298 AGGREGATES Hence, »»' = c. As w' = e, we have 1), using 1) in 257. 2. Tite result obtained in 247, 2 mai/ be stated : ce = c. (3 3. ee = c. (4 For obviously w« < e« < c«. But by 3), c« = c and by 1) w« = c. 261. 1. The cardinal number t of all functions f {xi ••• x^) which take on but two values in the domain of definition 21, of cardinal num- ber a, is 2 2t. Moreover, 2 21 > a. This follows at once from the reasoning of 249. 2. Let f be the cardinal number of the class of all functions de- fined over a domain,% whose cardinal number is c. Then f = c':=2c>c. (1 For the class of functions which have but two values in 31 is by ■ 1, 2c. On the other hand, obviously But cc = (2e)c, by 260, 1) = 2« by 259, 1) = 2S by 253, 2). Thus, cc = 2 c. That f>c follows from 250, since the class of functions there considered lies in the class here considered. 3. The cardinal number t of the class of limited integrable func- tions in the interval 21 is = f, the cardinal number of all limited functions defined over 21. NUMBEKS OF LIOUVILLE 299 For let 35 be a Cjmtor set in 31 [I, 272]. Being discrete, any limited function defined over J) is integrable. But by 229, Ex. 4, the points of 21 may be set in uniform correspondence with the points of 2). 4. The set of all functions ■ /(«^)=/i(^)+/2(^)+- (2 which are the sum of convergent series, and whose terms are continu- ous in 21, has the cardinal number c. For the set g of continuous functions in 31 has the cardinal number c by 248. These functions are to be distributed among the enumerable set @ of terms in 2). Hence the set of these functions is whose cardinal number is c' = c. Remarh. Not every integrable function can be represented by the series 2). For the class of integrable functions has a cardinal number > c, by 250. 5. The cardinal number of all enumerable sets in an m-way space 3J„ is c. For it is obviously the cardinal number of the distribution of 3J„ over an enumerable set (5, or Card ^l = e = c. JSPicmbers of Liouville 262. In I, 200 we have defined algebraic numbers as roots of equations of the type aoa;"+aia;"-i+ ••• + a„ = (1 where the coefficients a are integers. All other numbers in SR we said vrere transcendental. We did not take up the question whether there are any transcendental numbers, whether in fact, not all numbers in SR are roots of equations of the type 1). 300 AGGREGATES The first to actually show the existence of transcendental num- bers was lAouville. He showed how to form an infinity of such numbers. At present we have practical means of deciding whether a given number is algebraic or not. It was one of the signal achievements of Hermite to have shown that e = 2.71818 ••• is transcendental. Shortly after lAndemann, adapting Hermite's methods, proved that IT = 3.14159 ••• is also transcendental. Thereby that famous problem the Quadrature of the Oircle was answered in the negative. The researches of Hermite and Lindemann enable us also to form an infinity of transcendental numbers. It is, however, not our pur- pose to give an account of these famous results. We shall limit our considerations to certain numbers which we call the numbers of Liouville. In passing let us note that the existence of transcendental num- bers follows at once from 235, 2 and 244, 2. For the cardinal number of the set of real algebraic number is e, and that of the set of all real numbers is c, and c > e. 263. In algebra it is shown that any algebraic number a is a root of an irreducible equation, /(a;) = a^x" + a^x'"-^ -\ h a„ = (1 whose coefficients are integers without common divisor. We say the order of a is m. We prove now the theorem Let r„ = — , p^, q^^ relatively prime. In = a, an algebraic number of order m, as n = oo. Then In For let « be a root of 1). We may take S>0 so small that f(x')^0 in D^*(oC), and « so large that r„ lies in D^Qi), for n>8. Thus l/(OI = NUMBERS OP LIOUVILLE 301 for n>s, since the nifhnerator of the middle member is an integer, and hence > 1, ' On the other hand, by the Law of the Mean [I, 397], /(»•»)-/(«) = (»•„- «)/'(/3) where /3 lies in -Z)a(a). Now /(a)=0 and /'(/3)< some M. Hence ^r \ i on using 3). But. however large M is, there exists a v, such that qn>M, for any n>v. This in 4) gives 2). 264. 1. The numbers 7 = -^ 4- -^ 4- _Ea_ -L n ' 101' 102' 103! '^■^ where a„ < 10", and not all of them vanish after a certain index, are transcendental. For if L is algebraic, let its order be m. Then if L^ denotes the sum of the first n terms of 1), there exists a v such that 7]=\L — LJ> -r— , for «, > V. (2 But . v' being taken sufficiently large. But 8) contradicts 2). The numbers 1) we call the numbers of Liouville. 2. The set of Liouville numbers has the cardinal number c. For all real numbers in the interval (0*, 1) can be represented where not all the J's vanish after a certain index. The numbers 101' ^ 102' ^ 103! ^ can obviously be put in uniform correspondence with the set {/8f . Thus Card {\| =c. But \L\ > |X.|, hence Card \L\>t. On the other hand, the numbers \L\ form only a part of the numbers in (0*, 1). Hence Card \L\j states that a succeeds 5. Example 1. The aggregates 1, 2, 3, ... 2, 4, 6, ... ...-3, -2, -1,0,1,2,3,... ... c are ordered. Example 2. The rational number system R can be ordered in an infinite variety of ways. For, being enumerable, they can be arranged in a sequence „ „ „ Now interchange r-^ with r„. In this way we obtain an infinity of sequences. Example 8. The points of the circumference of a circle may be ordered in an infinite variety of ways. For example, let two of its points Pj, P^ make the angles a+^j, a + d^ with a given radius, the angle 6 varying from to 2 tt. Let Pj precede Pj when ^j < 6^. 302 ORDERED SETS 303 Hxample 4- The paeitive integers 3i niay be ordered in an infi- nite variety of ways besides their natural order. For instance, we may write them in the order 1, 3, 5, ... 2, 4, 6, ... so that the odd numbers precede the even. Or in the order 1, 4, 7, 10, ... 2, 5, 8, 11, ... 3, 6, 9, 12, ... and so on. We may go farther and arrange them in an infinity of sets. Thus in the first set put all primes ; in the second set the products of two primes; in the third set the products of three primes; etc., allowing repetitions of the factors. Let any number in set m precede all the numbers in set n>m. The num- bers in each set may be arranged in order of magnitude. Example 5. The points of the plane Sij ™^y ^^ ordered in an infinite variety of ways. Let Ly denote the right line parallel to the avaxis at a distance y from this axis, taking account of the sign of y. We order now the points of 5R2 by stipulating that any point on i^, precedes the points on any L^, when «/' < y" , while points on any L^ shall have the order they already possess on that line due to their position. 266. Similar Sets. Let 91, Sb be ordered and equivalent. Let a'^h, a'^ ^. If wh-en a < a in 21, 6 < ;8 in 53, we say 21 is similar to SS, and write gr ~ iw Thus the two ordered and equivalent aggregates are similar when corresponding elements in the two sets occur in the same relative order. Example 1. Let 21 = 1 9 3 S = «!, «2' "'&- •■• In the correspondence 21 ~ 48, let n be associated with a„. Then Example 2. Let ; = 1, 2, 3, •■• 304 ORDINAL NUMBERS In the correspondence 21 ~ S3, let a^ ~r for ?• = 1, 2, ••• m; also let 5„~m + »,« = !, 2 - Then 31=^35. Examples. Let 31=1 2 3 Let the correspondence between 31 and 33 be the same as in Ex. 2. Then 31 is not similar to 58- For 1 is the first element in 21 while its associated element a^ is not first in S3. Example 4. Let 9f — 1 2 3 S8 = ^i, a^ ■■■ Jj, ^2 ■■■ Let a„ ~ 2 w, 6„ ~ 2 w - 1. Then 31 ~ 93 but St is not =. SB- 267. iet 31 ^ 93, 93 =^ 5. Then 31 ^ S. For let a ~ J, a' ~ 6' in 21 ~ 93. Let J ~ c, J'~c' in 93 ~ g. Let us establish a correspondence 31 ~ S by setting a ~ c, a' ~c/ . Then if a a^> a^> ■■■ (1 For 0 = ••• fflj, flj' «i is a part of 91 whose elements occur in the same relative order as in 21, and 48 has no first element. 2. A sequence as 1) may be called a decreasing sequence, while «! < a2 < ^s ■■■ may be called increasing. In every infinite well ordered aggregate there exist increasing sequences. 3. Let 21, S, S, ••• Se a well ordered set. Let 2t = \a\ he well ordered in the a's, SB = \h\ be well ordered in the b's, etc. The set U = 2l, «, S-- is well ordered with regard to the little letters a, b ■ • • For U has a first element in the little letters, viz. the first ele- ment of 21. Moreover, any part of U, as iB, has a first element in the little letters. For if it has not, there exists in 33 an infinite decreasing sequence t> s>r> — This, however, is impossible, as such a sequence would deter- mine a similar sequence in U as S>®>SR> - which is impossible as U is well ordered with regard to 21, 53 •■■ i. Let 2l< S < g < ••• (1 Let each element of 21 precede each element of SS, etc. SECTIONS 307 Let each 31, -33, • •• he well ordered. Let Then, S = 3l + 5, S = « + (7. q>p > ••• (2 Now r lies in some set of 1) as 9i. Hence q, p, ■■■ also lie in 9?. But in 9J there is no sequence as 2). 5. Let 31, S, S, ■•■ be an ordered set of well ordered aggre- gates, no two of which have an element in common. The reader must guard against assuming that 31 + S5 + S+ •••, keeping the relative order intact, is necessarily well ordered. For let us modify Ex. 5 in 265 by taking instead of all the points on each L^ only a well ordered set which we denote by 3ly Then the sum 21 = 221, V has a definite meaning. The elements of 21 we supposed arranged as in Ex. 5 of 265. Obviously 31 is not well ordered. Sections 271. We now introduce a notion which in the theory of well- ordered sets plays a part analogous to Dedekind's partitions in the theory of the real number system 9?. Cf. I, 128. Let 31 be a well ordered set. The elements preceding a given element a of 21 form a partial set called the section of 21 generated hy a. We may denote it by Sa, or by the corresponding small letter a. 308 ORDINAL NUMBERS Example 1. Let 91 = 1 2 3 ••• Then ;S'100 = 1, 2,... 99 is the section of 31 generated by the element 100. Example S. Let 21 = «!, a^ ■■■ Sj, Sj"" Then Sb^ = a^a^---b.fi2bgb^ is the section generated by b^. Sb-^ = a^a^ • • • that generated by Jj, etc. 272. 1. Every section of a well ordered aggregate is well ordered. For each section of 21 is a partial aggregate of 21, and hence well ordered by 269, l. 2. In the well ordered set 21, let aa. Let b'~a' and b"'^a". Since 21 ~: SB, we have b" ai>a2> •••, which is impossible by 270, 1. SECTIONS 309 275- Let 21, 58 bmoell ordered and SI ^ 33. Then to Sa in 21 can- not correspond two sections Sb, aS/S each ^ Sa. For let b<0, and Sa =^ Sb, Sa ^ SjS. Then Sb ^ S^, by 267. (1 But 1) contradicts 274. 276. Let 21, S3 be two well ordered aggregates. It is impossible to establish a uniform and similar correspondence between 21 and S3 in more than one way. For say Sa ^ Sb in one correspondence, and Sa ^ S^ in an- other, b, /8 being different elements of SB. Then Sb c^ S^, by 267. This contradicts 275. 277. 1. We can now prove the following theorem, which is the converse of 273. Let 21, 93 be well ordered. If to each section of 2t corresponds one similar section of 93, and conversely, then 93 — 21. Let us first show that 21 ~ 93. Since to any Sa of 21 corre- sponds a similar section Sb in 93, let us set a ~ b. No other a' ~ b, and no other b' ~ a, as then Sa' ^ Sb or Sb' =i Sa, which contradicts 274. Let the first element of 21 correspond to the first of 93. Thus the correspondence we have set up between 21 and 93 is uniform and 21 ~ 93. We show now that this correspondence is similar. For let a ~ 5 and a' ~ J', a' < a. Then b' < 5. For a' lies in Sa ^ Sb and b' ~ a' lies in Sb. 2. From 1 and 273 we have now the fundamental theorem : In order that two well-ordered sets 21, 93 be similar, it is necessary and sufficient that to each section of 21 corresponds a similar section of 93, and conversely. 278. Let 21, 93 be well ordered. If to each section of 21 corre- sponds a similar section of 93i but not conversely, then 21 is similar to a section of 93. 310 ORDINAL NUMBERS Let us begin by ordering the sections of 31 and SB as in 272, 3. Let B denote the aggregate of sections of 83 to which similar sec- tions of 21 do not correspond. Then B is well ordered and has a first section, say Sb. Let ^ < b. Then to S^ in 83 corresponds by hypothesis a similar section Sa in 91. On the other hand, to any section Sa' of 21 corresponds a similar section Sb' of 83. Ob- viously b' < b. Thus to any section of 21 corresponds a similar section of Sb and conversely. Hence 21=^'$'^ by 277, l. 279. Let 21, 83 be well ordered. Hither 21 is similar to 83 or one is similar to a section of the other. For either : 1° To each section of 21 corresponds a similar section of 83 and conversely ; or 2° To each section of one corresponds a similar section of the other but not conversely ; or 3° There is at least one section in both 21 and S8 to which no similar section corresponds in the other. If 1° holds, 21 2=^ 83 by 277, l. If 2° holds, either 21 or 33 is similar to a section of the other. We conclude by showing 3° is impossible. For let A be the set of sections of 21 to which no similar section in 83 corresponds. Let B have the same meaning for 83. If we suppose 21, 83 ordered as in 272, 3, A will have a first section say Sa, and B a first section S/3. Let a6, and if a < b, it follows that b> a- Obviously they enjoy also the following properties. 2 If a = b , b = c , then o = c. For if c = Ord g, the first two relations state that 21^33 , ^^&. But then or ^ «- u nc-T 2I=^S , by 267. Hence o = c. 3 If •' a > b , b > c , then a > c. 281. 1, Let 21 be a finite aggregate, embracing say n elements. Then we set /-, j or Ord 21 = «. Thus the ordinal number of a finite aggregate has exactly similar properties to those of finite cardinal numbers. The ordinal num- ber of a finite aggregate is called finite^ otherwise transfinite. The ordinal number belonging to the well ordered set formed of the positive integers £> _ i 9 o we call ft). 2. The least transfinite ordinal number is (o. For suppose a = Ord 21 < o), is transfinite. Then 21 is =i a section of Q. But every section of 3^ is finite, hence the contradiction. 312 ORDINAL NUMBERS 3. The cardinal number of a set 21 is independent of the order in which the elements of 31 occur. This is not so in general for ordinal numbers. For example, let w— i 2 S ... 33=1,3,5,... 2,4,6,... ^^^^ Card 31 = Card 33 =So. ^^* Ord 3t < Ord 33, since 31 is similar to a section of 33, viz. the set of odd numbers, 1, 3, 5, ... 282. 1. Addition of Ordinals. Let 3t, 33 be well ordered sets without common elements. Let S be the aggregate formed by placing the elements of 33 after those of 21, leaving the order in 33 otherwise unchanged. Then the ordinal number of g is called the sum of the ordinal numbers of 21 and 33, or Ord 6 = Ord 21 + Ord 39, or c = a + b. The extension of this definition to any set of well-ordered aggre- gates such that the result is well ordered is obvious. 2. We note that , t ^ , t -^ t For 21 is similar to a section of S, and 33 is equivalent to a part of S. 3. The addition of ordinal numbers is associative. This is an immediate consequence of the definition of addition. 4. The addition of ordinal numbers is not always commutative. Thus if 21 =, (a^a^ ... in inf.), Ord 21 = «, let Then 33 = (iji^ ■•• 5„), Ord33 = w; S = (aja^ •■■ 61J2 •■• U' OrdS = c, !D = (Jj ... 6„aja2 ..■)- OrdS) = b. c = (0 + n , b = n + fo. ORDINAL NUMBERS 313 But 31 ^ a sectioi* of S, viz. : ^ Sb-^ , while 5D =ii 21. Hence ft) < c , ft) = b, or ft)+W>ft) , W + ft)=ft). 5. if a > 6, then c + o > c + 6, and a + c > b + c For let a = Ord 31, b = Ord S, c = Ord £. Since a>h, we can take for 58 a section Sb of 31. Then c + a is the ordinal number of ^j. , ^r ^i (a + 5!l, (1 and c + b is the ordinal number of S + Sb, (2 preserving the relative order of the elements. But 2) is a section of 1), and hence c + a > c + b. The proof of the rest of the theorem is obvious. 283. 1. The ordinal number immediately following a is a + 1. For let a = Ord 21. Let S3 be a set formed by adding after all the elements of 31 another element b. Then a + 1 = Ord 58 = b. Suppose now ah. Then there is one and only one ordinal number b such that , , 1, a = b + b. For let a = Ord 31 , b = Ord58. We may take 58 to be a section Sb of 31. Let 25 denote the set of elements of 31, coming after Sb. It is well ordered and has an ordinal number b- Then 31 = S + ®, preserving the relative order, and hence a = b + b. There is no other number, as 282, 6 shows. 314 ORDINAL NUMBERS 284. 1. Multiplication of Ordinals. Let 21, S3 be well-ordered aggregates having 0, 6 as ordinal numbers. Let us replace each element of 21 by an aggregate ==: S3. The resulting aggregate S we denote by SR . 91 As S is a well-ordered set by 270, 3 it has an ordinal number c. We define now the product b • a to be c, and write 6 • a = c. We say c is the result of multiplying a iy 6, and call a, h factors. ' We write a • a = o? , a-a-a = a^ , etc. 2. Multiplication is associative. This is an immediate consequence of the definition. 3. Multiplication is not always commutative. For example, let 21 = (dio^), i8 = (l, 2, 3 ••• in inf.). 58 • 21 = (6162*3 •••, CjCgCg •••)• 21- SB = (61, cj, 63,62, 63, C3, •••)• Hence Qrd (93 • 21) = w • 2 >«, Ord(2l-S3)=2ft) = (B. 4. Ifa2 by 285, 1. From this we get, using the first principle, as before, to2-|-l, to2 + 2, 0)2 4-3,... whose limit is toS. In this way we may obtain the numbers mm + n , m, n finite. The limit of any increasing sequence of these numbers as to , 0)2 , 0)8 , 0)4, ... is to • fo = afl, by 285, 1. From 0)2 we can get numbers of the type (oH + com + n I, m, n finite. Obviously we may proceed in this way indefinitely and obtain all numbers of the type ©"flo + (i)""^ai + (o"-\ + ■■■ +a„ (1 where a^, a^ ■■■ a^ are finite ordinals. LIMITARY NUMBERS 317 But here the proc*s does not end. For the sequence O) < O)^ < O)^ < ••• has a limit which we denote by <»". Continuing we obtain o)"", a)"""", etc. 288. It is intere'sting to see how we may obtain well ordered sets of points whose ordinal numbers are the numbers just con- sidered. In the unit interval 31 = (0, 1), let us take the points ^^ ' f ' 8 ' TB^ "■ (•'■ These form a well ordered set whose ordinal number is ySj, let ySj be the first number in it >/3{, etc. We get thus the sequence ' ^^<^l<^^<... (2 whose limit is \, say. Then X = « . For X is > any number in 1), which embraces all the numbers of Z^ < «. Moreover it is the least number which enjoys this property. 293. 1. The numbers of Z^ are not enumerable. For suppose they were. Let us arrange them in the sequence «15 «2' «3 ••• (1 Then, as in 292, there exists in this sequence the infinite enu- merable sequence a^ < „; < «^' < . . . (2 such that there are numbei-s in 2) greater than any given number inl). 320 ORDINAL NUMBERS Let «!,=«'. Then a' lies in Z^ by 291. On the other hand, by 285, a' is > any number in 2), and therefore > any number in 1). But 1) embraces all the numbers of Zj, by hypothesis. We are thus led to a contradiction. 2. We set .. n i /^ Sj = Card Z^. 294. There is no cardinal number between Kq and i/3>7> ■•• This, however, is not possible. For /3, 7, ••• form a part of Sa which is well ordered. There is thus one well ordered set having Kj as cardinal num- ber. Let r, r\ A fz LI = Ord Zj . Let now 21 be an enumerable well ordered set whose ordinal number is a. The set „ , „, the elements of 21 coming after Z^, has the cardinal number Kj by 241, 3. It is well ordered by 270, 3. It has therefore an ordinal number which lies in Z3, viz. li + « by 282, l. Thus Z^ embraces an infinity of numbers. 2. The least number in Z^ is fi. For to any number «< fi corresponds a section 21 of Z^. Hence a lies in Z^. CLASSES OF ORDINALS 321 296. 1. An aggregate formed of an Kj set of Nj sets is an Sj set. Consider the set a< D, «21' '«22' «13 • «23 ■ ••• «2a • Q!„i, ^i»2' ••• "ua • «al, «a2' ftaa •• ■ «a,« ••• «aa • Here each row is an H^ set. As there are an Xj set of rows, J. is an Kj set of Sj sets. To show that A is an Xj set, we associate each a„ with some number in the first two number classes. In the first place the elements a^, where l, k <. a may be associ- ated with the numbers 1, 2, 3, ••• < oa. The elements a.„, a„, lying just inside the to"* square and which are characterized by the condition that t= 1, 2, ••• w; «= 1, 2 •■• < w form an enumerable set and may therefore be associated with the ordinals £0, o) + 1, ••• < 0)2. For the same reason the elements just inside the CD + 1°' square may be associated with the ordinals (b2, (»2 + 1, • •• < 0)3. In this way we may continue. For when we have arrived at the «'" row and column (edge of the a"' square) we have only used up an enumerable set of numbers in the sequence 1, 2, ... o) ... «j. The proof is entirely similar to 293, 1. Suppose, m fact, that a = Kj . Let us arrange the elements of Zg in the Xj sequence "i 1 ttj ...«„...«„ ... (1 As in 292, there exists in this sequence an Kj increasing sequence ai'j, Z>2 ••• be a sequence of superposed cubical divisions of norms = 0. Any cell of Dj which lies within some S and which contains a point of 31 we call a black cell ; the other cells of D we call white. The black cells are not further subdivided. The division B^ divides each white cell. Any of these subdivided cells which lies within some S and contains a point of 31 we call a black cell, the others are white. Continuing we get an enumer- able set of non-overlapping cubical cells {cj. Each point a of 31 lies within some c. For a is the center of some S. But when n is taken sufficiently large, a lies in a cell of i>„, which cell lies within S. Let now % be limited and complete. Each a lies within a cube c, or on the faces of a finite number of these c. With a we associ- ate the diagonal 8 of the smallest of these cubes. Suppose MinS = in 31. As 31 is complete, there is a point « in 31 such that Min S = 0, in any Vt^Qo). This is not possible, since if i; is taken suificiently small, all the points of F^ lie in a finite number of the cubes c. Thus Min S>0. As the c's do not overlap, there are but a finite number. 2. In the foregoing theorem the points of 31 are not necessarily inner points of the cubes c. Let a be a point of 31 on the face of one of these c. Since a lies within some g, it is obvious that the 324 PANTAXIS 325 cells of some i)„, ^^sufficiently large, which surround a form a cube e, lying within S. Thus the points of 21 lie within an enumerable set of cells {c|, each c lying within some S- The cells c of course will in general overlap. Obviously also, if 21 is complete, the points of 21 will lie within a finite number of these c's. 302. If 21 is dense^ 21' is perfect. For, in the first place, 21' is dense. In fact, let a be a point of 21'. Then in any D*{a) there are points of 21. Let a be such a point. Since 21 is dense, it is a limiting point of 21 and hence is a point of 21'. Thus in any i)*(«) there are points of 21'. Secondly, 21' is complete, by I, 266. 303. Let ^ be a complete partial set of the perfect aggregate 21. Then S = 21 — S3 is demise. For if S contains tlie isolated point c, all the points of 21 in I)*{c) lie in 33, if r is taken sufficiently small. But SB being com- plete, c must then lie in 33. Remark. We take this occasion to note that a finite set is to be regarded as complete. 304. 1. -Z/"2l does not embrace all 9?„, it has at least one frontier point in ?lin ■ For let a be a point of 21, and b a point of 9?„ not in 21. The points on the join of a, b have coordinates x, = a. + ^(6.-aJ = a;.(0), 0<^<1, i=l, 2,...w. Let 6' be the maximum of those ^'s such that x (6) belongs to 21 if ^ < 6'. Then a;(^') is a frontier point of 21. 2. Let 21, S have no point in common. If Dist (21, S3)>0, we say 21, S3 are exterior to each other. 305. 1. Let 21 = 5aS be a limited or unlimited point set in SR„. "We say S3 < 21 is pantactio in 21, when in each i>s(a) there is a point S3. We say S3 is apantactic in 21 when in each Ds(a) there is a point « of 21 such that some !>,(«) contains no point of S3. 326 POINT SETS Example 1. Let 21 be the unit interval (0, 1), and 58 the ra- tional points in 21- Then S3 is pantactic in 21- Example 2. Let 21 be the interval (0, 1), and S the Cantor set of I, 272. Then 33 is apantactic in 21. 2. 7/" 93 < 21 is pantactic in 21, 21 contaircs no isolated points not inSd. For let a be a point of 21 not in 93. Then by definition, in any Di(ji) there is a point of 93- Hence there are an infinity of points of 95 in this domain. Hence a is a limiting point of 21. 306. Let 21 be complete. "We say 93 < 21 is of the 1° category in 21, if Sd is the union of an enumerable set of apantactic sets in 21. If 93 is not of the 1° category, we say it is of the 2° category. Sets of the 1° category may be called Baire sets. Example. Let 21 be the unit interval, and 93 the rational points in it. Then 93 is of the 1° category. For 93 being enumerable, let 93 = \h^\. But each 6„ is a single point and is thus apantactic in 21. The same reasoning shows that if 93 is any enumerable set in 21, then 93 is of the 1° category. 307. L Jf 93 is of the 1° category m 21, 21 - 93 = 5 is >.0. For since 93 is of the 1° category in 21, it is the union of an enumerable set of apantactic sets {93„i. Then by definition there exist points a^, a^, ••■ in 21 such that A.(«i)>A.(«2)>- , s„ = o (1 where I>(a^ contains no point of 93i, ^(a^) ^^ point of 932, ®*°* Let b be the point determined by 1). Since 21 is complete by definition, J is a point of 21. As it is not in any •93„, it is not in 93- Hence B contains at least one point. 2. Let 21 be the union of an enumerable set of sets {2l„S, each 2l„ being of the 1° category in SB. Then 21 is of the 1° category in 93- This is obvious, since the union of an enumerable set of enu- merable sets is enumerable. PANTAXIS 327 3. Let 58 he of tJlk 1° category in 21. Then 5 == 1 -S is of the 2° category in 21. For otherwise ^ + B would be of the 1° category in 21- But 21 - (33 + 5) = 0, and this violates 1. 4. It is now easy to give examples of sets of the 2° category. For instance, the irrational points in the interval (0, 1) form a set of the 2° category. 308. Let % he a set of the- 1° category in the cuhe Q. Then A = Q. — % has the cardinal numher c. If A has an inner point, I>s(^a'), for sufficiently small 2, lies in A. As Card Dj = c, the theorem is proved. Suppose that A has no inner point. Let 21 be the union of the apantactic sets 2li < 2(3 < •••in Q. Let J^j = Q — 2l„. Let q^ be the maximum of the sides of the cubes lying wholly in A^. Ob- viously Qn = 0, since by hypothesis A has no inner points. Let Q be a cube lying in A-^ . As 5„ = 0, there exists an Wj such that Q has at least two cubes lying in A„^ ; call them Qq, §j . There ex- ists an Wg > Wj such that Q^, Q^ each have two cubes in A^; call or more shortly Qt^^i,- Each of these gives rise similarly to two cubes in some A„^, which may be denoted by §.^_ ,^^ .3, where the indices as before have the values 0, 1. In this way we may continue getting the cubes Let a be a point lying in a sequence of these cubes. It obvi- ously does not lie in 21, if the indices are not, after a certain stage, all or all 1. This point a is characterized by the sequence which may be read as a number in the dyadic system. But these numbers have the cardinal number c. 309. Let 21 Se a complete apantactic set in a cuhe Q. Then there exists an enumerable set of cubical cells \ q | such that each point of 21 lies on a face of one of these q, or is a limit point of their faces. 328 POINT SETS For let Dj > 1)3 > ••• be a sequence of superimposed divisions of Q, whose norms B„ = 0. Let (^11, di2, di3." (1 be the cells of Z>j containing no point of 21 within them. Let *21' *22' 23 *■* V'" denote those cells of D^ containing no point of 31 within them and not lying in a cell of 1). In this way we may get an infinite se- quence of cells H) == {dj^nU where for each m, the corresponding n is finite, and m = 00. Each point a oi A lies in some d„^ „. For 21 being complete, Dist (a, 21) > 0. As the norms S„ = 0, a must lie in some cell of 2>„, for a sufficiently large n. The truth of the theorem is now obvious. 310. Let SS be pantactic in 21. Then there exists an enumerable set S <. 93 which is pantaetio in 21. For let 2>i > 2)2 > • " • ^^ ^ set of superimposed cubical divisions of norms d„= 0. In any cell of i)^ containing within it a point of 21, there is at least one point of 58. If the point of 21 lies on the face of two or more cells, the foregoing statement will hold for at least one of the cells. Let us now take one of these points in each of these cells; this gives an enumerable set @j. The same holds for the cells of D^. Let us take a point in each of these cells, taking when possible points of (Sj. Let @2 denote the points of this set not in S^ . Continuing in this way, let ■ g = gi-|-(S2+ - Then @ is pantactic in 21, and is enumerable, since each (5„ is. Corollary. In any set 21, finite or infinite, there exists an enumer- able set (S which is pantactic in 21. For we have only to set 93 = 21 in the above theorem. 311. 1. The points S where the continuous functiork f(x-^-" x^ takes on a given value g in the complete set 21, form a complete set. For let Cj, Cg "• ^^ points of S which = c. We show c is a point of S. For j;/- s x^ \ PANTAXIS 329 As/ is continuous*, ^, ^ . „^ . Hence ,., . and c lies in S. 2. LetfQjc-^ ••• xj) be continuous in the limited or unlimited set 21. J^ the value of f is known in an enumerable pantactic .set 6 in 21, which contains all the isolated points of 21, in case there he such, the value off is known at every point of 21. Eor let a be a limiting point of 21 not in ©. Since (S is pantactic in 21, there exists a sequence of points ej, eg ••• in S which = a. Since / is continuous, /(e„)=/(a). As / is known at each e„, it is known at a. 3. Let g= l/j be the class of one-valued continuous functions defined over a limited point set 21. Then f=Carclg = c. For let 9?_^ be a space of an infinite enumerable number of dimensions, and let , .. denote one of its points. Let/ have the value tjj at gj, the value JJ2 at ^2 ••• for the points of @ defined in 2. Then the complex ('?!' % •••) completely determines /. But this complex determines also a point 7) in 3Joo whose coordinates are ?;„. We now associate / with ''• H^'^^^ fc, since in g there is the function f(xi • ■ ■ x^) = ^ in 21, where ff is any real number. 312. Let S denote the class of complete or perfect subsets lying in the infinite set 21, which latter contains at least one complete set. ^^" 6 = Card 33 = 0. For let flj, a^, ■•• = a, all these points lying in 21. Then But for fj we may take any number in Q-^ = (1, 2, 3, •••) ; for tj we may take any number in 3^2 = ('1 + !> h + 2' "')' ^^°- 330 POINT SETS Obviously the cardinal number of the class of these sequences l)ise^ = c. But (a, «„«,,«,...) is a complete set in 21. Hence b>c. On the other hand, bD,>... (2 be a sequence of superimposed cubical division of norms = 0. Each Dn embraces an enumerable set of cells. Thus the set of divisions gives an enumerable Set of cells. Each cell shall have assigned to it, for a given set in SB, the sign + or — according as 93 is exterior to this cell or not. This determines a distribution of two things over an enumerable set of compartments. The cardinal number of the class of these distributions is 2' = c. But each Sg determines a distribution. Hence 6 < c. Transfinite Derivatives 313. 1. We have seen, I, 266, that 2I'>21">21"'> ••• Thus gj(„, ^ ^^^gj,^ gj^„ _ _ gjf„,^_ ^^ Let now 21 be a limited point aggregate of the second species. It has then derivatives of every finite order. Therefore by 18, 2)1^(21', 21", 21'", -) (2 contains at least one point, and in analogy with 1), we call the set 2) the derivative of order a)of% and denote it by 2l<">, or more shortly by 21". Now we may reason on 21" as on any point set. If it is infinite, it must have at least one limiting point, and may of course have more. In any case its derivative is denoted by 5{(u+i) or 21""'"'- The derivative of 21""'"' is denoted by 2l(«.+2) or 21""'"'' , etc. Making use of a> we can now state the theorem : TRANSFINITE DERIVATIVES 331 In order that the phint set 21 is of the first species it is necessary and sufficient that SI''"-' = 0. 2. We have seen in 18 that SI" is complete. The reasoning of I, 266 shows that Sl"+', 2l"+^ •••, when they exist, are also complete. Then 18 shows that, if Sl"+" w = 1, 2, ••• exist, i)t)(3l" > 2I"+i > Sl"+2 > • ■ • ) (3 exists and is complete. The set 3) is called the derivative of order o) ■ 2 and is denoted by g(("2) or Sl"^ Obviously we may continue in this way indefinitely until we reach a derivative of order « containing only a finite number of points. Then ^^^^ ^ ^ That this process of derivation may never stop is illustrated by taking for 31 any limited perfect set, for then 3l = 3l'=2l"= ... = 31" = St"-'' = ■•• 3. We may generalize as follows : Let « denote a limitary ordi- nal number. If each Sl^ > 0, ^ < a, we set when it exists. 4. If 21" > 0, while SI +i = 0, we say SI is of order a. 314. 1. Let a be a limiting point of 31. Let aj= Card F{(a). Obviously «s is monotone decreasing with B. Suppose that there exists an a and a Sq > 0, such that for all < 8 < Sg a = Card r(a). We shall say that a is a limiting point of rank a. If every «5 > «, we shall say that Rank a > a. If every ag > a, we shall say that Rank a > «. 332 POINT SETS 2. Let %hea limited aggregate of cardinal number a. Then there is at least one limiting point of^,of rank a. The demonstration is entirely similar to I, 264. Let Sj > 82 > ••• =0. Let us effect a cubical division of 21 of norm Sj. In at least one cell lies an aggregate 21^ having the cardinal num- ber «. Let us effect a cubical division of 2li of norm S^. In at least one cell lies an aggregate 2I2 having the cardinal number a, etc. These cells converge to a point a, such that Card FjCa) = «, however small B is taken. 3. If Card 21 > e, there exists a limiting point of^ of rank > e. The demonstration is similar to that of 2. 4. If there is no limiting point of%of rank > e, 21 is enumerable. This follows from 3. 5. Let Card 21 be >(.. Let S3 denote the limiting points of 21 whose ranks are > e. Then SB is perfect. For obviously 53 is complete. We need therefore only to show that it is dense. To this end let S be a point of SB. About b let us describe a sequence of concentric spheres of radii r„ = 0. These spheres determine a sequence of spherical shells \S^\, no two of which have a point in common. If 2l„ denote the points of 21 in S„, we have y ^ ^,^j^ ^ gi^ + gi^ + 21^ ^. ... Thus if eacli 2I„ were enumerable, V is enumerable and hence Rank b is not > e. Thus there is one set 2l„ which is not enu- merable, and hence by 3 there exists a point of S3 in S^. But then there are points of SB in any FJ.*(&), and b is not isolated. 6. A set 21 which contains no dense component is enumerable. For suppose 21 were not enumerable. Let ^ denote the proper limiting points of 21. Then ^ contains a point whose rank is > e. But the set of these points is dense. This contradicts the hy- pothesis of the theorem. 315. Let « lie in Z„. if 21° > 0, it is complete. For if « is non-limitary, reasoning similar to I, 266 shows that 21° is complete. Suppose then that « is limitary, and 21° is not TRANSFINITE DERIVATIVES 333 complete. The de^vatives of 31 of order < a which are not com- plete, form a well ordered set and have therefore a first element 31^, where /S is necessarily a limitary number. Then But every point of 21^ lies in each %y. Hence every limiting point of W is a limiting point of each W and hence lies in 31^- Hence 31^ is complete, which is a contradiction. 316. Let a he a limitary number in Z^. If '^^>0 for each y8 < «, 31" exists. For there exists an S„, m j< w — 2, sequence 7< S has no points in Slj, 2I2 ••• 3l„-i, while the point 1 lies in every ©f,"'. Thus jg,„, ^ J Hence jg^„+i, ^ ^^ and S8„ is of order ^, etc. 318. L Let 21 be limited or not, and let 2l/^> denote the isolated points of 21^. Then For Thus 21' = 22I<^> + 21" , 13 = 1,2,... 0 a 0, there is a non-enumerable set of terms in 1), if no 21^^' is perfect ; and as each term contains at least one point, 21' is not enumerable. If some 21^ is perfect, 21' contains a per- fect partial set and is therefore not enumerable by 245. 4. From 2, 3, we have : -For 21' to be enumerable, it is necessary and sufficient that there exists a number a in Z^ or Z^ such that 21" = 0. 5. If 21 is complete, it is necessary ayid sufficient in order that 21 he enumerable, that there exists an a in Z^ or Z^ such that 21" = 0. ^°'' 21 = 21. -F 21', and the first term is enumerable. 6. If 21^ = for some /3 < fl, we say 21 is reducible, otherwise it is irreducible. 319. 7/ 21" > 0, it is perfect. By 315 it is complete. We therefore have only to show that its isolated points 2lf = 0. Suppose the contrary ; let a be an isolated point of 21". Let us describe a sphere *S^ of radius r about a, containing no other point of 21". Let S3 denote the points of 21' in S. Let r>r.>r„> ■■• = 0. 336 POINT SETS Let S„ denote a sphere about a of radius r„. Let 93„ denote the points of 33 lying between S^-^, *S'„, including those points which may lie on S^.^. Then g3=»i + S32+®s+ ■•• +«■ Each 58m is enumerable. For any point of 33" is a point of 93" = a. Hence S3° = and S8m is enumerable by B18, 2. Thus 93 is enumerable. This, however, is impossible since 93° = a, and is thus > 0. 320. 1. In the relation 21' = 221^ + 21" /3=1, 2, ... r^+i • If 93^ includes all points of 93 whose distance from 21° is > r^, we have 93 = 93o + 93i+932+- Each 93„ is enumerable. For if not, 93° > 0. Any point of 93° as h lies in 21°. Hence Dist (J, 21°) = 0. On the other hand, as h lies in 93J,, its distance from 21° is > r„+i, which is a contradiction. 2. If '^' is not enumerable, there exists a first number a in Z^ or Z^ such that 21° is perfect. This is a corollary of 1. 3. if 21 is complete and not enumerable, there exists a first number a in Zj -I- Zg such that 21" is perfect. 4. if 21 is complete, 21 = S + ^ • where @ is enumerable, and '^ is perfect. If 21 is enumerable, ^ = 0. COMPLETE SETS 337 • Complete Sets 321. Let us study now some of the properties of complete point sets. We begin by considering limited perfect rectilinear sets. Let 21 be such a set. It has a first point a and a last point b. It therefore lies in the interval I=(a, b). If 21 is pantactic in any partial interval J= (a, ^8) of /, 21 embraces all the points of J, since 21 is perfect. Let us therefore suppose that 21 is apantactic in I. An example of such sets is the Cantor set of I, 272. Let i) = I S J be a set of intervals no two of which have a point in common. We say D is pantactic in an interval I, when I con- tains no interval which does not contain some interval 8, or at least a part of some S. It is separated when no two of its intervals have a point in common. 322. 1. livery limited rectilinear apantactic perfect set 21 deter- mines an enumerable pantactic set of separated intervals D= |Sj, whose end points alone lie in 21. For let 21 lie in /=(c6, ^S), where a, /S are the first and last points of 21. Let S3 = /— 21. Each point J of 58 falls in some in- terval S whose end points lie in 21. For otherwise we could approach b as near as we chose, ranging over a set of points of 21. But then J is a point of 21, as this is perfect. Let us therefore take these intervals as large as possible and call them S. The intervals S are pantactic in I, for otherwise 21 could not be apantactic. They are enumerable, for but a finite set can have lengths > I/n + 1 and < I/n, w = 1, 2 ••• It is separated, since 21 contains no isolated points. 2. The set of intervals D = {Bl just considered are said to be adjoint to 21, or determined by 21, or belonging to 21. 323. Let 21 be an apantactic limited rectilinear perfect point set, to which belongs the set of intervals D= {S\. Then 21 is formed of the end points JS= \e] of these intervals, and their limiting points JE' . For we have just seen that the end points e belong to 2t. More- over, 21 being perfect, JS' must be a part of 21. 338 POINT SETS 31 contains no other points. For let a be a point of 21 not in JE, E' . Let a be another point of 21. In the interval (a, a) lies an end point e of some interval of D. In the interval (a, e) lies an- other end point e^ In the interval (a, e{) lies another end point gg, etc. The set of points e, Sj, e^--- =a. Hence a lies in ^', which is a contradiction. 324. Conversely, the end 'points E=\e\ and the limiting points of the end points of a pantactic enumerable set of separated intervals D= \Bl form a perfect apantactic set 21. For in the first place, 21 is complete, since 21 = (^, JP')- 21 can contain no isolated points, since the intervals S are separated. Hence 21 is perfect. It is apantactic, since otherwise 21 would em- brace all the points of some interval, which is impossible, as D is pantactic. 325. Since the adjoint set of intervals D = \h\ is enumerable, it can be arranged in a 1, 2, 3, ••• order according to size as follows. Let S be the largest interval, or if several are equally large, one of them. The interval 8 causes J to fall into two other intervals. The interval to the left of S, call i^, that to the right of S, call Jj. The largest interval in ij,, call 8q, that in Jj, call Sj. In this way we may continue without end, getting a sequence of intervals ^' ^0' ^1' ^00' V' ^10' ^n"- 0- and a similar series of intervals J-i Jfl' A' -'oo' -'oi ■■■ The lengths of the intervals in 1) form a monotone decreasing sequence which = 0. If V denote a complex of indices yV • • • 326. 1. The cardinal number of every perfect limited rectilinear point set 21 is c. For if 21 is not apantactic, it embraces all the points of some in- terval, and hence Card 21 = c Let it be therefore apantactic. COMPLETE SETS 339 Let Z>= JS^I be its»adjoint set of intervals, arranged as in 325. Let g be the Cantor set of I, 272. Let its adjoint set of intervals be.ir= J?7,|, arranged also as in 325. If we set S„~ ■>;„, we have B^H. Hence Card 21 = Card g. But Card S = c by 244, 4. 2. The cardinal number of every limited rectilinear complete set 21 is either e or c. For we have seen, 320, 4, that 21 = @ + ^, ^50, where (S is enumerable and ^ is perfect, If ^ = 0, Card 21 = e. If ^ > 0, Card 21 = c. For Card 21 = Card @ + Card 'ip = e + c = c. 327. The cardinal number of every limited complete set 91 in 9?„ is either e or c. It is c, if 21 has a perfect component. The proof may be made by induction. For simplicity take m = 2. By a transformation of space [242], we may bring 21 into a unit square S. Let us therefore suppose 31 were in ^S* originally. Then Card 21 < c by 247, 2. Let S be the projection of 21 on one of the sides of S, and SS the points of 21 lying on a parallel to the other side passing through a point of g. If S3 has a perfect component. Card S3 = c, and hence Card 21 = c. If S3 does not have a perfect component, the cardinal number of each S3 is e. Now S is complete by I, 717, 4. Hence if S contains a perfect component, Card S = c, otherwise Card g = e. In the first case Card 21 = c, in the second it is e. 328. 1. Let 21 be a complete set lying within the cube Q. Let i>i > i>2 > ••• denote a set of superimposed cubical divisions of Q of norms = 0. Let d^ be the set of those cubes of D^ containing no point of 21. Let d^ be the set of those cubes of D^ not in dj, which contain no point of 21. In this way we may continue. Let SB = \d„l. Then every point of ^ = Q — 21 lies in S3. For 21 being 340 POINT SETS complete, any point a of A is an inner point of A. Hence D/^a) lies in A, for some p sufficiently small. Hence a lies in some d^. We have thus the result : Any limited complete set is uniquely determined hy an enumerable set of cubes { d„ j , each of which is exterior to it. We may call SS = \dJi the border of 21, and the cells d„, border cells. 2. The totality of all limited perfect or complete sets has the car- dinal number c. For any limited complete set S is completely determined by its border \d„\. The totality of such sets has a cardinal number < c'= c. Hence Card fSj < c- Since among the sets S is a c-set of segments, Card g > c- 329. If 21. denote the isolated points of 21, and 21^ its proper limiting points, we may write 2l = 2I. + 2lv Similarly we have 21. =2Ix. +2lx», 2Ia. = 21... + 2(^3, etc. We thus have 21 = 21. + 21., + 21.,. + ••• + 2I.-I. + 2t.n. At the end of each step, certain points of 21 are sifted out. They may be con'sidered as adhering loosely to 21, while the part which remains may be regarded as cohering more closely to the set. Thus we may call 21."-!., the »i* adherent, and 21." the w'" coherent. If the n'* coherent is 0, 21 is enumerable. If the above process does not stop after a finite number of steps, let 2l„ = i>H2f., 21.^, 2I..-)- If 2l„ > 0, we call it the coherent of order a. Then obviously gj = s 2I.». + 2I„ . We may now sift 21a, as we did 21. COMPLETE SETS 341 If a is a limitary ifumber, defined by «i< «2 < «3 ••• =«' we set 3L = i>t);2I;,"»f and call it, when it exists, the coherent of order a. Thus we can write 3t = 22r,a^ + 2l,3 «=1, 2,... 0. For if not, 21 = 3, and 3^ is enumerable. 331. S = a'. (1 For let 2> be a cubical division of space. As usual let denote those cells of D containing a point of 21, 21' respectively. The cells of Hi not in 21/) will be adjacent to those of 21^, and 342 POINT SETS these may be consolidated with the cells of i>, forming a new di- vision A of norm S which in general will not be cubical. Then The last term is formed of cells that contain only a finite number of points of 21. These cells may be subdivided, forming a new division JE such that in Ms = % + S^* (2 the last term is < e/3- Now if B is sufficiently small, Hence from 2), 3) we have 1). 332. If^>0, Card 31 = c. For let S3 denote the sifted set of 21 [I, 712]. Then $8 is per- fect. Hence Card S3 = c, hence Card 21 = c. 333. Let 21= Joj, where each a is metric and not discrete. If no two of the as have more than their frontiers in common, 21 is an enumerable set in the a''s. 21 mai/ he unlimited. Let us first suppose that 21 lies in a cube Q. Let a denote on removing its proper frontier points. Then no two of the a's have a point in common. Let ?i>?2>---=0' where the first term q-^ = Q. There can be but a finite number of sets a, such that their contents lie between two successive ^''s. For if a, we have H ' «..•••>?, But the sum on the left is < Q, for any n. As n may = oo, this makes Q = oo, which is absurd. If 21 is not limited, we may effect a cubical division of 9?^. This in general will split some of the a's into smaller sets b. In each cube of this division there is but an enumerable set of the b's by what has just been proved. CHAPTER XI MEASURE Upper Measure 334. 1. Let 21 be a limited point set. An eniimerable set of metric sets D= jcZJ, such that each point of 21 lies in some d^, is called an enclosure of 21. If each point of 21 lies within some d^ , J) is called an outer enclosure. The sets d^ are called cells. To each enclosure corresponds the finite or infinite series td, (1 which may or may not converge. In any case the minimum of all the numbers 1) is finite and < 0. For let A be a cubical division of space, 21^ is obviously an enclosure and the corresponding sum 1) is also 21a > since we have agreed to read this last symbol either as a point set or as its content. with respect to the class of all possible enclosures D, the upper measure o/2l, and write |=Meas2I = Min2£?,. D 2. The minimum of the sums 1) is the same when we restrict our- selves to the class of all outer enclosures. For let 2)= ld^\ be any enclosure. For each d^, there exists a cubical division of space such that those of its cells, call them d,^, containing points of d^ have a content differing from d^ by < h; • Obviously the cells \d^-\ form an outer enclosure of 21, and 343 344 MEASURE As e is small at pleasure, Min 2d, over the class of outer en- closures = Min Si, over the class of all enclosures. 3. Two metric sets whose common points lie on their frontiers are called non-overlapping. The enclosure D = 2c?, is called non- overlapping, when any two of its cells are non-overlapping. Any enclosure D may he replaced by a non-overlapping enclosure. For let U{d^., d^ = c^j -|- eg, Z7(t?i, d^, rfg) z=d-^-\- e^-\- eg, U(d^ d^ = ld^\, H= \ej be two non-overlapping enclosures of 21. Let B^, = Dv(d^,eJ. Then A=SS,.|, f, « = 1, 2, ... is a non-overlapping enclosure of 21. For S„ is metric by 22, 2. Two of the S's are obviously non- overlapping. Each point of 21 lies in some d, and in some e,, hence a lies in S„. 2. We say A is the divisor of the enclosures D, E. 336. //•2IS; hence afoHiori Hence _ ^ «>93. But := ^ 21>S8, by 336, >S-e, by2). (3 On the other hand, ^ _ 2r<2la<2I + e, by 2). (4 From 3), 4) we have 1), since e is arbitrarily small. 338. If 31 is complete, = _ For by definition =. 2l=Min2(f„ with respect to all outer enclosures D = \d^\. But 21 being com- plete, we can replace ^ by a finite set of cells F=\f\ lying in D, such that F is an enclosure of 31. Finally the enclosure F can be replaced by a non-overlapping enclosure G = \g^\ by 334, 3. Thus 21 = Min 2^„ with respect to the class of enclosures Cr. But this minimum value is also 31 by 2, 8. 346 MEASURE 339. Let the limited set 21 = |2I„| he the union of a finite or infinite enumerable set of sets 2l„. Then i<2i„. .(1 For to each 2I„ corresponds an enclosure i)„ = f (f„. J such that Idn, < 2l„ + — , e > 0, arbitrarily small. But the cells of all the enclosures 2>„, also form an enclosure. Hence _ /_ \ 2l<2J„.<2(2r„ + |;j This gives 1), as e is small at pleasure. 340. Let 21 lie in the metric set Tl. Let A = W — % he the complementary set. Then f+I>m. For from m = '& + A, follows ^ ^ = a«<2l + ^, by 339. But 1 ^ m = m, by 337. 341. J/ 21 = 53 + S, and S3, E are exterior to each other, i = i + f . (1 For, if any enclosure I)=\d^\ of 21 embraces a cell containing a point of SB and E, it may be split up into two metric cells d[, d[', each containing points of 93 only, or of E only. Then d, = d[ + d['. Tlius we may suppose the cells of D embrace only cells D' = ld[\ containing no point of ®, and cells i)" = fc?('| con- taining no point of 93. Then 25. = 25; + 2<'. (2 UPPER MEASURE 347 By properly chodling D, we may crowd the sum on the left down toward its minimum. Now the class of enclosures B' is included in the class of all enclosures of 33, and a similar remark holds for Z>". Thus from 2) follows that i>i+s. This with 339 gives 1). 342. 7/ 1 = S + sot, 5K heing metric, l = i + ^. (1 For let 2) be a cubical division of norm d. Let n denote points of SJi in the cells containing points of Front $0?. Let m denote the other points of SSt- Then m and 93 are exterior to each other, and by 337 and 341, Mea;s(93 + m) = S + m. ^s 2l = 93 + m + n, Meas (93 + tn) < 1 by 336. ^^^"^ . S is arbitrarily small. 343. 1. Let 21 lie in the metric set 93, and also in the metric set ®- ^'* jB = 93-21 , C=e-2l. For let © = i)t>(93, S) , 93 = © + 93i , e = 2) + ei 5 = 93i + i> , c = ei + i>. 348 MEASURE Thus _ _ _ g _ ^ = § + Sj _ (gj + 5) = S - J. 2. If SI<58, the complement of 21 with respect to 93 will frequently be denoted by the corresponding English letter. Thus A = C(n'), Mod SB Loiuer Measure 344. 1. We are now in position to define the notion of lower measure. Let 21 lie in a metric set 2)?. The complementary set ^ = SD? — 21 has an upper measure A. We say now that 9K — -4 is the lower measure of 21, and write 2I = Meas 21 = ^-3. By 343 this definition is independent of the set 9K chosen. When l^gj we say 21 is measurable, and write A set whose measure is is called a null set. 2. Let H= [e,] he an enclosure of A. Then 21 = Max (§J - Si,) , with respect to the class of all enclosures E. 3. If @ = {e.} is an enclosure of 21, the enclosures E and @ may obviously, without loss of generality, be restricted to metric cells which contain no points not in 3)?. If this is the case, and if (S, ^are each non-overlapping, we shall say they are normal enclosures. If @, 55 are two normal enclosures of a set 21, obviously their divisor is also normal. LOWER MEASURE 345. 1. • 2r>o. For let 31 lie in the metric set 9W. Then n = m-I. But by 336, A<§i, hence m-I>o. 2. n^ Hence 2l = ^-IA. Hence, -^ _ ,-. = m-A 5. Hence, by 336, = — Thus, ^ = .^ = m-A 21, and -4 = 9JJ — 21. It is sufficient, for then 1) shows that But the right side is by definition 21 ; hence 21 = 21. It is necessary as 347 shows. 350. Let 21 = Ja„J be the union of an enumerable set of non- overlapping metric sets. Then 21 is measurable, and S = So„. (1 LOWER MEASURE 351 Let S denote tha* infinite series on the right of 1). As usual let S„ denote the sum of the first n terms. Let 2l„ = (tti, ••• a„). Then 2l„ < 31 and by 336, ^« = 'S'„ SI, and 2l„ + ^„ = 3JJ as usual. Then ^ ^ ^ 2t„ + A„ = m. (6 But A< A„ , hence Z < A„. Thus 6) gives = ^ ^ A + n, 21. Let A, B, be the comple- ments of 21, 33, S, with reference to 3Jf. Let Il=\e^\ , F= ]fj be normal enclosures of B, C. Let dmn = Dvie^, /„), and Z> = \d„„\ the divisor of E, F. 352 " MEASURE As all the points of A are in 5, and also in C, they are in both E and F^ and hence in the cells of i), which thus forms a normal enclosure of A. Let Let us set , „ n , ■, Then by 360, ^ ' ^ 7m = 2 is <. Z. As in defining Cantor's set, I, 272, let us place a black interval of length l^ in the middle of SI. In a similar manner let us place in each of the re- maining or white intervals, a black interval, whose total lengths = l^. Let us continue in this way; we get an enumerable set of black intervals 33i and obviously 58 =X. If we omit the end points from each of the black intervals we get a set SB*, and obviously SB* = X. The set § = 21 - 0, and whose lower content is 0. 2. Within each of the black intervals let us put a set of points having the end points for its first derivative. The totality of these points form an isolated set ^ and 3' = §• But by 331, 3 = 3'- If iiow § is not discrete, ^ is not. We have thus the theorem : There exist isolated point sets which are not discrete. LOWER MEASURE 355 3. It is easy to exWnd Harnack sets to 9t„ . For example, in 3J2' let S be the unit square. On two of its adjacent sides let us place congruent Harnack sets §. We now draw lines through the end points of the black intervals parallel to the sides. There results an enumerable set of black squares ® = lS„]. The sides of the squares @ and their limiting points form obviously an apan tactic perfect set S. Let 9,0, af + aj+ ••• = m be a series whose sum < m< 1. We can choose § such that the square corresponding to its larg- est black interval has the area aj ; the four squares corresponding to the next two largest black intervals have the total area a^, etc. He°«e 1 = 1 _^ = ^. 355. 1. If S = \e^l is an enclosure of SI such that 2e„-i there exists an S such that f < e, then 21 is measurable. For let g = e + f. Then e < 21. Hence e < H by 348. But iy(2ti, 2I2) is measurable by 358. ^^* 2I, = 3) + 0i , 2l2 = 3)-|-a2. Then Oj, Oj are measurable by 352, 2. ^^ U = (2li,2l2) = 35+01 + 02, U is measurable. As U and S81 are measurable, so is S2. In a similar manner we show that Sg, S4 ■•• are measurable. As 2l = 293„, 21 is measurable by 358, 1, and the relation 1) holds by the same theorem. 360. Let 211 < STj < ••• 5e « s«* °-^ measurable aggregates whose union 21 is limited. Then 21 is measurable, and S=limS„. 358 MEASURE ^°'^^^ 02 = 2l2-SIi , a3 = 2ls-Sl2... For uniformity let us set a^ = 21. Then As each a„ is measurable ^ = lim(ai+ •■• +a„) = Umi„. 361. Let SIj, ^••- be measurable and their union SI limited. If S) = Dv [2I„| > 0, it is measurable. For let 2t lie in the metric set SD?; let S) + i> = 9w , 2r„+A=a« as usual. Now 35 denoting the points common to all the 3l„, no point of D can lie in all of the 2l„, hence it lies in some one or more of the A„. Thus i). (2 From 1), 2) we have D= ^A \ As each A„ is measurable, so is D. Hence S) is. 362. -Z/ 2li>2l2> ••• i« an enumerable set of measurable aggre- gates, their divisor 35 is measurable, and S) = lim n ' n=oo For as usual let D, A„ be tlie complements of 3), 2l„ with respect to some metric set SD?. Then 2>=f^l , ^„ 0, there are standard e-enclosures of any limited set 21. For let S = {e„| be any ^/-enclosure of 21. Then 2e„-i<7?. (2 Each e„ being metric, may be enclosed in the cells of a finite standard outer enclosure #„ , such that -Fn-e„A^. (g = m-c=m- c^|. (3 For let Ad be an e-outer enclosure of A, formed of standard non-overlapping cells all of which, after having discarded certain parts, lie in iDZ. LOWER MEASURE 361 Let ^ ^ = m-A^ + %, (4 where g denotes the frontier points of A^ lying in 21. Obviously K is complete. Since each face of i) is a null set, g is a null set. Thus each set on the right of 4) is measurable, hence i = m-Ao + § = m-A-e' , 0f = 1 > 21 - e, from which follows 3), since e is small at pleasure. 365. 1. 7^21 is complete, it is measurable, and S = i. For by 364, On the other hand, 1 = 1, by 338. 2. Xiet 33 be any measurable set in the limited set 21. Then I = Max S. (1 For |>§ = 5. Hence, |>MaxS. (2 But the class of measurable components of 21 embraces the ■ class of complete components S, since each 6 is measurable by 1. Thus Max»>Max(f. (3 From 2), 3) we have 1), on using 364. 366. Van VTeck Sets. Let @ denote the unit interval (0, 1), whose middle point call M. Let 3 denote the irrational points of (g. Let the division i)„, w = 1, 2, ••• divide @ into equal intervals S„ of length 1/2"- 362 MEASURE We throw the points 3 into two classes 21 = |a|, 33 = [bl having the following properties : 1° To each a corresponds a point b symmetrical with respect to M, and conversely. 2° If a falls in the segment S of D„, each of the" other seg- ments S' of i>„ shall contain a point a' of 21 such that a' is situated in S' as a is situated in S. 3° Each B of D„ shall contain a point a' of 21 such that it is situated in B, as any given point a of 21 is situated in (g. 4° 21 shall contain a point a situated in S as any given point a' of 21 is in any B„. The 1° condition states that 21 goes over into SB on rotating @ about M. The 2° condition states that 21 falls into n=l, 2, 2", 2^, ••• congruent subsets. The 3° condition states that the subset 2t„ of 21 in S„ goes over into 21 on stretching it in the ratio 2" : 1. The condition 4° states that 21 goes over into 2l„ on contracting it in the ratio 1 : 2". We show now that 21, and therefore 93 are not measurable. In the first place, we note that _ _ 21 = S, by 1°. As 3 = 21 + 95, if 21 or 93 were measurable, the other would be, and «, «, 2l = 93 = f Thus if we show 21 or 93 = 1, neither 21 nor 93 is measurable. We show this by proving that if 21 = a< 1, then 93 is a measurable set, and 93 = 1. But when 93 is measurable, 93 = ^ as we saw, and we are led to a contradiction. Let € = €j + e^+ ••• be a positive term series whose sum e is small at pleasure. Let @j = Je„} be a non-overlapping Cj-enclosure of 21, lying in (5. Then (gl = 2e„ = a + ei = ai , 0] — a — Cj. LOWER MEASURE 363 Each interval e„4jontains one or more intervals ri„i, r]„^, ... of some D„ such that '^Vnm = e„~a„ , 0(l-a)|l + « + a2+ ...|-2e„ >l-€. As e is small at pleasure, S3 = 1. 367. (TF. JT. Young.) Let 2ii , sr^ , 2I3... (1 he an infinite enumerable set of point sets whose union 21 is limited. Let ^„>«>0 , w = l, 2... Then there exists a set of points each of which belongs to an infinity of the sets 1) and of lower measure > a. 364 MEASURE For by 365, 2, there exists in the sets 1), measurable sets Si , ^2 , £3- (2 each of whose measures 6„ > a. Let us consider the first n of these sets, viz. : rr nr nr .c, The points common to any two of the sets 3) form a measurable set ©I, by 358, 1. Hence the union Ei„ = f!D«} is measurable, by 359. The difference of one of the sets 3), as Ej and Dv(^^^, £j„), is a measurable set Cj which contains no point in common with the remaining sets of 3). Moreover Cj > a - fii„. In the same way we may reason with the other sets Sg, E3 ■•• of 3). Thus 21 contains n measurable sets Cj, Cj ••• c„ no two of which have a common point. Hence c = d + ••• + c„ is a measurable set and l>c>«(a-6i„). The first and last members give iin > « - Thus however small a > may be, there exists a fi such that ti,. (l-|)«. (4 Let us now group the sets 2) in sets of /i. These sets give rise to a sequence of measurable sets Sj^ , 62,1 , Esjit ••■ (5 such that the points of each set in 5) belong to at least two of the sets 1) and such that the measure of each is > the right side of 4). We may now reason on the sets 5) as we did on those in 2). We would thus be led to a sequence of measurable sets ASSOCIATE SETS 365 such that the points '«f each set in 6) lie in at least two of the sets 5), and hence in at least 2^ of the sets 1), and such that their measures are. / „ \ / , \ >(i-o)i-^V>a-o«- 2A 22 In this way we may continue indefinitely. Let now SB^ be the union of all the points of 21, common to at least two of the sets 1). Let 352 ^^ ^^^ union of the points of 21 common to at least 2^ of the sets 1), etc. In this way we get the sequence Si>932^ - each of which contains a measurable set whose measure is >(1 - e)«. We have now only to apply 25 and 364. 368. As corollaries of 367 we have: 1. Let Qj, Qj ■■■ ^^ ^^ infinite enumerable set of non-overlapping cubes whose union is limited. Let each Q„ > a > 0. Then there exists a set of points b whose cardinal number is C, l^ing in an infin- ity of the Q„ and such that b > a. 2. (^Arzeld.} Let y^ , y^ ■■■ =r). On each line y^ there exists an enumerable set of intervals of length 8„. Should the number of inter- vals v^ on the lines y„ be finite, let i'„ = oo. In any case 8„ > a > 0, w = 1, 2, ••• and the projections of these intervals lie in 21 = (a, b). Then there exists at least one point x = ^ in 21, such that the ordinate through | is cut by an infinity of these intervals. Associate Sets 369. 1. Let ei>e2>e3 ••• =0. (1 Let g„ be a standard e„-enclosure of 2r„. If the cells of @„+i lie in (g„, we write @i > @2 > '•• (^ and call 2) a standard sequence of enclosures belonging to 1). Obviously such sequences exist. The set n,=i>v\<&,l is called an outer associated set of 21. Obviously 2l<2le- 366 MEASURE 2. EcLch outer associated set 21^ is measurable, and f=i,= lim|„. (1 For each @„ is measurable; hence 21^ is measurable by 362, and 2l, = limg„ ^limCi + O, 02I; Thus a + Sa>i> (2 byl). Also S,=a + S=i by 369, 2. This with 2) gives Hence ©„ = ©. (3 But 35>S)„ + ©6. This with 3) gives ©6=0. In a similar manner we find that ®„ = 0. Hence S) is a null set by 3). 368 MEASURE 373. 1. -Zy 21, 93 are, separated, then '^ = Dv {%, SB) is a null set. For S), = Bv (31,, 93,) is a null set by 372. But © < ©,. 2. Let 21, 93 be the Van Vleck sets in 366. We saw there that 1 = i = 1. Then by 369, 2, S, = 5, = 1. The divisor of 21,, 93, is not a null set. Hence by 372, 2t, 93 are not separated. Thus the condition that ^ be a null set is necessary, but not sufficient.. 374. 1. Let J2l„(, i93„i be separated divisions of 21. Let g.» = i>t)(2li, 93«). Then jS.,| is a separated division ofSH also. We have to show there exists a null enclosure of any two of the sets S«, S„„ . Now S„ lies in 21, and 93, ; also 6„„ lies in 2I;„, 93„. By hypothesis there exists a null enclosure @ of 21., 21^ ! and a null enclosure g of 93,, 93„. Then ® = i>w(®, g) is a null enclosure of 2I„ 2L and of 93., 93„. Thus those cells of ®, call them ®„, con- taining points of both 2It, 2l„ form a null set; and those of its cells @i,, containing points of both 93,, 93„ also form a null set. Let G-=^\g\ denote the cells of @ that contain points of both S«, Sm„. Then a cell g contains points of 2li 2l„ 93, 93„. Thus g lies in ©„ or ©,,. Thus in either case G^ is a null set. Hence [S,,} form a separated division of 21. 2. Let i) be a separated division of 21 into the cells dj, d^--- Let E be another separated division of 21 into the cells ej, eg "• We have seen that F = {/.,5 where /„ = Dv(d^, e,) is also a sepa- rated division of 21. We shall say that F is obtained by superim- posing U on D OT D on E, and write F=I) + E= F+ D. 3. Let ^ be a separated division of the separated component 93 of 21, while ^ is a separated division of 21. If d^ is a cell of D, e, a cell of E, and d,^ = Dv(^d^, e,), then («. = «„<,...) +8.. Thus superposing E on D causes each cell d^ to fall into sepa- rated cells d,,, djj ••■ S,. The union of all these cells, arising from different d,, gives a separated division of 21 which we also denote hy D + E. 375. Let J2l„| be a separated division of 21. Let 93 < 21, and let 93„ denote the points of S in 2l„. Then 593„| is a separated division of^. SEPARATED SETS 369 For let S) be a nwU enclosure of 3I„, ?I„. Let 3D^ denote the cells of 35 containing points of both ^^, 2l„. Let g denote the cells of 35 containing points of S3 ; let (g„_ j denote the cells con- taining points of both 33^ , S3„ . Then So* < 2)^- As 3)ai, is a null set, so is (i^. 376. 1. Let 21 = (58, S) he a separated division of^. Then S = i + f . (1 For let 6j > €2 > ••• = 0. There exist 6„-nieasurable enclosures of 2t, 93, S; call them respectively A^, -B„, C\. Then @„ = J.„-|- -Sn + C'n is an e„-enclosure of* 21, 58, 6 simultaneously. Since 93, S are separated, there exist enclosures £, C of 93, E such that those cells oi D = B + containing points of both 93 and (S form a null set. Let us now superpose D on @„ getting an e„-enclosure -B„= 5e„si of 21, 93, £ simultaneously. Let ej„ denote the cells of JE^ containing points of 93 alone ; e„„ those cells containing only points of S ; and e^^ those cells containing points of both 93, 6. Then 2e„, = S^j, + 2e,„ + 2?6, . (2 s As Sejc = 0, we see that as w = t», Hence passing to the limit w= 00, in 2) we get 1). 2. Xe« 21= 193„| ie a separated division of limited 21. Then l = 2i„. (1 For in the first place, the series £ = 2i„ (2 is convergent. In fact let 2l„ = (93i, 932 ••• ^n)- Then 2l„ < 21, and hence !„ < 1. 370 MEASURE On the other hand, by 1 the sum of the first n terms of the series 2). Thus and hence B is convergent by 80, 4. Thus -B%. The last two relations give 1). CHAPTER XII LEBESGUE INTEGRALS General Theory 377. In the foregoing chapters we have developed a theory of integration which rests on the notion of content. In this chapter we propose to develop a theory of integration due to Lebesgue, which rests on the notion of measure. The presentation here given differs considerably from that of Lebesgue. As the reader will see, the theory of Lebesgue integrals as here presented differs from that of the theory of ordinary integrals only in employing an infinite number of cells instead of a finite number. 378. In the following we shall suppose the field of integration 21 to be limited, as also the integrand SI lies in 3?^ and for brevity we set /(a;) =f(x-^ ■■• x^). Let us effect a separated division of SI into cells Sj, Sg •••■ I^ each cell h, lies in a cube of side d, we shall say 2) is a separated division of norm d. As before, let M, = Maxf , m. = Min/ , (o,= Oscf=M,-m. in S.. the summation extending over all the cells of 31, are called the upper and lower sums off over 31 with respect to B. The sum „ j, - f is called the oscillatory sum with respect to B. 379 . Ifm = Min f, M=Ma.xf in St, then mf'„ + Z>'„'. Obviously 2), 3) hold simultaneously for the sequence ID„\. 2. The sequence 1) is called an extremal sequence. 3. Let ]Dn} ^fi <«w extremal sequence, and E any separated divi- sion of 21. Let U„ = I>„ + U. Then H-^, -Ej ••• *s <*w extremal sequence also. 384. Let f be L-integrable in f[. Then for any extremal sequence r/=limS/(f)<. (1 where d^ are the cells of D„, and |, any point ofSH in d^. ^'"'' s,^<^fiOl„ j , lira fij /= 0; and it is sufficient if there exists a sequence of superimposed separated divisions {E^l, such that lim €IeJ = 0. 7l=a> It is necessary. For / = lim Sjy^ , I = lim Sj)^. As/ is i-integrable, 0= f- f= lim (,Sj,^ - Sj,J = lim n^J. It is sufficient. For _ ^^„ 0, there exists a separated division D of ^, for "^^'"^ nj,f some m. ■ Thus we may take 2>„ for D. GENERAL THEORY 377 It is sufficient. Bor let ej >e2 > ••• =0. Let \D„\ be an extremal seqaence for which o3 ■•• Then {A„j is a set of superimposed separated divisions, and obviously Hence / is i-iutegrable by 386. 388. In order that f he L-integrable, it is necessary and sufficient that, for each pair of positive numbers a, a there exists a separated division D of 21, such that if ri^, i)^, ■•■ are those cells in which Oscf> CO, then 2^^ < 0-. (1 It is necessary. For by 387 there exists a separated division D = 1 81} for which Oj)/ = Sco^S, < (na. (2 If 0^, ^2 ••• denote the cells of B in which Osc/ < w, D,j)f = "Ldjji, + Scfli^i > (oLij, . (3 This in 2) gives 1). It is sufficient. For taking e > small at pleasure, let us then take o" = , o> = —^1 (4 2Xi 22f where D, = Osc /in 21. From 1), 3), and 4) we have, since m, < XI, £lj)f< 2Ilf. + ^(ofi, < o-Xl + 20)^, < o-fl + tot = e. We now apply 887. 389. 1. If f is L-integrable in 21, it is in S3 < 21. For let si)„J be an extremal sequence of /relative to 21- Then by 386, Ii^„/=0. (1 378 LEBESGUE INTEGRALS Bat the sequence {D„{ defines a sequence of superposed sepa- rated divisions of 93, which we denote by {J^„|. Obviously Hence by 1), and / is i-integrable in 58 by 386. 2. If / is L-integrahle in 21, so is \f\. The proof is analogous to I, 507, using an extremal sequence for/. 390. 1. Let jSInf he a separated division of ?I into a finite or in- finite number of subsets. Letf be limited in 21. Then ff=ff+ff+- (1 Xa gt^ai Xaia For let us 1° suppose that the subsets SIj •■• 21, are finite in num- ber. Let \D^\ be an extremal sequence of/ relative to 21, and ID„„\ an extremal sequence relative to 2l„. Let Then \U„\ is an extremal sequence of /relative to 21, and also relative to each 21™- Now _^8r,£„=f'ai, £„+ ••• +^a„£„- Letting n = cc, we get 1), for this case. Let now r be infinite. We have i=fi™. (2 ^^* 93„ = (2Ii-2r„) , 6„ = 2l-33n- Then 93„, E„ form a separated division of 21, and f=C+S„- If K is taken large enough, 2) shows that S„<-^ , n>v , il!f=Max|/| in 21. where c is a constant. fcf=cff. GENERAL THEORY 379 Thus by case 1°,'« ff= ff+ ff = f+ - + /"+€', (3 4!%! 4::^n where by 885, 2 I e' I < Mi„ < e , n>v. Thus 1) follows from 3) in this case. 2. Let J2I„} be a separated division ofH. Then iff is Lintegrable in 21, or if it is in each 2l„, and limited in 21. 391. 1. Letf = g in% except at the points of a null set St. Then T* /» f= 9- (1 ^°''^^* 21 = 33+91. Then f /= f /+ ff= ff (2 Xa gtr© ■*::!» 4i^ Similarly 7=^,? ^, But/ = ^ in 93. Thus 2), 3) give 1). 392. 1. //OO; ['■^='l^- Ifor.l be an extremal sequence common to both/ and g. Letting w = oo , we get 1). We consider now the general case. Let 21 = 93 + St- Then ff=ff ' fff=fs^ since But in SS,fOinn. Then Mmg- ff< ff-g 0. We set 2l = e + 5«. Then / is i-integrable in £ by case 1°. It is i-integrable in 5R by 880, 2. Then it is i-integrable in 21 by 390, l. 2. If / is i-integrable in % we cannot say that the points of discontinuity of/ form a null set. Example. Let/= 1 at the irrational points ^^^ in 21 = (0, 1) ; = at the other points 9?, in 21. Then each point of 21 is a point of discontinuity. But here since 9? is a null set. Thus/ is i-integrable. 400. Iff(x^ ••• x^) has limited variation in 21, it is L-integrahle. For let Z) be a cubical division of space of norm d. Then by I, 709, there exists a fixed number F, such that for any B. Let w, and let the number of these cells be v. Let ?;, denote the points of 21 in d[ . Then 384 LEBESGUE INTEGRALS Hence Thus V= ^ J-m.^ ^^^ 1, o\ 2'?.<''^"'<^;j^;;^ , by 2), <^<. , byl). Hence/ is i-integrable by 388. 401. Let ^=f,mn 0, w, = 0, and 3) holds. IfJf.<0, M^, 0, 4) still holds, since M = N^. Thus 2) holds in all these cases. Summing 2) gives 2w,^< C f<^Nfi^ for the division @,, since in a cell e of @, containing no point of 21, ^ = 0. Letting s = oo, we get 1), since the end members INTEGRAND SETS 386 On the 2° hypoth^s, f= c^+ r<^= C4>= ff, ■J^ss otat J^A oLa cL% since /or»w a nuZ2 set. For let 2r = 3 + 'iP, so that/= in 3. By 401, Let €j > 62 >•••== 0. Let ^„ denote the points of ip where />£„. Then r>r = 0, byl). Each ^„ is a null set. For _ ...,. = 0. Hence f „ = 0. Then ^ = l^„S= @2 > ■•• form a standard sequence of enclosures of O, such that S ^ F) r'2 Let us project each section of (g„ corresponding to a given value of x„+i on dim, and let 2l„ be their divisor. Then 2l„ > 21. Thus 60 bemL-integrable in '&. Then ff = % (1 where 3 is the integrand set corresponding tof. For let {Sj be a separated division 3 of 21. On each cell S, erect a cylinder g^ of height M, = Max /in S.. Then by 404, Let E= [Sj ; the S^ are separated. Hence, 6>0 being small at pleasure, for a properly chosen D. Thus 5< f/- (2 Similarly we find ff<^. (3 From 2), 3) follows 1). 406. Letf'>0 be L-integrdble over the measurable field 21. Then the corresponding integrand set 3 is measurable, and ^ /^ 3= f- (1 For by 2) in 405, S< ff. Using the notation of 405, let c„ be a cylinder erected on S„ of height m„ = Min / in S„. Let c = |c„| . Then c < 3, and hence £<§• (2 But 21 being measurable, each c„ is measurable, by 404. Hence c is by 859. Thus 2) gives c<3. (3 Now for a properly chosen D, ~e+ I f< SjwA = c. =0H 388 LEBESGUE INTEGRALS Hence r\ form a measurable set for each X, f is measurable in 21. P'or 21a^ having the same meaning as in 407, 2tA^ = 2Ix-StM- Each set on the right being measurable, so is 21a,i • MEASURABLE FUNCTIONS 389 409. 1. If f is mSdsurahle in ^, it is L-integrahle. For setting m = Min /, M= Max / in 21, let us effect a division I) of the interval g = (m, M) of norm d, by interpolating a finite number of points Let us call the resulting segments, as well as their lengths, Let 31^ denote the points of 21 in -which ™i-i small at pleasure, and then fix it. Let us denote by Sl„ the points of SI at whicli - e < FXx-) < 6. (2 Of course 3I„ may not exist. We are thus led in general to the «^*^ Sli , STa , %- (3 The complementary set A„ = 21 — 2l„ will denote the points where i s? ^ n i ^ /-a I F^(.x) \>€. (4 If now F is convergent at x, there exists a v such that this point ^^""^'^ 21. , 2I..1 , 21.+2- (5 The totality of the points of convergence forms a set which has this property : corresponding to each of its points x, there exists a V such that x lies in the set 5). A set having this property is called the semi- divisor of the sets 3), and is denoted by SdvfSlJ. Suppose now, on the other hand, that 1) does not converge at the point x in 21. Then there exists an infinite set of indices n^%^>--- we have n = © = S). (5 For denoting the complementary sets by the corresponding Roman letters, we have D = A^ + Dvi^^,A^) + Dvi%,As')+ - But Q has precisely the same expression. Thus O = SD, and hence by 4), © = S). 392 LEBESGUE INTEGRALS 4. Letnn + A„ = $i, n=l, 2, ••• Then Qdv{3l„i+SdvM„|=S3. For each point J of 93 lies either 1° only in a finite number of 2l„, or in none at all, or 2° in an infinite number of 2l„ . Ill the 1° case, b does not lie in §l„ Sl,+j ••• ; hence it lies in A„ J.,+1 ••• In the 2° case b lies obviously in Qdv |2t„|. 5. Zf 2lj, 2I2 ■■■ '^^*' measurable, and their union is limited, = QdvS2l„i , ® = Sdv{2l„| are measurable. For let ©„= i)v(Sr„, 2I„+j ••■) . Then © = SS)„J . But © is measurable, as each S)„ is. Thus Sdv \A„l is measur- able, and hence Q is by 4. 6. Let Q = Qdv J2I„} , each 2l„ being measurable, and their union limited. If there are an infinity of the 2l„, say %,.\- ; h<'2<- whose measure is > a, then ^ Q>a. (6 For let 93„= (21^., 3r.„^, ••■), then «„>«. Let ^ = J)v\^„l . As93„>S8„+i, 5 = limS„>a (7 by 362. As Q>S8 we have 6) at once, from 7). Limit Functions 411. Let ,■ J,. ^ J. \ J ^ N lim/(a;i ••• x^, t^ — t„) = (}>(z^ ... a;„), i=T as X ranges over 21, t finite or infinite. Let f be measurable in 21 and numerically e2> ••• ==0. Let S„^, denote the points of 21 where ^--enSo. Let S„ = Sdv {6„.| . Then93<6„. But the 6„, being measurable, S„ is by 410, 5. Finally 93 = -Z>« iS„| , and hence 53 is measurable. lim/(a;i ■■■ x^,t^--- t^') = {x^--- x^), for X in 21, and t finite or infinite. Let t', t" •■■ =r. Let each /, =/(a;, i^"') he measurable, and numerioally —fa+g,' Let ®, denote the points where Then for each e > 0, ^.^^ g^ ^ q_ ^j For by 411, (^ is measurable, hence g, is measurable in 21, hence ®, is measurable. Suppose now that 1) does not hold. Then im®, = Z>0. J — CO Then there are an infinity of the ®„ as ®„, ®,, ••• whose measures are >\>0. Then by 410, 6, the measure of ®=Qdv|®.i is >X. But this is not so, since/, = ^, at each point of 21. 413. 1. Let iim_^(2;i-a;„, «i-0=K^i-^m)» for X in 21, and t finite or infinite. ^^^ t\t"-=T. (1 Ifeachf=f(x, «''') is measurable, and numerically = \imff(x,ty (2 t=rj^^ 394 LEBESGUE INTEGRALS For set =f.+9., ^°dlet |^^|=ff,+ f9.- (3 JLii J^% J^u Let S, denote the points of 21, at which \ff.\>^l and let 33, + -B, = SI. Then 33,, B, are measurable, since ff, is. Thus by 390, ^ ^ ^ / 5'.= / 9.+ I 9.- Jbsn ots. =4/5. Hence , . /» i «. -=. «, «, / 9,\u}S„|, where each S„ is a standard enclosure, each of whose cells e„„ is rectangular. But the sections tnm(?') are also rectangular. Hence 2I,(x) = i>t>{e„„(a;)}, being the divisor of measurable sets, is measurable. 415. Let 2lj be an outer associated set of 21, both lying in the stand- ard cube Q. Then 2le(a;) is L-integrable in q, and i= fica;). (1 For let {@„} be a sequence of standard enclosures of 21, and g„=5e„„S. Then @„ = 2e„„ (2 771 ''"^ i„(a;) = 2e„„(a'). (3 Now e„„ being a standard cell, e„m(a;) has a constant value > for all X contained in the projection of c„m on q. It is thus con- tinuous in q except for a discrete set. It thus has an ^-integral, and This in 2) gives by 3). ^q , = 2 j e„„(a;) = fllU^-), by 413, 2, = finix-), (4 =i/q ITERATED INTEGRALS 397 On the other haiM, ©(a;) is a measurable function by 411. Also i = i, = lim i„ = Aim S„(a;), by 413, 1. (5 '^°^ i(a;) = lim§„(:r). Thus this in 5) gives 1). 416. Let SI lie in the standard cube Q. Let % be an inner asso- ciated set. Then fi^(x') is L-integrable in q, and %= r%ix-). ^'^^^ n.Cx) = 0{x} - A^Qxy Hence %(^x') is i-integrable in q, and f^ix} = fQ(x) - fl^x-) owq owq owq = 0-A , by 415, = i = a by 370, 2. 417. Let measurable 31 Ke in the standard cube Q. Then ^ /»= 21=/ 21(2;). (1 ^O'' 3I.(a;)<3l(a:) form a null set. 420. Let 21 = S3 • E 6e measurable. Let b denote the points of 93 /or wAicA the corresponding sections S are measurable. Then For by 419, S8=b + 9i, and S'l is a null set. Hence by 418, 421. Let f> iw 21. If the integrand set ^, corresponding to f be measurable, then f is L-integrable in 21, and 5 =/./■• For the points of ^ lying in an m + 1 way space 9i„^.i may be denoted by ^ = (y,... y„,.), where y = {yx — y^ ranges over 9J„, in which 21 lies. Thus 21 may be regarded as the projection of ^f on 9i„. To each point y ITERATED INTEGRALS 399 of 21 corresponds a section 0(«/), which for brevity may be denoted by M. Thus we may write ^ = 2l-t. As S is nothing but an ordinate through y of length /(«/), we have by 419, ^^ r% ^= / «= / /• 422. Let f he L-integrahle over the measurable field 21 = 33 • S- Let b denote those points of 33, for which f is L-integrahle over the corresponding sections E. Then ff = fff- (1 Moreover "iR = SQ — b is a null set. Let us 1° suppose f> 0. Then by 406, 3 is measurable and 3 = ff- (2 Let /S denote the points of S3 for which 3(a;) is measurable. Then by 420, 3f=j3(a;). (3 By 419, the points IP = S3 - /3 (4 form a null set. On the other hand, QQx) is the integrand set of/, for 21 (a;) = £. Hence by 421, for any x in y8, and y8 < b. (6 From 2), 3), 5) we have ff= f ff- o From 6) we have 9^ = «8-b 0, such that the auxiliary function 9=f+O>0, in 21. Then /, g are simultaneously i-integrable over any section 6. Thus by case 1° fcf+0-)^ f Cif+oy (8 Now ^ n n n ^ (f+0:,= f+ C= f+On, (9 C(if+C^= ff+ol. (10 By 418, 6 is i-integrable in 58, and hence in 6. Thus r fif+o)= f Cf+oct. (11 Xb o(/S Xb Xe Jjb As b differs from S3 by a null set, r^= rt = % (12 0. We begin by showing that the set of points 21^^ of 21 at which f>\, is measurable. Then by 408, 3, / is measurable in 21. Now/ being i-integrable in 31, its integrand set 3 is measur- able by 406. Let 3f;i be the section of 3 corresponding to a;„+i= X.. Then the projection of 3;^ on SR^ is 21^. Since 3 is measurable, the sections 3^ are measurable, except at most over a null set L of values of X, by 419. Thus there exists a sequence 1 a * " ^ X none of whose terms lies in L. Hence each 3a„ is measurable, and hence 91a„ is also. As 21a„^i < 31a„) each point of 31^ lies in ^ = Dvm,J, (1 so that 2j^ < 2). (2 On the other hand, each point c? of S) lies in 2lx- For if not, fid) X„ which contradicts 3). But not lying in 31^,, d cannot lie in 2), and this contradicts our hypothesis. Thus S)<21a. (4 From 2), 4) we have S) = 31a. But then from 1), 31a is measurable. Let the sign off be now unrestricted. 402 IMPROPER L-INTEGRALS Since /is limited, we may choose the constant (7, such that g = fCx)+C>0,inn. Then g is i-integrable, and hence, by case 1°, g is measurable. Hence/, differing only by a constant from g, is also measurable. 2. Let 31 be measurable. Iff is L-integrable in % it is measur- able in 21, and conversely. This follows from 1 and 409, i. 3. From 2 and 409, 3, we have at once the theorem : When the field of integration is measurable, an L-integrable func- tion is integrable in Lebesgues sense, and conversely; moreover, both have the same value. Remark. In the theory which has been developed in the fore- going pages, the reader will note that neither the field of integra- tion nor the integrand needs to be measurable. This is not so in Lebesgue's theory. In removing this restriction, we have been able to develop a theory entirely analogous to Riemann's theory of integration, and to extend this to a theory of upper and lower in- tegration. We have thus a perfect counterpart of the theory developed in Chapter XIII of vol. I. 4. Let 21 Je metric or complete.. J//(a;j ••• a;„j) is limited and M-integrable, it is a measurable function in 21. For by 381, 2, it is i-integrable. Also since 21 is metric or complete, 21 is measurable. We now apply 1. IMPROPER L-INTEGRALS Upper and Lower Integrals 425. 1. We propose now to consider the case that the integrand /(xj ••• a;^) is not limited in the limited field of integration 91. In chapter II we have treated this case for i2-integrals. To extend the definitions and theorems there given to i-integrals, we have in general only to replace metric or complete sets by measurable sets; discrete sets by null sets; unmixed sets by separated sets ; UPPER AND LOWER INTEGRALS 403 finite divisions by^iseparated divisions; sequences of superposed cubical divisions by extremal sequences; etc. As in 28 we may define an improper i-integral in any of the three ways there given, making such changes as just indicated. In the following we shall employ only the 3° Type of definition. To be explicit we define as follows : Let/(a;j .•• x^) be defined for each point of the limited set 31. Let 2l„p denote the points of 31 at which -(^0. (1 The limits ^ ^ lim / / , lim / / (2 in case they exist, we call the lower and upper (improper) L-in- teffrals, and denote them by In case the two limits 2) exist and are equal, we denote their common value by ^ and say/ is (improperly) L-integrahle in 81, etc. 2. In order to use the demonstrations of Chapter II without too much trouble, we introduce the term separated function. A func- tion / is such a function when the fields 2l„p defined by 1) are separated parts of 31. We have defined measurable functions in 407 in the case that / is limited in 31. We may extend it to unlimited functions by requiring that the fields 3la/3 are measurable however large a, yS are taken. This being so, we see that measurable functions are special cases of separated functions. In case the field 31 of integration is measurable, 3l„/3 is a meas- urable part of 31, if it is a separated part. From this follows the important result : Iff is a separated function in the measurable field 31. it is L-in- tegrable in each %^. 404 IMPROPER L-INTEGRALS From this follows also the theorem: Let f he a separated function in the measurable field SI. If either the lower or upper integral off over 21 is convergent, f is L-integrable in 21, and n /» / /= lim / /. 426. To illustrate how the theorems on improper ^-integrals give rise to analogous theorems on improper i-integrals, which may be demonstrated along the same lines as used in Chapter II, let us consider the analogue of 38, 2, viz. : If f is a separated function such that j f converges, so do l f. Let {^„| be an extremal sequence common to both Let e denote the cells of JS^ containing a point of '^^ ; e' those cells containing a point of ^p' ; S those cells containing a point of 2l,p but none of %: Then r cL%c = lim f SIT/ • e + ^Ml ■ e' + ^M', • Sj . In this manner we may continue using the proof of 38, and so establish our theorem. 427. As another illustration let us prove the theorem analogous to 46, viz. : Let 2li, 2I2, ••• 2l„ form a separated division of %. If f is a separated function in 21, then ff= ff+-+ ff, oka 4rai 4;an provided the integral on the left exists, or all the integrals on the right exist. For let 21,, op denote the points of 21.^ in 21,. Then by 390, 1, In this way we continue with the reasoning of 46. L-INTEGRALS 405 428. In this w^ we can proceed with the other theorems ; in each case the requisite modification is quite obvious, by a con- sideration of the demonstration of the corresponding theorem in ^-integrals given in Chapter II. This is also true when we come to treat of iterated integrals along the lines of 70-78. We have seen, in 425, 2, that if SI is measurable, upper and lower integrals of separated functions do not exist as such ; they reduce to i-integrals. We may still have a theory analogous to iterated ^-integrals, by extending the notion of iterable fields, using the notion of upper measure. To this end we define : A limited point set at 31 = SB • S is suhmeasurahle with respect to 93, when 1= f 1. We do not care to urge this point at present, but prefer to pass on at once to the much more interesting case of i-integrals over measurable fields. L-Integrals 429. These we may define for our purpose as follows : Let/(a;i ■•• x^) be defined over the limited measurable set 31. As usual let 2l„^ denote the points of 31 at which -«0. Let each 3Iap be measurable, and let / have a proper i-integral in each Slap. Then the improper integral of/ over 31 is / /= lim f /, (1 when this limit exists. We shall also say that the integral on the left of 1) is convergent. On this hypothesis, the reader will note at once that the dem- onstrations of Chapter II admit ready adaptation ; in fact some of the theorems require no demonstration, as they follow easily from results already obtained. 406 IMPROPER L-INTEGRALS 430. Let us group together for reference the following theo- rems, analogous to those on improper ^-integrals. 1. If f is (improperly') L-iviegrahle in 21, it is in any measurable part of 21. 2. If g,h denote as usual the non-negative functions associated withf, then ■ ff=fff-fh. (1 Xsi d/n oLn 3. If I f is convergent, so is l \f\, and conversely. 4. When convergent, \ff\ 0, o- > 0, r/| some /3q. This shows that i is convergent. Similarly we show oi-sp the other integral converges. 2. This form of proof could not be used in 38, 2, since 1) in general is not an unmixed division of ^..p. 3. In a similar manner we may establish the theorem analo- gous to 39, viz. : If I f and i f converge, so does j f. 4. Let us look at the demonstration of the theorem analogous to 43, 1, viz. : fff= ff ; fh = - ff provided the integral on either side of these equations converges. 408 IMPROPER L-INTEGRALS Let us prove the first relation. Let Sp denote the points of SI at which/ C* form a null set. We may therefore suppose that g = C everywhere. Then 6 — C is a null set at each point ; we may thus adjoin them to Q. Thus we may suppose that 6 = C at each point of 93, and that 93 = -B is the union of an enumerable set of complete sets B^. As we shall suppose that f. is convergent, let «i< tta < ••• = CO , Let us look at the sets 3l„^, 93^^, which we shall denote by 2l„. These are measurable by 429. Moreover, the reasoning of 72, 2 shows that without loss of generality we may suppose that 21 is such that 93„ = 93. We may also suppose that each S„ is measur- able, as above. 2. Let us finally consider the integrals Xd /■ (1 These may not exist at every point of 93, because / does not admit a proper or an improper integral at this point. It will suffice for our purpose to suppose that 1) does not exist at a null set in 93. Then without loss of generality we may suppose in our demonstrations that 1) converges at each point of 93. On these assumptions let us see how the theorems 73, 74, 75, and 76 are to be modified, in order that the proofs there given may be adapted to the present case. 410 IMPROPER L-INTEGRALS 433. 1 . The first of these may be replaced by this : Let Ba, n denote the points of^at which c„ > o". Then lim5,.„ = 0. For by 419, 'fft n ■ 21= / e, oLfs as by hypothesis the sections S are measurable. Moreover, by hypothesis is .a separated division of S, each set on the right being measur- able. Thus the proof in 73 applies at once. 2. The theorem of 74 becomes : Let the integrals be limited in the complete set S3. Let @„ denote the points of SB at which p Then = «■ limg„ = S. The proof is analogous to that in 74. Instead of a cubical division of the space SR^,, we use a standard enclosure. The sets SB„ are now measurable, and thus b = Dv\^,] is measurable. Thus b„ = b. The rest of the proof is as in 74. 3. The theorem of 75 becomes : Let the integral ff , f be limited in com,plete 93. Then . >0 lim / |/=0. ITERATED INTEGRALS 411 The proof is entii^ly similar to that in 75, except that we use extremal sequences, instead of cubical divisions. 4. As a corollary of 3 we have Let the integral ^ f , />o be limited and L-integrahle in 33. Let ^ = \B^] the union of an enumerable set of complete sets. Then lim r ff=0. For if S8„ = (A, B^... B^), and » = 93„, + ©„, we have Xsdlc,, i'iB^A. i'®„2c„ But for m sufficiently large, ©„ is small at pleasure. Hence We have now only to apply 3. 434. 1. We are now in position to prove the analogue of 76, viz. : Let 31 = 33 • (S be measurable. Let I f be convergent. Let the integrals I f converge in SB, except possibly at a null set. Then Xis. Cf=C Cf (1 provided the integral on the right is convergent. We follow along the line of proof in 76, and begin by taking / > in 31. By 428, we have hence ff=limf Cf (2 412 IMPROPER L-INTEGRALS Now e > being small at pleasure, -e+ / / /< / / / , for a > some G-q, Since we have seen that we may regard 58 as the union of an enumerable set of complete sets, we see that the last term on the right = 0, as n = 00, by 433, 4. Thus f r 0. Z%e general case is now obviously true. For 21 =

in S^, and < in 9?. Here ^ and SR are measurable. We have therefore only to use 1) for each of these fields and add the results. 2. The theorem 1 states that if // , f ff. both converge, they are equal. Mohson* in a remarkable paper on Lebesgue Integrals has shown that it is only necessary to assume the convergence of the first integral ; the convergence of the second follows then as a necessary consequence. * Proceedings of the London Mathematical Society, Ser. 2, vol. 8 (1909), p. 31. ITERATED INTEGRALS 413 I 435. We close this chapter by proving a theorem due to Lebesgue, which is of fundamental importance in the theory of Fourier's Series. Letf(jc) he properly or improperly L-integrable in the interval 2l=:(a<6). Then lim J, = lim C\ / (« + S) - / (a;) | dx = \\xa C\^f\dx=Q, a<^<^ + S(?. Then by 4), J, = 'Y in S, and elsewhere = ; let •^ = 7 in g, and else- where = 0. Thus using 4), since A = (^ in (a, /3), except at points of measure < o-/4. Similarly Thus Jh(«) =/(a; + 2 w) +f(x -2v^- 2f(x) is continuous at m = 0, and has the value 0, at points of continuity, and at points of discontinuity of 1° kind of/. Finally let us sup- pose that / is (properly or improperly) i-integrable in 21 ; this last condition being necessary, in order to make the Fourier co- efficients a„, S„ have a sense. F=F (^x) = \%-\- aj cos a; + a^ cos 2x+ ••• • -f Jj sin a; -f- 62 sin 2 a; -|- • • • (1 CO = I Oq -h 2(a„ cos nx + b„ sin nx'), 1 where we will now write a„ = — / f (x") COS nxdx, (2' b„ =— j fix) sin nxdx. (2" Since/ (a;) is periodic, the coefficients a„, S„ have the same value however c is chosen. If we make c = — tt, these integrals reduce to those given in 436. We may write F= - If'it^dt j i-H 2(cos nx cos nt + sin nx sin nt) \ = - rii + i cos n(t -x-)\f (f)dt. (3 Thus F„ = ^CK-f(.t-)dt, where p„ = 1 -I- i cos m(« - x). (4 Provided sinlG-:c)^0, (5 we may write P„ = i |sinK*-a;) + 22sin|(*-a;)cos7M(«-a;) " 2sinl(«-a;)' ^^ 1 1 2sin|(« — a;) sinl(«-a;) -H2 1 sin 2^(e-.)-sin2i^O-.))]. 422 FOURIER'S SERIES p _ sinl(2n + l)(t-x) ,„ "~ 2siniat-x) ' ^ if 5) holds. Let us see what happens when 5) does not hold. In this case ^(t — a;) is a multiple of tt. As both t and x lie in (c, c + 2 tt), this is only possible for three singular values : t = x ; t = 0, x= c + 27r ; t = o + 2 "jr, x= c. For these singular values 4) gives A = ^. ■ (7 As P„ is a continuous function of t, x, the expression on the right of 6) must converge to the value 7) as x, t converge to these singular values. We will therefore assign to the expression on the right of 6) the value 7), for the above singular values. Then in all cases . ^ „ . , „ . , ■jrXc 2 sin ^(^t - x) '^ ^ ■' Let us set o , i Then 1 f i'-'-^'+v , , „ . sin I'M , J; = - f(x + 2u) — du. Trjbi(c-x) sin M Let us choose e so that C — X= — IT, then 2 2 Replacing m by — m in the first integral on the right, it becomes £ '^y o N sin vM , /(a; — z u) du. sinu Thus we get IT F„=- r\f„, if we set sin u 2 To evaluate J„ let us break 58 up into the intervals These intervals are equal except the last, which is shorter than the others unless 6 — a is a multiple of ir/n. We have thus Jn=f +[+■■■+[ If we set n we see that while v ranges over S32s, u ranges over Sj^.j. This substitution enables us to replace the integrals over SBzj by those over S32J-H since / g (v) sin nvdv =— / ^(m + — )sin nudu. Hence grouping the integrals in pairs, we get '^n= I g (u') sin nudu + V / \g{u') — g(u + —j sin nudu + I g (u) sin nudu, * The presentation given in 439-448 is due in the main to Lebesgue. Cf. his classic paper, Mathematisehe Annalen, vol. 61 (1905), p. 251. Also his Leipns sur les Series Trigonometriques, Paris, 1906. VALIDITY OP FOURIER'S DEVELOPMENT 425 where 93' is 33, or %_i + 93,, depending on the parity of r. Now < »n 1^1 (2 g(u)-g{u + '!^ sin nudu < r'"~\ ^1 .>' M + < ff(u) du du. (3 (4 Thus J^ = 0, if the three integrals 2), 8), 4) = 0. Moreover, if these three integrals are uniformly evanescent with respect to some point set S < 93, t/i, is also uniformly evanescent in S. In particular we note the theorem t/n = 0, if g is L-integrahle in 93- We are now in a position to draw some important conclusions with respect to Fourier's series, o 440. 1. Let f(x) be L-integrahle in (c, c+27r). Then the Fourier constants a„, 6„ = 0, as w = oo. For «n = - I /(^) COS nxdx is a special case of the t/„ integral. As/ is i-integrable, we need only apply the theorem at the close of the last article. Similar reasoning applies to J„. 2. For a given value o/a; m 31 = (— tt, tt) let Sin u (1 be L-integrable iw 93 = ( 0, - ). Then Fourier s development is valid at the point x. 426 FOURIER'S SERIES For by 438, Fourier's series =f(x) at the point x, if D„(jc) = 0. But Dn is a special case of «/„ for which the g function is in- tegrable. We thus need only apply 439. 3. For a given a; m 21 = (— tt, tt), let XW = ^ (2 be L4ntegrable in S3 = ( 0, ^ j. Then Fourier g development is valid at the point x. For let S > 0, then Jjs ato Isinwl Jifs u <0+v)£\x(^)\du = , as S = , by hypothesis. 4. For a given x in 21 = ( — tt, tt), let ^^)^/C^+^)-/(^) (3 6e Lintegrahle in 21. jTAen Fourier s development is valid at the point X. ^°' .. ^ f(x + 2u')-fCx) f(x-2u~)-fix-) ^^ ^ u u = 2[<»(2M) + a)(-2M)]. Thus X is i-integrable in f 0, ^J, as it is the difference of two integrable functions. 441. (^Lebesgue'). For a given a; in 21 = ( — tt, tt) let 1° limw r\(u)\du=0; 2° lim r\^fr(u+B}-yfr(u)\du = for some -n such that „ 0i \'i>(u) ■ + I sin vu du as w = oo. ie< us first consider D'. Since 0t. But this is indeed so. For 4 — 4 Hence g>^/'i_ zr"j >l>( , if ^>5. TT Thus D'" = 0. 5=0 Finally we consider D'" . But the integrand is an integrable function in f ^, ^ J . Thus it = as n = x. 442. 1. The validity of Fourier's development at the point x de- pends only on the nature off in a vicinity of x, of norm S as small as we please. For the conditions of the theorem in 441 depend only on the value of/ in such a vicinity. 2. Let us call a point x at which the function (^(m) =f{x + 2 w) +fix -2m)- 2f(x) is continuous at m = 0, and has the value 0, a regular point. In 438, we saw that if a; is a point of discontinuit}' of the first kind for /(a;), then a; is a regular point. 3. Fourier's development is valid at a regular point x, provided for some 77 lim r'|i^(M + S)-i/r(tt)| «fM=0 , 0 0, -Ni,= <0. Then y^^^ Uia^^,-) -f^a^) \==Pj> + Nj,. (2 Also !/(«!) -/(a) \ + {/(a^) -/(ai) J + ... + [/(&) -/(a„) | =i'^ - N^. On the left the sum is telescopic, hence f{b)-f(a}=Pj,-Nj,. (3 From 2), 3) we have F^ = 2 P^ +/(a)-/(6) = 2 Nn +f(b) -/(fl). (4 L^*^°^ MaxP^ = P , Maxi^^ = i7 with respect to the class of finite divisions D. We call them the positive and negative variation of /(a;) in 21. Then 4) shows that r=2P+/(«)-/(S) , F=2i^+/(6)-/(a). (5 ing these, we From 5) we have f(h:,-fia) = P-N. (7 Instead of the interval 21 = (a < 6), let us take the interval (a < x), where x lies in 21. Replacing J by a; in 7), we have fix)=fia-) + P(x:,-Nix-). (8 Adding these, we get y=P + N. (Q (9 LIMITED VARIATION 431 Obviously -P(«^ N{x) are monotone increasing functions. Let ti(x) be a monotone increasing function in SI. If we set g{x)=f(:a)+P(ix) + f.(_x-) h(x) = N(x)+^{x), we get 1) from 8) at once. 4. From 8) we have \f(x)\<\f(a)\+P(x) + Nix-) < |/(«) I + Vix). (10 5. We can now show that when f(x) has finite variation in the interval 21 = (a < J) it has limited variation and conversely. For if / has finite variation in SI we can set where ^, yjr are monotone increasing in 21. Then if 21 is divided into the intervals Sj, fig •" ^^® have Osc/ < Osc ^+ Osc i|r , in S^. Osc (j> = A(j> , Osc ylr = Ai/r , in B^ since these functions are monotone. Hence summing over all the intervals S,, 2 Osc/ < S A V^f Let now Cr be obtained from a division F by adding the point *• ^^^^" V,f>V,f. ^^"°^ Max Fi,>Max Vp. E F Hence to find Va^^f, we may consider only the class F. Let now El be a division of (a, 6), and E^ a division of (6, c). Then E-^ + E^isA division of class E. Conversely each division of class Ogives a division of (a, J), (J, e). Now From this 11) follows at once. 444. We establish now a few simple relations concerning the variation of two functions in an interval 21 = (a < 6). ^- V(if+c)=Vf (1 where for brevity we set » /., ^ 2- F(c/)=|e|F/. (2 LIMITED VARIATION 433 3. Letf, g be monotone increasing functions in 21. Then nf+ff)= Vf+ Vg. (3 ^'"' 2 I (/,i + ^.+1) _ (/ + 5,j I = S I (/,! -/O + (^^,j _ ^0 1 = '^\f.+i-M + ^\9,+i-g.\- 4. J'or aw«/ two functions f, g having limited variation, Vif+g^ M — i M* > ri(l — Tf^). LIMITED VARIATION 435 Thus, if ^^ = Max \ <}> \ in (t), rj + S), vO- - v)aL, vO- - •7) with S. Thus, when i/r is monotone in some (0, J^), Fourier's develop- ment is valid. But obviously when yjr is monotone, the condition that %F(w)=0 is satisfied. Our theorem is thus established in this case. Let us now consider the ease that the variation F(m) of ■\}r is limited in (u, f ). From 443, 10), we have \y|r(u}\<\^jrCO\+Viuy As before we have |'^| 0, a S(, > 0, such that uV(u')<€ , for any 0(u) has limited variation in some interval (0 < f), ?< — . For let < M < 7 < f, then 1 Now ■Jr=(f>(u^ SHIM Hence F„, f < j F„,<^ + | Cy) \ i r.-J-+- 1 But sin u being monotone, 1 1 1 sin u sin 7 V^y Thus Similarly, Now sin M sin u sin 7 ^' '^ — Sin 7 ^ 0<^^\=0 , withS. oLs 448. (Lipschits-Dini.y At the regular point x, Fourier's devel- opment is valid, if for each e > 0, there exists a 8^ > 0, such that for each < S < Sq , I (u + S) - <^(m) I < , , ^ - , , for any u in (S, S,,). I log 6 I For ^ |.^(m + 8)-0(m)| , g |0W |_ M M^ Now a: being a I'egular point, there exists an rj' such that I <^(") I < «' for w in any (S, ?;'). Thus taking ^ , < 2 e, for any S < t;. Thus V • n s • A X = 0, as 6 = 0. Uniqueness of Fourier's Development 449. Suppose /(a;) can be developed in Fourier's series / (2^) = i ''^o + 2(a„cos nx + ft„sin nx), (1 1 a„=-/ f (x) cos vxdx , S„ = _/ /(a;) sin7ia;ia;, (2 UNIQUENESS OP FOURIER'S DEVELOPMENT 439 valid in SI = (— TJ? tt). We ask can /(a;) be developed in a simi- lar series ^^ni/,^// ,i/- ^ ,o / {X) = ^ a^ + z^i^a^ cos nx + 6; sin nx), (3 also valid in 31, where the coefficients are not Fourier's coefficients, at least not all of them. Suppose this were true. Subtracting 1), 3) we get = K«o — *o) + ^ 1 («*" — "''n) COS nx + (6„ - 6;^) sin nx\=0, Cq + 2 [(?„ COS nx + (Z„ sin nx\ = 0, in 21. (4 Thus it would be possible for a trigonometric series of the type 4) to vanish without all the coefficients e„, d^ vanishing. For a power series Po+PiX + P-i^+ — (5 to vanish in an interval about the origin, however small, we know that all the coefficients p^ in 5) must = 0. We propose to show now that a similar theorem holds for a trigonometric series. In fact we shall prove the fundamental Theorem 1. Suppose it is known that the series 4) converges to for all the points of ^ = (— tt, tt), except at a reducible set 9J. Then the coefficients c„, d^ are all 0, and the series 4) = a^ all the points of 21. From this we deduce at once as corollaries : Theorem 2. Let ^ be a reducible set in 21. Let the series 00 a^+'lla^ cos nx + /3„ sin nxl (6 converge in 21, except possibly at the points 9?. Then 6) defines a function F(^x') in 21 — 9?. If the series ^^, ^ ^ f «i cos nx + ^l sin nx] converges to F(x) in 21 — 9i, its coefficients are respectively equal to those in 6). Theorem S. If fix) admits a development in Fourier's series for the set 21 — 9t, any other development of fix) of the type 6), valid in 21 — 9? is necessarily Fourier s series, i.e. the coefficients «„, ^„ have the values given in 2). 440 FOURIER'S SERIES In order to establish the fundamental theorem, we shall make use of some results due to Miemann, Gr. Cantor, Harruich and Schwarz as extended by later writers. Before doing this let us prove the easy Theorem 4- If /(a;) admits a development in Fourier's series which is uniformly convergent m 21 = ( — tt, tt), it admits no other development of the type 3), which is also uniformly convergent in 31. For then the corresponding series 4) is uniformly convergent in 21, and may be integrated termwise. Thus making use of the method employed in 436, we see that all the coefficients in 4) vanish. 450. 1. Before attempting to prove the fundamental theorem which states that the coefficients a„, b„ are 0, we will first show that the coefficients of any trigonometric series which converges in 21, except possibly at a point set of a certain type, must be such that they = 0, as w = oo. We have already seen, in 440, l, that this is indeed so in the case of Fourier's series, whether it con- verges or not. It is not the case with every trigonometric series as the following example shows, viz. : 2 sin w ! X. (1 1 When x= — - all the terms, beginning with the r !*, vanish, and hence 1) is convergent at such points. Thus 1) is conver- gent at a pantactic set of points. In this series the coefficients a„ of the cosine terms are all 0, while the coefficients of the sine terms b„ , are or 1. Thus 6„ does not = 0, as w = oo. • 2. Before enunciating the theorem on the convergence of the coefficients of a trigonometric series to 0, we need the notion of divergeyice of a series due to Hamack. Let A = a.^^ + a^+ ••• (2 be a series of real terms. Let g^, (r„ be the minimum and maxi- mum of all the terms where as usual A^ is the sum of the first n terms of 2). Obviously UNIQUENESS OF FOURIER'S DEVELOPMENT 441 Thus the two sequences \gj, {Cr„\ are monotone, and if limited, their terms converge to fixed values. Let us say The difference b = a-g is called the divergence of the series 2). 3. For the series 2) to converge it is necessary and sufficient that its divergence b = 0. For if A is convergent, -e + ^<^„+p<^ + e , ^ = 1,2... Thus -e + A is small at pleasure. Suppose now b = 0. Then by hypothesis, G; g exist and are equal. There exists, therefore, an w, such that 9-e 0, there exists a subinterval of 2l = (-7r, tt) at each point of which its divergence b < S. Then a„, 6„ = 0, as n = cc. For, as in 450, there exists for each a; an w^,, such that I a„ cos nx + b„sinnx\<- , n>m^ (1 442 FOUlilER'S SERIES for anj' point x in some interval S3 of 81. Thus if h is an inner point of S3, a; = 6 + yS will lie in 93, if )8 lies in some interval B = (p, q). Now a„ cos n(6 + /8) + 6„ sin m(i + ;8) = (a„ cos nb + 6„ sin m6) cos m/S — (a„ sin nb — b„ cos m6) sin m/8. a„ cos w (6 — j8) + J„ sin n(b — ^^ = (ffl„ cos nb + b„ sin m6) cos m^S + (a„ sin nb — 6„ cos w6) sin w/3. Adding and subtracting these equations, and using 1) we have I (a„ cos nb + J„ sin m6) cos wyS | < -, s I (a„ sin w6 — 6„ cos nJ) sin w/8 I < -, for all w>TOj.. Let us multiply the first of these inequalities by- cos nb sin w/S, and the second by sin nb cos w/3, and add. We get I a„ sin WiSi I < S , /Sj = 2;8 , w >m^. (2 Again if we multiply the first inequality by sin nb sin wyS, and the second by cos nb cos n^, and subtract, we get I b„ sin w/Sj I < S , n> m^. (3 From 2), 3), we can infer that for any e > I «n I < e , I J„ I < e , w > some m, (4 or what is the same, that a„, J„ = 0. For suppose that the first inequality of 4) did not hold. Then there exists a sequence Wj < Wj < ••• =00 (5 such that on setting we will have ««,>«'• (6 If this be so, we can show that there exists a sequence "1 < ^2 < ■■■ = °° in 5), such that for some yS' in B, |a,,sin./,yS'| >S, (7 UNIQUENESS OF FOURIER'S DEVELOPMENT 443 which contradicts ^ ™*y be chosen so small that for any r and any | 7 | < 79 , 1 a„^ I cos 7 > (S + S') cos 7o > S. (8 Let us take the integer I'j so that >7r + 2 7,_ (9 Tl^«^ ^(-x(?-p)-27o)>2. IT Thus at least om. odd integer lies in the interval determined by the two numbers 2 2 Let mj be such an integer. Then 2 2 -(i»'i + 7o)<»»i<-(9''i-7o)- (10 7r TT If we set we see that the interval -Si = (^i, g'j) lies in B. The length of ^1 is 2 7o/j'i- Then for any j8 in 5j, i'i/3 = Wi| + 7i > |7il<7o- Thus by 8), I a„j sin i/j/3 I = I a„, | cos 7^ > S. (12 But we may reason on B^ as we have on B. We determine v^ by 9), replacing j», ? by ^j, jj . We determine the odd integer m^ by 10), replacing^, q, v^ by ^j, jj, i-g- 'The relation 11) deter- mines the new interval B^ = (p^, q^), on replacing Wj, v^ by m^, v^. The length of B^ is 2 70/^2' *^ii S. In this way we may continue indefinitely. The intervals B^>B2> ••• = to a. point /3', and obviously for this /3', the rela- 444 FOURIER'S SERIES tion 7) holds for any x. In a similar manner we see that if b„ does not = 0, the relation 3) cannot hold. 452. As corollaries of the last theorem we have : 1. Let the series 00 2(a„ cos nx + 6„ sin nx') (1 be such that for each 5 > 0, the points in 2l = (— tt, tt) at which the divergence of 1) is >S, form an apantactic set in 21. Then a„, h„ = 0, as n = ao. 2. Ziet the series 1) converge in 21, except possibly at the points of a reducible set SR. Then a„, 6„ = 0. For 9J being reducible [318, 6], there exists in 21 an interval 33 in which 1) converges at every point. We now apply 451. F (a;) = S(a„ cos nx + b„ sin nx") at the points of% = (^—ir, tt), where the series is convergent. At the other points of 21, let F(x') have an arbitrarily assigned value, lying between the two limits of indetermination g, Cr of the series. If F is R-integrable in 21, the coefficients a„, J„= 0. For there exists a division of 21, such that the sum of those in- tervals in which Osc JP > &) is < o-. There is therefore an interval 3f in wliich Osc F (m) _ V 4 / sin wmV 4 M^ ^\ nu J^ if we agree to give the coefficient of ^^ the value 1. Let us give X an arbitrary but fixed value in S3. Then for each e > 0, there exists an m such that A + A + - + ^n-i = F{x) + e„ , I e„ I < e, n>m. Thus A„ = e„+i — e„. Hence ^ » ./ sinntt V 1 iL (w-1)m J L WW J I = J'(a;)+ S 446 FOURIER'S SERIES The index m being determined as above, let us take u such that M < — , so that m < — ; m u and break S into three parts S, = l. , ,^2 = 2 , ^3=2, m+l >c+l where k is the greatest integer < tt/m, and then consider each sum separately, as % = 0. Obviously lim S. = 0. tt=0 As to the second sum, the number of its terms increases indefin- itely as M = 0. For any u, 1 l'S'3|<6- or KU > TT — u. ; + Hence (tt — m)^ tt — MJ ^S = yS*! + /Sj + ^Sg = 0, as M = 0, which proves the limit 2), on using 4). To prove the limit 3), we have 4m q \ nu J Let us give « a definite value and break T into three sums. m where m is chosen so that I ^„ I < e , « > w ; ^2 = 2, m+I where X is the greatest integer such that \u < 1; and Obviously for some M, Also since 2'3 = 2. A+l irj - • - , since - < \, +^^\- ^> - ^^^^> = 0,asu=0, (1 u^ except possibly at an enumerable set (g iw 21- At the points @, let uS{x, m) = as M = 0. (2 Then/ is a linear function in 31. Let us ^rs< suppose with Schwarz that @ = 0. We introduce the auxiliary function, g^x') = ■»?i(a;) - | c(a: - a)(a; - S), where i(z) =/(a;) _/(a)- p^ !/(J) -/(a)j, — a 7j = ± 1, and c is an arbitrary constant. The function ^(a:) is continuous in 21, and ^(a) = ^(6) = 0. Moreover /■ , \ , ^ \ o / % 2-5^ — ■ ' ' » \ 1 S-S — L = 0, as M = 0. u^ Thus for all < m < some S, a = gix + u')+ff{x-u}-2g(x^>0. (3 From this follows that ^(a;)<0 in 21. For if ^(2;)>0, at any point in 21, it takes on its maximum value at some point ^ within 21. for < M < S, 8 being sufficiently small. Adding these two in- equalities gives 6r< 0, which contradicts 3). Thus ^ <0 in 21. Let us now suppose L^O for some x in 21. We take c so small that T r sgng = sgn r]L = r) sgn L. UNIQUENESS OF FOURIER'S DEVELOPMENT 449 But 7] is at pleasijire ± 1, hence the supposition that L=^0 is not admissible. Hence i = in 21, or /(^) = /(«)- |^S/(5)- /(a) j (4 is indeed a linear function. Let us now suppose with Liiroth that (£ > 0. We introduce the auxiliary continuous function. h(x)=L(x}+cCx-ay , e>0. Suppose at some inner point f of 21 LQ)>0. (5 This leads to a contradiction, as we proceed to show. For then KD - Ki) = i(D +c\i^-ay-(h-ay\> o, provided T^i-\ 0= i^^ >c. We shall take c so that this inequality is satisfied, i.e. c lies in the interval e = (0*, (7*). Thus A(^)>A(5) >/*(«). Hence hCx) takes on its maximum value at some inner point e of 2[. Hence for S > sufficiently small, A(e + M)- A(e)<0 , A(e-w)-%)<0 ,"00. But this contradicts 7), which requires that lim2r(e, m)<0. 450 FOURIER'S SERIES Hence e is a point of @. Hence by 2), TiQe + m) - A(e) ^ hje - w) - A(e) ^ q ^ asM = 0. By 6), both terms have the same sign. Hence each term = 0. Thus f or M > ^ lim ^^' ^ ^^ ~ ^'^'^ - lim f^" =^ ^> ~ f^"^ f^^ )-Aa) «=o ±u ±u b — a + 2c(e — a). /'(.) = A61zl£(^) + 2 <.-«). (8 — a Thus to each c in the interval S, corresponds an e in (S, at which point the derivative oif(x) exists and has the value given on the right of 8). On the other hand, two different c's, say c and c', in S cannot correspond to the same e in S. For then 8) shows that c(^e — a)= c'(e — a), or as ^ , e> a, c = c. Thus there is a uniform correspondence between S whose cardi- nal number is c, and @ whose cardinal number is e, which is absurd. Thus the supposition 5) is impossible. In a similar manner, the assumption that i < at some point in 21, leads to a contradiction. Hence i = in 21, and 4) again holds, which proves the theorem. 456. Cantor s Theorem. Let 00 ^a^ + S(a„ cos nx + 6„ sin nx) (1 1 converge to in 2l = (— tt, tt), except possibly at a reducible set JR, where nothing is asserted regarding its convergence. Then it con- verges to at every point in 21, and all its coefficients a^i «!, a^ ••■ hi hi h '" — ^■ For by 452, 2, a„, 6„ = 0. Then Riemann's function / (a;) = -|- affe^ — 2J -^ («n cos nx + 6„ sin wa;) UNIQUENESS OF FOURIER'S DEVELOPMENT 451 satisfies the conditiaiis of the Schwarz-Liiroth theorem, 455, since 5R is enumerable. Thus /(a;) is a linear function of x in 21, and has the form « + /Sa;. Hence a+^a; — I aoa;2 = — 2^ — (a„ cosM2;+6„ sinnx). * (2 1 ** The right side admits the period 2 tt, and is therefore periodic. Its period to must be 0. For if <» > 0, the left side has this period, which is absurd. Hence w = 0, and the left side reduces to a constant, which gives /S=0, aQ= 0. But in 21 — SK, the right side of 1) has the sum 0. Hence « = 0. Thus the right side of 2) vanishes in 21. As it converges uniformly in 21) we may deter- mine its coefficients as in 436. This gives a„ = , J„ = , « = 1, 2... CHAPTER XIV DISCONTINUOUS FUNCTIONS Properties of Continuous Functions 457. 1. In Chapter VII of Volume I we have discussed some of the elementary properties of continuous and discontinuous functions. In the present chapter further developments will be given, paying particular attention to discontinuous functions. Here the results of Baire * are of foremost importance. Le- besgue f has shown how some of these may be obtained by sim- pler considerations, and we have accordingly adopted them. 2. Let us begin by observing that the definition of a continu- ous function given in I, 339, may be extended to sets having iso- lated points, if we use I, 339, 2 as definition. Let therefore f(x-^ ■ ■ ■ «;„) be defined over 21, being either limited or unlimited. Let a be any point of 21. If for each e > 0, there exists a S > 0, such that |/(a=) -/(«) I < «. for any z in Vs(d), we say f is continuous at a. By the definition it follows at once that / is continuous at each isolated point of 21. Moreover, when a is a proper limiting point of 21, the definition here given coincides with that given in I, 339. If /is continuous at each point of 21, we say it is continuous in 21. The definition of discontinuity given in I, 347, shall still hold, except that we must regard isolated points as points of con- tinuity. * " Sur les Functions de Variables reeles," Annali di Mat., Ser. 3, vol. 3 (1899). Also his Lemons sur les Functions Discontinues. Paris, 1905. t Bulletin de la Societe Mathemalique de France, vol. 32 (1904), p. 229. 452 PROPERTIES OF CONTINUOUS FUNCTIONS 458 3. The reader w?ll observe that the theorems I, 350 to 354 inclusive, are valid not only for limited perfect domains, but also for limited complete sets. 458. 1. If f(x-^ •■• x^) is continuous in the limited set 21, and its values are known at the points o/ SB < 21, then f is known at all points of S3' lying in 21. For let Jj, Jg, Jg •■• be points of S3, whose limiting point h lies in 21. Then lim/(fi„)=/(J). 71=00 2. If f is known for a dense set S3 in 21, and is continuous in 21, / is known throughout 21. For 33, > 5j_ 3. If f(x^ ■■■ x^) is continuous in the complete set 21, the points S3 in 21 where /= c, a constant, form a complete set. If 21 is an interval, there is a first and a last point ofSS. For/= e at a; = ttj, a^ ■•• which = a ; we have therefore /(a) = lim/(a„) = c. n=oo 459. The points of continuity S of f(x^---x^) in 21 lie in a deleted enclosure ^. If% is complete, ^ == S. For let ej > 62 > ••• = 0. For each e„, and for each point of continuity e in 21, there exists a cube O whose center is c, such that Osc/ sufficiently small and n sufficiently great. 454 DISCONTINUOUS FUNCTIONS Oscillation 460. Let ^^=OBcfix^-x„) in FsCa). This is a monotone decreasing function of S. Hence if wj is finite, for some S > 0, Q) = lim the oscillation off at x = a, and write (o = Osc/. Should is taken, we say w = + oo. When to = 0, / is continuous at a; = a, if a is a point in the domain of definition of /. When o) > 0, / is discontinuous at this point. It is a measure of the discontinuity off at x = a; we write « = Disc/(a;i--- x„). x—a 461. 1. Let d = Disc/(a:i ••• 2;„) , e = I>iscg(x^ ••• a;™), at x = a. Then \d - e\< Disc {f ± g') < d + e. For in Fj (a), I Osc/- Osc^l < Osc(/±^) < Osc/+ Osc^r. 2. If f is continuous at x = a, while Disc^ = d, then Disc (/+kfor7n a complete set. For let a^, a^---he points of ^ which = a. However small S >0 is taken, there are an infinity of the a„ lying in FsCa). But at any one of these points, Osc/> *. Hence Osc/> A; in Vs (a), and thus a lies in B. 4. Letf(x^ ■■■ x^) be limited and R-integrahle in the limited set 21. The points S at which Oscf>kform a discrete set. For let 2) be a rectangular division of space. Let us suppose i^ > some constant c> 0, however 2> is chosen. In each cell 8 oiB, Osc f>k. Hence the sum of the cells in which the oscillation is>k can- not be made small at pleasure, since this sum is t^. But this contradicts I, 700, 5. 5. Let f(x^ --x^) be limited in the complete set 21. If the points S in 21 at which Oscf>k form a discrete set, for each k, then f is B-integrable in 21. 456 DISCONTINUOUS FUNCTIONS For about each point of 21 — S as center, we can describe a cube S of varying size, such that Osc/< 2 A; in S. Let 2> be a cubical division of space of norm d. We may take d so small that ^^ = 2d, is as small as we please. The points of 21 lie now within the cubes E and the set formed of the cubes c?.. By Borel's theorem there are a finite number of cubes, say J?! 1 Vi — such that all the points of 21 lie within these 7;'s. If we prolong the faces of these 17's, we effect a rectangular division such that the sum of those cells in which the oscillation is > 2 A is as small as we choose, since this sum is obviously < ^^. Hence by I, 700, 5, /is i2-integrable. 6. Letf(x^ ■■• a;„) be limited in 21; let its points of discontinuity in 21 6e 3). If f is M-integrable, 35 is a null set. If 21 is complete and 3) is a null set, f is R-integrable. Let / be iZ-integrable. Then 3) is a null set. For let Cj > e^ > ... = 0. Let 3)„ denote the points at which Osc/> e„. Then 3) = {3?„S. But since /is ^-integrable, each 33„ is discrete by 4. Hence 35 is a null set. Let 21 be complete and 35 a null set. Then each 35„ is complete by 3. Hence by 365, 35„ = 35„. As S = 0, we see 2)„ is discrete. Hence by 5, /is ^-integrable. If 21 is not complete, / does not need to be jB-integrable when 35 is a null set. Example. Let %^ = gl , n=l, 2... ; m<2». *2 Let2l=2li + 2l2. 3' ^=\^.\ ' « = 1, 2...; r<3'. 1 m Let fix) = - , at a; = 2^ = 1 in 2I2 . Then each point of 21 is a point of discontinuity, and 21 = 35. But 2li, 2I2 are null sets, hence 21 is a null set. POINTWISE AND TOTAL DISCONTINUITY 457 On the other hand, J/=i , r/=o, and / is not i2-integrable in 21. Pointioise and Total Discontinuity 463. Let/(3;i ••• x„) be defined over 21. If each point of 21 is a point of discontinuity, we say /is totally discontinuous in 21. We say/ is pointwise discontinuous in 21, if/ is not continuous in 21= {a j, but has in any V(^a) a point of continuity. If/ is continuous or pointwise discontinuous, we may say it is at most pointwise discontinuous. Example 1. A function /(ajj ••• x^) having only a finite number of points of discontinuity in 21 is pointwise discontinuous in 21. Example S. Let /(a;) =0 , for irrational a; in 21 = (0, 1) 1 J m = - , tor x = — n n = 1 , for x=Q, 1. Obviously/ is continuous at each irrational x, and discontinuous at the other points of 21. Hence / is pointwise discontinuous in 21. Example S. Let 5) be a Harnack set in the unit interval 21 = (0, 1). In. the associate set of intervals, end points included, let/(a:)=l. At the other points of 21, let /= 0. As 35 is apantactic in 21, /is pointwise discontinuous. Example 4-. In Ex. 3, let S) = @ + %, where @ is the set of end points of the associate set of intervals. Let/= 1/w at the end points of these intervals belonging to the w* stage. Let/= in g. Here / is defined only over !D. The points % are points of continuity in 3). Hence/ is pointwise discontinuous in ©. Example 5. Let /(a;) be Dirichlet's function, i.e. /= 0, for the irrational points 3 in 21 = (0, 1), and = 1 for the rational points. 458 DISCONTINUOUS FUNCTIONS As each point of 31 is a point of discontinuity,/ is totally discon- tinuous in 31. Let us remove the rational points in 21 ; the deleted domain is 3. In this domain/ is continuous. Thus on removing certain points, a discontinuous function becomes a continuous function in the remaining point set. This is not always the case. For if in Ex. 4 we remove the points i5, retaining only the points (g, we get a function which is totally discontinuous in @, whereas before / was only pointwise discontinuous. 464. 1. Iff(x^ ••• x„) is totally discontinuous in the infinite com- plete set 31, then the points b„ where Disc/>a) , (o>0, form an infinite set, if m is taken sufficiently small. For suppose b„ were finite however small <» is taken. Let a)j>a)2>"- =0. Let Dj, 2>2, ••• be a sequence of superposed cubical divisions of space of norms d„ = 0. We shall only con- sider cells containing points of 21. Then if cZj is taken sufficiently small, i)j contains a cell Sj, containing an infinite number of points of 21, but no point at which Disc/>o)j. If d^ is taken sufficiently small, D^ contains a cell S2a>2. In this way we get a sequence of cells, which = a point p. As 21 is complete, p lies in 31. But / is obviously continuous at p. Hence / is not totally discontinuous in 31. 2. If 31 is not complete, b„ does not need to be infinite for any o) > 0. Example. Let 21 = j — I , w = 1, 2, ••• and m odd and <2". At ^, let/= — • Then each point of 21 is a point of discontinuity. But b„ is finite, however small qj > is taken. 3. We cannot say /is not pointwise discontinuous in complete 21, when b„ is infinite. m EXAMPLES OF DISCONTINUOUS FUNCTIONS 459 Example. At the points • - i = S'i, let / = ; at the other in] points of 21 = (0, 1), let/=l. Obviously / is pointwise discontinuous in 21. But b„ is an infinite set for w < 1, as in this case it is formed of 91, and the point 0. Examples of Discontinuous Functions 465. In volume I, 330 seq. and 348 seq., we have given ex- amples of discontinuous functions. We shall now consider a few more. Example 1. Riemann's Function. Let (a;) be the difference between x and the nearest integer ; and when x has the form n + ^, let (a;) = 0. Obviously (a;) has the period 1. It can be represented by Fourier's series thus : (^)=- sin 2 TTx sin 2 • 2 -rrx sin 3 • 2 tts; ). a Miemann' s function is now Fix-y^t^jf. (2 This series is obviously uniformly convergent in 21 = (— oo, oo). Since (a;) has the period 1 and is continuous within ( — -J, ^), we see that (wa;) has the period -, and is continuous within — , — ). The points of discontinuity of (wa;) are thus 2n 2nj ^„=(^ + -l , s = 0, ±1, ±2, (An 71 ) Let @= |@„|. Then at any x not in @, each term of 2) is a con- tinuous function of x. Hence F{x) is continuous at this point. On the other hand, F is discontinuous at any point e of @. For F being uniformly convergent, BlimF{x-)=-2Bhm^ (3 L\imFCx) = -ZL\im^. (4 x=e x=e n'- 460 DISCONTINUOUS FUNCTIONS We show now that 3) has the value jfCe) '^!^—-, for e = \ , e irreducible. (5 16 w^ 2w and 4) the value 16 n^ F(.e-) + ^. (6 Hence DiscJ'(2:) = ^. (7 To this end let us see when two of the numbers f- -, and H— + - m4=n 2mm zn n are equal. If equal, we have 2r+l 2s+l (8 Thus if we take 2 s + 1 relatively prime to n, no two of the num- bers in (g„ are equal. Let us do this for each n. Then no two of the numbers in @ are equal. Is Let now x= e = h - • Then (mx) is continuous at this point, 2n n unless 8) holds; i.e. unless ?w is a multiple of n, say m= In. In this case, 8) gives 2 r + 1 = ^(28 + 1). Thus I must be odd ; 1= 1, B, 5 ••• In this case {mx') = at e, while 72 liui (mx')= — ^. When m is not an odd multiple of w, x=e obviously R lim (mx) = (me). Thus when m = In, I odd, x=e to2 2 IV vrC- 2n^ V- When m is not a multiple of w, i21im^^^ = l^. x=t m? m^ EXAMPLES OF DISCONTINUOUS FUNCTIONS 461 Hence x=e 2n^ [ 1^ 3'' 5^ = i^(e)- 16 w2' by 218. This establishes 5). Similarly we prove 6). Thus -F(a;) is discontinuous at each point of (S. As \^(^-)\j be the corresponding set of intervals. Let 0^= Cj + Fj, where Fj is obtained by putting a 0-y set in each interval of D^ . Let B^ be the intervals correspond- ing to CIj. Let Q^= 0^+ Fg where Fj is obtained by putting a 0^ set in each interval of D^, etc. The zeros of/(3wa;) are obviously the points of C^. Let ^=SV(^«^)=^^»<^^)- The zeros of F are the points of S = { (7„s- Since each C„ is a null set, S is also a null set. Let J. = 21 — S. The points J., ® are each pantactic in 31. Obviously F converges uniformly in 21, since 0 X = , x=Q. is continuous, except at the points @. At a; = e„. Disc J' = — • Example 2. Let (S = !e„| denote the rational numbers. Let ^ N 1 • nx 1 j^ /-I g'(a;)=lim- =1 , x4^*J = , a; = 0, which we considered in \, 331. Then #(:.)= 2;;^^(^-«n) is continuous, except at the rational points, and at a;= e„, Disc^(a;)=^- m\ 469. In the foregoing gix) is limited. This restriction may be removed in many cases, as the reader will see from the following theorem, given as an example. Let E=\e^^...^,\ he an enumerable apantactic set in 21. Let @ = (E, i?'). For any x in §1, and any e^ in E, let x— e^ lie within a cube SB. Let g(x.^---x^ be continuous in S3 except at x—0, where g = +(x>, as x=0. Let Sc^j...,, be a positive term convergent series. 464 DISCONTINUOUS FUNCTIONS is continuous iw -4. = 31 — @. On the other hand, each point of ^ is a point of infinite discontinuity. For any given point x=a oi A lies at a distance > from (S. 'ri^"« Miu(a;-O>0, as X ranges over some V-^ia), and e^ over E. ^^^^^^ |^(a;-Ol< some iff, for X in F,(a), and e^ in E. Thus F is uniformly convergent at x = a. As each g(x — e^) is continuous at a; = a, jP is continuous at a. Let next x = e,. Then there exists a sequence x\ x" ... = e, (1 whose points lie in A. Thus the term g(^x — e,) = + oo as a; ranges over 1). Hence a fortiori ^ = + oo. Thus each point of ^ is a point of infinite discontinuity. Finally any limit point of ^ is a point of infinite discontinuity, by 462, 1. 470. Example. Let gCx) = - , a„ = , a > 1. x a" F(x)=l.c^{x-a„) ^ nll+ a"x is a continuous function, except at the points -I -1 -1... a a^ «■* which are points of infinite discontinuity. 471. Let us show how to construct functions by limiting processes, whose points of discontinuity are any given complete limited apantactic set S in an w-way space SR^. EXAMPLES OF DISCONTINUOUS FUNCTIONS 465 1. Let us for simplicity take m = 2, and call the coordinates of a point X, y. Let Q denote the square whose center is the origin, and one of whose vertices is the point (1, 0). The edge of Q is given by the points a;, y satisfying \x\ + \y\ = \. (1 Thus J [|i on the edge Q Cx, y-) = li.li ,^,, ,^1 .,„ =1, inside (2 of the square Q. Hence ^-^i + ci.l + l,!)" ,^^^^^.^^^ L{x,y') = J lim 1 — ^ L2_u_ n=oc. l+wjl — |a;| — |t/| '\, on the edge, ^o . 0, off the edge. 2. We next show how to construct a function g which shall = on one or more of the edges of Q. Let us call these sides Cj, ej, 63, e^, as we go around the edge of Q beginning with the first quadrant. If 6r = on e,, let us denote it by Gr^; if G^ = on e^, e< let us denote it by 6r„, etc. We begin by constructing G--^. We observe that 1 ii,n ^1^1 _fl, for< = 0, n=«l+n\t\ lO, fori^O. Now the equation of a right line I may be given the form X cos a. + y sin a = p where 00. Hence 1, on I, .0, off Z. If now we make I coincide with ej, we see that H, (x, y} = 2Z(x, y-)L{x, y) = | ^'^ ^^ ^i_ Hence ^ N TT ,- N f I1 in G except on e-., a,ix, y-) = Gix, y) - E,(,x. y) = [^^^^ .^andwithoutV w U c os « + y sin « — p _ Z(x, y)= 1 — lim !— I 7 ^ ^ < ^ -^"^ n=» 1 + w a; cos a + ^ sin a — JO 466 DISCONTINUOUS FUNCTIONS In the same way, <^1234 = G- — (-^1 + -^2 + -^8 + -^4)- By introducing a constant factor we can replace Q by a square Qc whose sides are in the ratio c : 1 to those of Q. Thus 1 f ^, on the edge of (>„ »=» 1 + [^ + '-^'j lO, outside. We can replace the square Q by a similar square whose center is a, h on replacing \x\, \y\ by \x— a , \y — h\. We have thus this result : by a limiting process, we can con- struct a function g (x, y') having the value 1 inside Q, and on any of its edges, and = outside Q, and on the remaining edges. Q has any point a, h as center, its edges have any length, and its sides are tipped at an angle of 45° to the axes. We may take them parallel to the axes, if we wish, by replacing \x\, \y\ in our fundamental relation 1) by \x-y\ , \x + y\. Finally let us remark that we may pass to wi-way space, by re- placing 1) by l^il +12^2 1 + - + |a;„| = l. 3. Let now O = Jq„| be a border set [328], of non-overlapping squares belonging to the complete apantactic set £, such that O -f- E = 8? the whole plane. We mark these squares in the plane and note which sides q„ has in common with the preceding q's. We take the gj^xy) function so that it is = 1 in q„, except on these sides, and there 0. Then Gr{x, z/) = "^gjxy^ has for each point only one term ^ 0, if a;, y lies in Q, and no term ^ if it lies in (5. p . \ _ I li for each point of O, 1 0, for each point of S. EXAMPLES OF DISCONTINUOUS FUNCTIONS 467 Since S is apaiffiactic, each point of 6 is a point of disconti- nuity of the 2° kind ; each point of Q is a point of continuity. 4. Let/(a;i/) be a function defined over 21 which contains the complete apantactic set E. I 0, in S. 472. 1. Let 2t = (0, 1), J8„= the points ll!L±A in 31. Then 53„, S3„ have no points in common. Let/„(a;) = 1 in 58„, and = in ^„ = SI - SB„ . LetS8=fS„S. Then J'(x)=2/„(a;) = ■ 1, in «, , 0, in 5 = 21 - 93. The function F is totally discontinuous in 53, oscillating be- tween and 1. The series F does not converge uniformly in any subinterval of 21. 2. Keeping the notation in 1, let G'(^)=2;-/n(*)- At each point of 93„, G=-. while a = Om£. n The function G- is discontinuous at the points of SB, but con- tinuous at the points £. The series Gr converges uniformly in 21, yet an infinity of terms are discontinuous in any interval in 21. 473. Let the limited set 21 be the union of an enumerable set of complete sets {2I„{. We show how to construct a function/, which is discontinuous at the points of 21, but continuous else- where in an »n-way space. JJet us suppose first that 2t consists of but one set and is com- plete. A point all of whose coordinates are rational, let us call rational, the other points of space we will call non-rational. If 21 has an inner rational point, let /= 1 at this point, on the frontier of 21 let /= 1 also ; at all other points of space let /= 0. Then each point a of 21 is a point of discontinuity. For if a; is a fron- 468 DISCONTINUOUS FUNCTIONS tier or an inner rational point of %f(x) = 1, while in any V(x) there are points where /= 0. If x is not in 21, all the points of some i)(2;) are also not in 21. At these points /= 0. Hence /is continuous at such points. We turn now to the general case. We have 21 = ^1+^2 + ^3+ — where ^j = 2Ii , -dj = points of 2I2 not in 2li, etc. Let/^ = 1 at the rational inner points of A-^, and at the frontier points of Slj ; at all other points let /j = 0. Let /j = ^ at the rational inner points of A^, and at the frontier points of A^ not in A^ ; at all other points let /j = 0. At similar points of A^ let/3 — h ^'^'^ elsewhere = 0, etc. Consider now iji vr /^ ^ \ Let x= a be a point of 21. If it is an inner point of some A,, it is obviously a point of discontinuity of F. If not, it is a proper frontier point of one of the A's. Then in any D (a) there are points of space not in 21, or there are points of an infinite number of the jI's. In either case a is a point of discontinuity. Similarly we see F is continuous at a point not in 21. 2. We can obviously generalize the preceding problem by sup- posing 21 to lie in a complete set 33, such that each frontier point of 21 is a limit point of ^ = SB - 21. For we have only to replace our m-way space by S3. Functions of Class 1 474. 1. Baire has introduced an important classification of functions as follows : Let /(xy-'-x^) be defined over 21 ; /and 21 limited or unlimited. If/ is continuous in 21, we say its class is in 21, and write Class /=0 , or Cl/=0 , Mod 21. " f=limUxy...x^), n=oo each/„ being of class in 21, we say its class is 1, if/ does not lie in class 0, mod 21. FUNCTIONS OF CLASS 1 469 2. Let the serial. F(x) = l.Ux) converge in 21, each term/„ being continuous in SI. Since we see F is of class 0, or class 1, according as F is continuous, or not continuous in 21. A similar remark holds for infinite prod- 3. The derivatives of a function /(a;) give rise to functions of class or 1. For let fix) have a unilateral differential coeffi- cient g (x) at each point of 21. Both / and 21 may be unlimited. To fix the ideas, suppose the right-hand differential coefficient exists. Let h-^>h^>--- = (). Then is a continuous function of x in 21. But 3(a;) = limg'„(a;) 71=00 exists at each a; in 21 by hypothesis. A similar remark applies to the partial derivatives dx^^ dx„ of a function /(a;j ••• x„). 4- Let f(x) = ]imUx,-x^), 71=00 each /„ being of class 1 in 21. Then we say, Cl/= 2 if / does not lie in a lower class. In this way we may continue. It is of course necessary to show that such functions actually exist. 475. Example 1. 1, for a; > 0, .0, for a: = 0. Let ^/- N T nx n=co 1 + nx This function was considered in I, 331. In any interval 21= (0 < 6) containing the origin a; = 0, Cl/= 1 ; in any inter- val (a < 6), a > 0, not containing the origin, Cl/= 0. 470 DISCONTINUOUS FUNCTIONS Example 2. ^^* /(:.) = lim^ = 0, in 21= (-00, 00). The class of /(a;) is in 21. Although each /„ is limited in 21, the graphs of /„ have peaks near x = Q which = oo, as w = oo. Hxample S. If we combine the two functions in Ex. 1, 2, we /(a;) = Hm | — ^ — + — 1 ! ^^* r 1 1 1 fl, foi-rc^O, 1 0, for a; = 0. Hence CI /(a;) = 1 for any set 53 embracing the origin ; = for any other set. Example Jf. _l_ ^®* /(a;) = Km a;/""" , in 21 = (0, 1). n=0D '^^^^ fix) = Q , fora;=0 1 = a;e^ , for a; > 0. We see thus that / is continuous in (0*, 1), and has a point of infinite discontinuity at a; = 0. Hence Class /(a;) = 1, in 21 = 0, in (0*, 1). Example 5. ^^^ f (a;) = lim -^ in 21 = (0, oo). "=" x + - n Then /(a:)=l , for :. > X = + 00 , for a; = 0. Here lim/„(a;) fl=oo does not exist at a; = 0. We cannot therefore speak of the class of /(a;) in 21 since it is not defined at the point a; = 0. It is defined in 93 = (0*, oo), and its class is obviously 0, mod 93. FUNCTIONS OF CLASS 1 471 Example 6. Let /(a;) = sin - , for a; ^ z = a constant a , for x = 0. We show that CI/ = 1 in 21 = (- 0, oo). For let f„Cx)=c(l--^) + \ 1 + nxj 1 nx + nx sm = ffni^) + KCx). r 1 1 a; + n ) Now by Ex. 1, 1 • / N f 0, for a; > 0, hni cr„(a;) = -^ , „ ^"^ ^ U, fora;= 0; lim A„ (a;) = sin -, for a; > 0, X while iim rt„ {^xj = . 0, for x=0 As each /„ is continuous in 21, and lini/„(a;)=/(a;)in2l, we see its class is < 1. As / is discontinuous at a; = 0, its class is not in 21. Example 7. Let .^ ^ ,. 1 .1 ^ / (x) = nm - • sin - . „=«, w X Here the functions /„(a;) under the limit sign are not defined for x = 0. Thus /is not defined at this point. We cannot there- fore speak of the class of / with respect to any set embracing the point x=0. For any set SB not containing this point, CI /= 0, since /(a;) = in S3. Let us set Let (_x} = sin - , for a; =^ = a constant c for x=0. gipc) = lim - (/.(a;) = lim ^„(a;). 472 DISCONTINUOUS FUNCTIONS Here ^ is a continuous function in 21 = (— oo, oo). Its class is thus in 21. On the other hand, the functions 0 is taken, 21 is an @, set, we shall say 21 is an ®, set, e = 0, which we may also express by (gj^o. 2. Let f(x^--- x^ he continuous in the limited complete set 21. Then 21 is an ©» set, e = 0. For let e > be taken small at pleasure and fixed. By I, 353, there exists a cubical division of space D, such that if 2l„ denote the points of 21 in one of the cells of D, Osc/< e in 2l„. As 2l„ is complete, since 21 is, 21 is an S^ set. 3. An enumerable set of points 21 = {a„i is an @,io set. For each a„ may be regarded as a complete set, embracing but a single point. But in a set embracing but one point, Osc/= 0. 4. The union of an enumerable set of 6^ sets 21 = f 21^} is an (S^ set. For each 21^ is the union of an enumerable set of limited sets 2t„ = [2fm,„|, »» = 1, 2, ... and Osc/0 be taken small at pleasure and fixed. About each point of S) we describe a sphere of radius p. Let % denote the points of 21 not mthin one of these spheres. Obviously 2Ip is com- plete. Let p range over r-^>r^> ■•■ =0. If we set 2( = J. + 2), obviously A = l%J. As/ is continuous in 2lr„, it is an (g^ set. Hence 21, being the union of A and ©, is an @^ set. 478. 1. Let 21 be an @, set. The points 3) o/ 21 common to the limited complete set SQform an @, set. For 21 is the union of the complete sets 2l„, in each of which Osc/somejB, and we may take e and n so that m, + -p. FUNCTIONS OF CLASS 1 476 But as/„(a) =jQ(a), we have |/(«) -/<,(«)! somey. Hence if e is sufficiently small, and thus a is in St^. Thus 1) is established. But S)„p is a divisor of complete sets, and is therefore complete. Thus % is the union of an enumerable set of complete sets j58J, in each of which Osc/ 0. For in any subset 93 of 21 containing the point x, Osc f>k. Hence Osc/ is not iBj > ••■ — 0- ^7 hypothesis 21 is the union of a sequence of complete sets in each of which Osc /< Wi- 21 is also the union of a sequence of complete sets 476 DISCONTINUOUS FUNCTIONS in each of which Osc/< Wj. If we superpose the division 1) of 21 on the division S^, each SI,, will fall into an enumerable set of complete sets, and together they will form an enumerable sequence in each of which Oscf^. Continuing in this way we see that 21 is the union of the complete sets sr„i , 2i„2 , 2i„3-. (6; such that in each set of S„, Osc/< 2l2.,>->2l„.„. To 2l„.„ belongs the field constant <7„,^ ; this we associate with the lattice point I and call it the corresponding lattice constant. Let now S be a cell of 2)„ containing a point of 21. It has 2" vertices or lattice points. Let P, denote any product of a differ- ent factors x„, Xr,, ••• x^^. We consider the polynomial 4> = AP„+ ^BF„_, + 2 CP„_, + ... + IKP, + L, the summation in each case extending over all the distinct products of that type. The number of terms in is, by I, 96, (:0H")+©*--^^=^-- FUNCTIONS OF CLASS 1 477 We can thus detOHiiine the 2" coefficients of function. We now define f„(x^ ■■■ x^) by stating that its value in any cube S of i)„, containing a point of 21, is that of the correspond- ing 0, V, |/(a;)-/„(z)|v. (3 Let Wj < e/8. Let Slu^ be the first set in S^ containing the point x, 2I2.J the first set of S^ lying in 2lu, and containing x. Continuing we get 2lu,>2r,,>2l3.,>->3re.,- Let ^e be the union of the sets in S-^ preceding 21,. ; of the sets in S^ preceding 212. and lying in 2li, , and so on, finally the sets of S^ preceding 2lei,, and lying in 2le_,,.^_,. Their number being finite, S= Dist (2l„^, ^e) is obviously > 0. We may therefore take i- > e so large that cubes of Z>„ about the point x lie wholly in D^(x), J? < S. Consider now/„(a;), n> v, and let us suppose first that x is not a lattice point of 2)„. Let it lie within the cell S of i>„. Then fn(x^ is a mean of the values of where I is any one of the 2" vertices of E, and C„y^ is the corre- sponding lattice constant, which we know is associated with the set2l„,„. We observe now that each of the '*»«n <2l..,. (4 For each set in S„ is a part of some set in any of the preceding sequences. Now 2l„j„ cannot be a part of Slij,, 4 < tj, for none of 478 DISCONTINUOUS Jj'UNCTIONS these points lie in Dr,(x)- Hence 3l„j_^ is a part of Slj.j. For the same reason it is a part of Slj.j, etc., which establishes 4). Let now x' be a point of 2l„y„ . Then I ^.,„ - t\^ I < I C„,, -fix') I ■+ |/(z')- O;, I 6/2 form a complete part of 21, by 462, 3. But U, being a part of @, is enumerable and is hence an g, set by 477, 3. Let us turn to i8 = 21 — -£^. For each of its points J, there exists a S> 0, such that Osc/< e in the set 6 of points of i8 lying in DsCb). As 21 is complete, so is 6. As U is complete, there is an enumerable set of these b, call them 61, h^ •••, such that 35 = \b,\. As 21 = 33+ -E', it is the union of an enumerable set of complete sets, in each of which Osc/ < e. This is true however small e>0 is taken. We apply now 480, 1. 2. We can now construct functions of class 2. Example. Let /„(a;j ••• a;,„)= 1 at the rational points in the unit cube Q, whose coordinates have denominators < n. Else- where let/„ = 0. Since /„ has only a finite number of discontinu- ities in O, Cl/„ = 1 in Q. Let now fi^i ■■■ a:„) = lim/„. n— some s. Hence at such a point /= 1. Thus each point of O is a point of discontinuity and Disc/= 1. Hence 01/ is not 1. As / is the limit of functions of class 1, its class is 2. 483. Let f(x-^ ••• a;^) he continuous with respect to each x^, at each point of a limited set 21, each of whose points is an inner point. Then Glass f< I. 480 DISCONTINUOUS FUNCTIONS For let 21 lie within a cube Q. Then ^ = O — 21 is complete. We may therefore regard 21 as a border set of A ; that is, a set of non-overlapping cubes iq„J. We show now that Cl/<1 in any one of these cubes as q. To this end we show that the points 58„ of q at which a+~e2>-"=0. Let a^ be a point of continuity of /j. Then „ „ Osc/i < e , in some 2ri= V^la^). In 2li there is a point h of continuity of /j. Hence Osc/j < e^ in some F,(S), and vi^e may take h so that F,(J)<2li. But in F,(6) there is a point a^ at which /2 is continuous. Hence Osc/i<62 , Osc/2v|2l„| contains at least one point a of 21. But at this point each/^ is continuous. 485. 1. Let 21 = 33 + 6 le complete, let 93, 6 he pantactie with reference to 21. Then there exists no pair of functions f g defined over 21, such that if 93 are the points of discontinuity of f in 21, then 93 shall he the points of continuity of g in 21. This is a corollary of Volterra's theorem. For in any Vi(^a) of a point of 21, there are points of 93 and of S. Hence there are points of continuity of/ and g. Hence/, g are at most pointwise discontinuous in 21. Then by 484, there is a point in 21 where/ and g are both continuous, which contradicts the hypothesis. 2. Let 21= 93 +S he complete, and let 93, S each he pantactie with reference ^0 21. If^ is hypercomplete, S is not. For if 93, @ were, the union of an enumerable set of complete sets, 473 shows that there exists a function / defined over 21 which has 93 as its points of discontinuity ; and also a function g 482 DISCONTINUOUS FUNCTIONS which has @ as its points of discontinuity. But no such pair of functions can exist by 1. 3. The non-rational points 3? *w '"wy cube Q cannot be hyper- complete. For the rational points in Q are hypercomplete. 4. As an application of 2 we can state : The limited function /(a;j •••.a;„) which is < at the irrational points of a cube O, and > at the other points 3 of Q, cannot be of class or 1 in d. For if CI/ < 1, the points of Q where/ > must form a hyper- complete set, by 479, 4. But these are the points 3. 486. 1. {Baire.") If the class of f(x-y-x^) is 1 in the com- plete set 21, it is at most pointwise discontinuous in any complete S8<21. If CI/ = 1 in SI, it is < 1 in any complete S3 < §t by 481, 4 ; we may therefore take 93 = 21. Let a be any point of 21. We shall show that in any V = Vi(^a') there is a point c of continuity of /. Let ej > €2 > ••• = 0. Using the notation of 479, 1, we saw that the sets % = {m^ F2 > ••■ defines a point c in Fat which /is continuous. 2. If the class off(x-y ■•• a;„) is 1 in the complete set 21, its points of discontinuity 'S^form a set of the first category. FUNCTIONS OF CLASS 1 483 For by 462, 3, tte points 0„ of S at which Osc/> - form a n complete set. Each £)„ is apantactic, since / is at most pointwise discontinuous, and C)„ is complete. Hence S) = ]On\ is the union of an enumerable set of apantactic sets, and is therefore of the 1° category. 487. 1. Let f he defined over the limited complete set 21. If Class / is not < 1, there exists a perfect set 35 in 31, such that f is totally diseovMnuous in S). For if CI/ is not <1 there exists, by 480, an e such that for this e, 21 is not an Se set. Let now c be a point of 21 such that the points o of 21 which lie within some cube q, whose center is c, form an ©^ set. Let 33= \a\, 6= \e\. Then S3 = S. For obviously S < S3, since each c is in some a. On the other hand, 58 < £. For any point h of 33 lies within some q. Thus h is the center of a cube q' within q. Obviously the points of 21 within q' form an @e set. By Borel's theorem, each point c lies within an enumerable set of cubes fc„|, such that each c lies within some q. Thus the points o„ of 21 in c„, form an @e set. As 6 = Ja„!, E is an g^ set. Let S) = 2t - S. If 3) were 0, 21 = S and 21 would be an g, set contrary to hypothesis. Thus 35 > 0. 35 is complete. For if I were a limiting point of 3) in E, I must lie in some c. But every point of 21 in c is a point of 6 as we saw. Thus I cannot lie in £. We show finally that at any point d of 35, Osc/>e, with respect to 35. If not, Osc/e at each point of 3), each point of 35 is a point of discontinuity with respect to S). Thus/ is totally discon- tinuous in 35. This shows that 35 can contain no isolated points. Hence 35 is perfect. 484 DISCONTINUOUS FUNCTIONS 2. Let f be defined over the limited complete set 21. If f is at most pointwise discontinuous in any perfect 33 < 21, its class is < 1 in 21. This is a corollary of 1. For if Class / were not 0, or 1, there exists a perfect set 35 such that/ is totally discontinuous in 3). 488. If the class off, g Q in ^, the class of ^=yfis<\. For example, let us consider the product h =fg. If CI h is not < 1, there exists a perfect set 35 in 21, as we saw in 487, l, such that h is totally discontinuous in S). But/, g being of class < 1, are at most pointwise discontinuous in 35 by 486. Then by 484, there exists a point of 35 at which /, g are both continuous. Then h is continuous at this point, and is therefore not totally discon- tinous in 35. Let us consider now the quotient is not < 1, ^ is totally discontinuous in some perfect set 3) in 21. But since/ > in 35,/ must also be totally discontinuous in 35. This contradicts 486. 489. 1. Let F=: ^f^-L^^i ••• s^m) converge uniformly in the com- plete set 21. Let the class of each termf be < 1, then Class f <1 inn. For setting as usual [117], F=F^ + F^ (1 there exists for each e > 0. a fixed rectangular cell R),, such that I ^;i I < €, as a; ranges over 21. (2 As the class of each term in F^ is < 1, CI J\ < 1 in 21. Hence 21 is an (§., set with respect to F^. From 1), 2) it follows that 21 is an @, set with respect to F. 2. Let ^ = n/j...^,(a;| ••• a;„) converge uniformly dn the complete set 21. If the class of eachf^ is < 1, then C\ F <\ in 21. SEMICONTINUOUS FUNCTIONS 485 Semicontinuous Functions 490. Let /(^i • • • x„) be defined over 21. If a is a point of 21, Max/ in Vs(^a} exists, finite or infinite, and may be regarded as a function of S. When finite, it is a monotone decreasing function of 8. Thus its limit as 8 = exists, finite or infinite. We call this limit the maximum off at x = a, and we denote it by Max/. Similar remarks apply to the minimum of /in Vs(^a}. Its limit, finite or infinite, as S == 0, we call the minimum of f at x = a, and we denote it by Min/. The maximum and minimum of /in Vsi^a) may be denoted by Obviously, 491. Example 1. Then Example 2. Then Example S. Then Max/ , Min/. a, 5 (/, 5 Max (-/)=- Min/ Min (-/)=- Max/. x=a x=a /(a;) = iin(-l, 1) , X for x^O = , for « = 0. Max/=+oo , Min/=-oo. x=0 x=o 1 . /(2;) = sin - in (-1, 1) for x^O = Max/= 1 for a; = 0. Min/=-l. X=Q X=0 /(:c) = lin (-1, 1) , forrc^O = 2 , for a; = 0. Max/= 2 X=Q Min/=1. x=0 486 DISCONTINUOUS FUNCTIONS "We observe that in Exs. 1 and 2, ii^/=Max/ , liin/=Mm/; while in Ex. 3, lim/= 1 , and hence Max/> lim/. 1=0 1=0 x=0 ^^^° lim/= Min/. x=0 x=0 Example 4- i /(a;) = (a?+l)sini in(-l, 1) , fora;=jt:0 X = -2 , fora;=0. ^^""^ Max/= 1 , Min/= - 2, x=0 x=0 lim/= 1 , lim/= — 1. 1=0 ;r=0 Examples. Let j^^,) ^ ^^ , for rational x in (0, 1) = 1 , for irrational x. ^^^*^ Max/=1 , Min/=0, 1=0 x=o ihii/=i. x=0 492. 1. For M to be the maximum of f at x = a, it is necessary and sufficient that 1° € > 0, S > 0, fipc) < M+ e, for any x in V6(a) ; 2° there exists for each e > 0, and in any ^{(a), a point a such that M-e0, o>0, Max/< iff +e. But for any x in Fj(a), /(x) < Max/. SEMICONTINUOUS FUNCTIONS 487 Hence • j?^ n ^^ . „ ^ f(x) < M+ e , a; in Vsia), which is condition 1°. As to 2°, we remark that for each e > 0, and in any TjCa), there is a point a, such that -e+ Max/ is small at pleasure. From 2° we have Max/> Jf-e, and hence letting 8=0, Max/ > M. (2 From 1), 2) we have M= Max/. 2. J'or m to he the minimum of f at x = a, it is necessary and sufficient that 1° e > 0, S > 0, m — e 0, and in any Fj (a), a 'point a such that f(^a) 0, there exists a S > 0, such that f(x) < /(a) + e , for any x in ViCja). (1 Similarly the condition for infracontinuity is f{a) - e < fix') , for any x in F{(a). (2 Let us prove 1). It is necessary. For when /is supracontinu- ous at a, /(a) = Max /(a;). Then by 492, l, e>0 , S>0 , f(x') 0, 8 > , /(a;) < /(a) 4- e' , iovanyxuiVsCa). Thus if e' < e _ r /(«) < /(«) + e , for any x in F, (a) , v = ^. 3. The sum of two supra (infra') continuous functions in 21 is a supra (infra) continuous function in 2{. For let/, g be supracontinuous in 21 ; let/ + ^ = h. Then by 1, /(z) c an arbitrary constant form a complete set. For let/> e at 6j, Sj ••• which = 6 ; we wish to show that h lies in SB. Since /is supiacontinuous, by 494, l, f{x) some m. Hence c is small at pleasure, and b lies in 93. 496. 1. The oscillatory function oi^x) is supraeontinuous. For by 493, „(^)= Max/- Min/ = Max/+Max(-/). But these two maximal functions are supraeontinuous by 494, 2. Hence by 494, 3, their sum w is supraeontinuous. 2. The oscillatory function to is not necessarily infracon- tinuous, as is shown by the following Example. /= 1 in (—1, 1), except for x = 0, where /= 2. Then (oQc) = 0, except at a; = 0, where w = 1. Thus Min €2> ••■ =0. Since Min to (a;) = 0, there exists in any Vi(^a) a ponit a^ such that a)(o(j)<|ej. Hence (o{x') < e^ in some F6,(ai) < Fj. In Fj, there exists a point «2 such that o)(a;) F{, > ... = a point a of 21, at which / is obviously continuous. Thus in each V^ia) is a point of continuity of/. Hence /is at most pointwise discontinuous. 497. 1. At each point X of^, cji = Min S/(z) -f(:x)\, and yjr = Min J/(a;) -/(z) J are both = 0. Let us show that = at an arbitrary point a of 21. By 494, 2, f(x) is supracontinuous ; hence by 494, l, /(z) is small at pleasure, this gives ,^(«) = 0. 492 DISCONTINUOUS FUNCTIONS 2. If f is semicontinuous in the complete set 21, it is at most point- toise discontinuous in 21. For «(a;) =/(a;) -fix) To fix the ideas let / be supracontinuous. Then <^ = in 21. Hence 1) gives Min a)(x) = Min ifr(x) = 0, by 1. Thus by 496, 3, / is at most pointwise discontinuous in 21. CHAPTER XV DERIVATES, EXTREMES, VARIATION Derivates 498. Suppose we have given a one-valued continuous function f{x) spread over an interval 21= (a<6). We can state various properties which it enjoys. For example, it is limited, it takes on its extreme values, it is integrable. On the other hand, we do not know 1° how it oscillates in 21, or 2° if it has a differ- ential coefficient at each point of 21. In this chapter we wish to study the behavior of continuous functions with reference to these last two properties. In Chapters VIII and XI of volume I this subject was touched upon ; we wish here to develop it farther. 499. In I, 363, 364, we have defined the terms difference quo- tient, differential coefficient, derivative, right- and left-hand dif- ferential coefficients and derivatives, unilateral differential coeffi- cients and derivatives. The corresponding symbols are Arc i/'(a) , RfXx-) , Lf'(x). The unilateral differential coefficient and derivative may be de- noted by Tlf'ia) , Uf'ix-). (1 When ^. lim-^ A=o Aa; does not exist, finite or infinite, we may introduce its upper and lower limits. Thus /'(a) = Ii^|^ , /'(a)=lim^ (2 A=o Ax - -^srAa; always exist, finite or infinite. We call them the upper and lower differential coefficients at the point x=a. The aggregate of values 493 494 DERIVATES, EXTREMES, VARIATION that 2) take on define the upper and lower derivatives of /(a;), as in I, 363. In a similar manner we introduce the upper and lower right- and left-hand differential coefficients and derivatives, Bf , Bf , Lf , Lf. (3 Thus, for example, ^/'(a)= i2li^^^^^±%==^^, A=o h finite or infinite. Cf. I, 336 seq. li f(x) is defined only in 21 = (a < j8), the points a, a + h must lie in SI. Thus there is no upper or lowet right-hand differential coefficient at a: = /3 ; also no upper or lower left-hand differential coefficient at a; = a. This fact must be borne in mind. We call the functions 8) derivates to distinguish them from the deriva- tives Bf, Lf. When Bf (a} = Bf (a), finite or infinite, Bfia) exists also finite or infinite, and has the same value. A similar remark applies to the left-hand differential coefficient. To avoid such repetition as just made, it is convenient to in- troduce the terms upper and lower unilateral differential coeffi- cients and derivatives, which may be denoted by Uf , Ef>. (4 The symbol U should of course refer to the same side, if it is used more than once in an investigation. When no ambiguity can arise, we may abbreviate the symbols 3), 4) thus: B , B , L , L , U , U. The value of one of these derivates as -B at a point x = a may similarly be denoted by _ i2(«). The difference quotient a — h may be denoted by A(a, by DERIVATES 495 Example 1. * f(x) = xsin- , a;#=0 in (- 1, 1) X ^ = , x = 0. A sin - Here for :c=0, ^=__i = sini. Ax h h Hence _ fi/'(0) = + l , ^/'(0)=-l, X/'(0)=4-l , i/-'(0)=-l, /'(0) = +l , /'(0)=-l. Example 2. f{x) = 2;^ sin i , 2; ^ in ( - 1, 1) X = , a;=0. . 1 Af ^'"^ Here for x=0 . — = Az Hence ^/'(0)= + oo , ^/'(0)=-oo, i/'(0)= + oo , i/'(0)=-oo, 7'(0)= + oo , /'(0)=-OO. Example 3. /(*) = 2; sin - , for < a; < 1 X = x^ sin - , for — 1 < a; < X = , for2;=0. Here ^/'(0)=+l , ^/'(0) = -l, Af'(0)= + oo , i/'(0)=-oo, /'(0)= + oo , /'(0)=-oo. 500. 1. Before taking up the general theoiy it will be well for the reader to have a few examples in mind to show him how complicated matters may get. In I, 367 seq., we have exhibited functions which oscillate infinitely often about the points of a set 496 DERIVATES, EXTREMES, VARIATION of the 1° species, and which may or may not have differential co- efficients at these points. The following theorem enables us to construct functions which do not possess a differential coefficient at the points of an enumer- able set. 2. Let (S = {e„} he an enumerable set lying in the interval 21. For each X in 21, and e„ in (5, let a; — e„ lie in an interval 58 containing the origin. Let g(x) be continuous in 33. Let g'{x) exist and be numerically < M in 58, except at x=0, where the difference quotients are numerically < M. Let A = 2a„ converge absolutely. Then F{x}=1.a„g(x-e„) is a continuous function in 21, having a derivative in S = 21 — (S. At the points of S, the difference quotient of F behaves essentially as that of g at the origin. For g(_x) being continuous in 58, it is numerically < some con- stant in 21. Thus F converges uniformly in 21. As each term g(^x — e„) is continuous in 21, F is continuous in 21. Let us consider its differential coefficient at a point x of £• Since g'(^x — e„) exists and is numerically < M, F'(x)=^a„g\x-e,^ , by 156, 2. Let now a; = e,„, a point of @, F(ix) = a^g(ix - e^) + ^*a^(x - e„) = anSi^ - «m) + G-(x). The summation in 2* extends over allw=ifcm. Hence by what has just been shown, Cr has a differential coefficient at a; = e^ . Thus behaves at a; = e^, essentially as — at a; = 0. Hence Aa; Aa: UF'ie^^ = a^Ug'(Q-) + (?'(e„). (1 501. Example 1. Let g(x) = ax , x> J < < a. = bx , a; < 0, DERIVATES 497 Then • is continuous in any interval 21, and has a derivative iit the points of SI not in @. At the point e^, ^ ■tlV' Let @ denote the rational points in 21. The graph of FQc) is a continuous curve having tangents at a pantactic set of points ; and at another pantactic set, viz. the set (g, angular points (I, 366). A simple example of a ^ function is g(x') =\x\ = + y/y?. Example 2. Let g^x) = x^ sin — , xi=(i = , a;=0. This function has a derivative a' (a;) = 22; sin tt cos— , a;=^0 X X = , 2; = 0. Thus if S(?„ is an absolutely convergent series, and @= 5e„j an enumerable set in the interval 21 = (0, 1), F(x) =^c„g(x- ej is a continuous function whose derivative in 21 is J"(a;) = 2cy(a;-e„). Thus F has a derivative vi^hich is continuous in 21 — 6, and at the point x = e„ since Disc ^' = 2 e^TT, Disc g'C^x^ = 2 TT. x=0 498 DERIVATES, EXTREMES, VARIATION If (5 is the set of rational points in 21, the graph of F{x) is a continuous curve having at each point of 21 a tangent which does not turn continuously as the point of contact ranges over the curve; indeed the points of abrupt change in the direction of the tangent are pantactic in 21. Example S. Let gCx) = a; sin log x^ , a; ^ = , x=0. Then g' (x) = sin log x^ -\- 2 cos log t? , X'^U. At a; = 0, -3. = sin log P Ax ^ which oscillates infinitely often between ± 1, as A = Ax = 0. Let @ = |e„| denote the rational points in an interval 21. The series J' = ^ - (a; - e„) sin log (x - e^f satisfies the condition of our theorem. Hence Fix) is a continu- ous function in 21 which has a derivative in 21 — @. At a: = e„, m?"(a:) = i-+G"(0 , Cr^'(a;) = -!+(?' (O. m'' ~ yrr Thus the graph of J' is a continuous curve which has tangents at a pantactic set of points in 21, and at another pantactic set it has neither right- nor left-hand tangents. 502. Weierstrass' Function. For a long time mathematicians . thought that a continuous function of x must have a derivative, at least after removing certain points. The examples just given show that these exceptional points may be pantactic. Weierstrass called attention to a continuous function which has at no point a differential coefficient. This celebrated function is defined by the series F(x') = S a" cos b^TTx — cos irx + a cos hwx + cfi cos b^vx -f- ••• (1 u where < a < 1 ; J is an odd integer so chosen that a6 > 1 + I ■TT. (2 DERIVATES 499 The series F cftiverges absolutely and uniformly in any interval 21, since , „ , , ^ |a"coso"7ra; | < «". Hence ^ is a continuous function in 21. Let us now consider the series obtained by diiferentiating 1) termwise, 0(x) = — 7r2(«5)''sin h^'irx. If ab < 1, this series also converges absolutely and uniformly, by 155, 1. In this case the function has a finite derivative in 2[. Let us suppose, however, that the condition 2) holds. We have ^F ■^ a" — — = 1. ^ 1 — ao ab — 1 Consider now _ „ „ §„ = 2 -=- J cos J"7r(2; 4- A) — cos t"7ra; j . m ft Up to the present we have taken h arbitrary. Let us now take it as follows ; the reason for this choice will be evident in a moment. where i„ is the nearest integer to b'^x. Thus _i and in particular the first is > 0. Thus sgn Q^ = sgn |! = sgn e„7;^. Thus if 2) holds, | ^„ | > | ^„ |. Hence from 3), sgn Q = sgn Q^ = sgn e„i7,„, and Let now m = oo . Since ?;„ = ± 1 at pleasure, we can make ^ = +00, or to — 00 , or oscillate between ± oo, without becoming definitely infinite. Thus F (x) has at no point a finite or infinite differential coefficient. This does not say that the graph of F does not have tangents; hut when they exists they must he cuspidal tangents. DERIVATES 501 503. 1. Voltejfa^s Function. In the interval 21 = (0, 1), let § = Jj/j be a Harnack set of measure 02. Hence /' (a;) is not J?-integrable, as § = A > 0. We have seen, in I, 549, that not every limited i2-integrable function has a primitive. Volterra's function .illustrates con- versely the remarkable fact that Not every limited derivative is R-integrahle. 2. It is easy to show, however, that The derivative of Volterra's function is L-integrahle. For let 2l;i denote the points of 21 at which /' (a;) > \. Then when \>l/m, m=l, 2, ••• %^ consists of an enumerable set of intervals. Hence in this case 21;^ is measurable. Hence 21;^, \>0, is measurable. Now 21 , X>0, differs from the foregoing by add- ing the points 3n in each S„ at which fix) = 0, and the points §. But each ^„ is enumerable, and hence a null set, and § is measur- able, as it is perfect. Thus 21;^, \>0, is measurable. In the same way we see '^^ is measurable when \ is. negative. Thus 21^^ is measurable for any X,, and hence i-integrable. 504. 1. We turn now to general considerations and begin by considering the upper and lower limits of the sum, difference, prod- uct, and quotient of two functions at a point x = a. Let us note first the following theorem : Letf(jc-^ ■•• z„) be limited or not in 21 which has x= a as a limiting point. Let 4>6 = Max/, <^j = Win fin Vi*(a'). Then lim/=lim<^s , lim/=lim4>s. This follows at once from I, 338. DERIVATES 503 2. Letf{x-^ ••'*Cm)) ^(a?! ••• x„) he limited or not in 21 which has x = a as limiting point. ^ lim f= a , lim g = ^ \imf=A , limff — B as x^ a. Then, fhese limits being finite, a + ^<^(f+g)0 , gix)>0, a^<'^fg0 , /3<0<5, Ay80 , ^(a;)>A>0, ^0 , g(ix)>0, a^ < lim fgk>(i. «0. , g{x) > 0. Let limg exist. Then lim fg = lira f- limg. (19 If also g(x) limfg = limf- limg. >k>0. (20 limf/g = \imf/limg. (21 Mmf /g = lim f /limg. (22 505. The preceding resiilts can be used to obtain relations be- tween the derivates of the sum, difference, product, and quotient of two functions as in I, 373 seq. DERIVATES 605 1. Let w(a;)«:M(a;)+t)(a;). Then Aw_Am Ai; ^ Az Ax Ax' ^ Thus from 504, 1), we get the theorem : Vu' + v' U< Uw' < Uu' + Uv'. (2 If u has a unilateral derivative Uu', Uw' = Uu' + Uv', (3 Uw' = Uu' + Uv'. (4 We get 3), 4) from 1), using 504, 15), 16). 2. In the interval 21, u, v are continuous, u is monotone, increasing, V is > 0, and v' exists. Then, if w = uv, we have Uw' = wv' + vUu', (1 Uw' =uv' + vUu'. (2 For from Aw , , . ^ Av , Am —- =(u + Au) ■T-+v~—, Ax Ax Ax we have ^r / i , TT^■ Au Uw' = uv' + U lim V — — — - Ax , , rTT Au = uv' + V U lim — Ax which gives 1). Similarly we establish 2). 506. 1. We show now how we may generalize the Law of the Mean, I, 393. Let f(^x) he continuous in 31= (a.<5). Let m, M he the mini- mum and maximum of one of the four derivates off in 21. Then for any «0. /S — a 506 UERIVATES, EXTREMES, VARIATION We introduce the auxiliary function cx}=f<:x}-<:M+c^x, (2 ^'^ere 0C^) - 4>(ia) ^ /(/3) -/(«) g Hence ,^(^) - ,^(«) = S(^ _ «) = ^. Consider now the equation It is satisfied for x= a. If it is satisfied for any other x in the interval ^"^0^, there is a last point, say x = y, where it is satisfied, by 458, 3. Thusfora;>7, ./.(a;) is ><^(«). M'^"^^ B' 0. (3 Now from 2) we have Rf'(y) = .R'(y:} + M+e >M. Hence M is not the maximum of Rf'(x') in 21. Similarly the other half of 1) is established. The case that ?w or iltf is infinite is obviously true. 2. Letf(x) be defined over 21 == (a < J). Let a^ is small at pleasure. We show now that _ ^ ^o For there exists an a in 21, such that E(a~) < /i + e. There exists therefore a /3 > a in 21, such that a— p Thus by 506, 1, m = Min L is taken. 5. If one of the. derivates of the continuous function f(x) is continuous in an interval 21, the derivative f'{x) exists, and is con- tinuous in 21. This follows from 3. 6. If one of the four derivates of the continuous function f(x) is = in an interval 2t, /(a;) = const in 21. This follows from 3. 508. 1. If one of the derivates of the continuous function f{x) is > in 21 = (a < J), f(x) is monotone increasing in 21. For then m = Min Bf > 0, in (a < a;). Thus by 506, i, /(^)-/(«)>o. 2. If one of the derivates of the continuous function f(x) is >^ in 21, fix) is monotone decreasing. 3. If one of the derivates of the continuous function f{x) is >.0 in 21, without heing constantly in any little interval of 21, f(x) is an increasing function in 21. Similarly f is a decreasing function in 21, if one of the derivates is <^ 0, without being constantly in any little interval of 21. The proof is analogous to I, 403. 509. 1. Let f(^x') be continuous in the interval'^, and have a deriv- ative, finite or infinite, within 21. Then the points where the deriva- tive is finite form a pantactic set in 21. For let a < /3 be two points of 21. Then by the Law of the ■ Mean, p — a As the right side has a definite value, the left side must have. Thus in any interval (a, /8) in 21, there is a point 7 where the differential coefficient is finite. 610 DERIVATES, EXTREMES, VARIATION 2. Let f(x) he continuous in the interval 2l = (a is a univariant function in 21. This is not possible, since <^ has the same value at a and h. 3. Let fips) he continuous in 21 = (a < J), and have a derivative, finite or infinite, in 21= (a*, 6). Then Min/(a;)<^'(a) a^> ••■ = a iw 21, such that /'(«) = lim/'(«„)- (1 *'°^ f(a+h}-f(a) .,, ^ ^ ^ , 1 /o ft Let now A range over h^> h2> ■•• == 0. If we set «„= «/, , the relation 1) follows at once from 2), since /'(a) exists by hypothesis. 510. 1. A right-hand derivate of a continuous function f(x') cannot have a discontinuity of the 1° kind on the right. A similar statement holds for the other derivates. DERIVATES 511 For let B(x) b« one of the right-hand derivates. It it has a discontinuity of the 1° kind on the right at a; = a, there exists a number I such that l—e in SI = - 1 , for a: < 0. Thus at a; = 0, i2 is continuous on the right, but has a discon- tinuity of the 1° kind on the left. 3. Let f{x) he continuous in % = (a, 5), and have a derivative, finite or infinite, in 21* = (a*, 5*). Then the discontinuities off'(x'y in 21, if any exist, must he of the second kind. This follows from 1. Example. ' r^.w^^;, 1 Then /(a;)=a^sin- , for a;^ in 21 = (0, 1) X = , for a; = 0. /'(a;) = 2 a; sin cos- , x + Q XX = , a:=0. The discontinuity of /'(a;) at a; = 0, is in fact of the 2° kind. 4. Let f{x) he continuous in 21 = (a< J), except atx=a, is a point of discontinuity of the 2° kind. Let fix) exist, finite or infinite, in (a*, J). Then x = a is a point of infinite discontinuity off'ix-). 512 DERIVATES, EXTREMES, VARIATION For if p = R lim/ (a:) , q = R lim/ (a;), there exists a sequence of points ai>a2 >•••=«, such that /(a„)=/>; and another sequence fi^> /32> ■■• =a, such that /()S„) = 3'. We may suppose an>/3„ , ora„<^„ , k=1, 2, ■•• Then the Law of the Mean gives where 7„ lies between «„, /8„. Now the numerator =p — q, while the denominator = 0. Hence Q„= + co, or — oo , as we choose. 5. Ziet f (a;) have a finite unilateral differential coefficient U at each point of the interval 21. Then U is at most pointwise discon- tinuous in 21. For by 474, 3, U'ls & function of class 1. Hence, by 486, 1, it is at most pointwise discontinuous in 21. 511. Let fix) he continuous in the interval (a < S). Let R(x) denote one of the right-hand derivates of f(x). If R is not con- tinuous on the right at a, then lm. Let us choose t], and c such that OT + 17 < c < a. (2 We introduce the auxiliary function <^(a;)= cx—f(x). ^" R'(x') = c-Rf'(x). (3 Now if S > is sufficiently small, Rf'(^x')(7 , o->0. Hence ^(a;) is an increasing function in 21*. But, on the other ^™'^' Bf'id) = Rf'ia), since a> m. Hence ^<^'(a)= e - ^/'(a)= c - «< 0. Hence ^ is a decreasing function at x = a. This is impossible since 7. N°^ Q(h,-h)=QQ>-h,h-). Hence ^^j - A, A) > 7. If now we fix h, Q (a;, A) is a continuous function of a;. As Q is < 7, for X = a, and > 7, for x = b — h, it takes on the value 7 for some a;, say for a; = |, between a, h — h. Thus But by the Law of the Mean, Thus/' (a;) = 7, at a; = ?? in 21. 2. ie« /(a;) be continuous in the interval 21, and admit a deriva- tive, finite or infinite. If f'(x) = in 21, except possibly at an enumerable set @, then /' = also in @. 514 DERIVATES, EXTREMES, VARIATION For if /'(a) = 0, and /'(/3) = i ^ 0, then f'(x) ranges over all values in (0, 6), as x passes from a to ;8. But this set of values ■ has the cardinal number c. Hence there is a set of values in (a, j8) whose cardinal number is c, where /'(«) =#= 0. This is contrary to the hypothesis. 3. Let /(a;), g(x) he continuous and have derivatives, finite or infinite, in the interval 21. If in 21 there is an a for which f'ia^>9Xa\ and a /3 for which f'i/3)0 , B'(/3)<0. Hence by 1 there is a point where 8' = 0. 513. 1. If one of the four derivates of the continuous function f(x') is limited in the interval 21, all four are, and they have the same upper and lower R-integrals. The first part of the theorem is obvious from 507, l. Let us effect a division of 21 of norm d. Then \U = lim 1.MA , M, = Max B, in d,. But the maximum of the three other derivates in d^ is also M^ by 507, 1. Hence the last part of the theorem. 2. Let fix) be continuous and have a limited unilateral derivate as Rin^= (a< 6). Then CRd*0. Tl^"« Min/<^0; also let us suppose the given derivate is i2 = Bf'(x'). ^^* gix,t')=fix-)-f(ia-)-t(x-a-) , « > 0. Obviously \g\is the distance/ is above or below the secant line, y = t(x— a')+f(a'). Thus in particular for any t, g{a,t)=0 , g{c,t) = p-t(e-a^. Let 9' > be an arbitrary but fixed number < p. Then g(_c, t)-q:=p -q-t(^c-a) = (.p-q) iit0, DERIVATES 517 Hence • , ^ for any t in the interval J = (t, 7), < t < T. We note that Card SL = c. Since for any t in J, ^(a, t) = 0, and g{e, t) > q, let a: = e, be the maximum of the points < c where ffQx, €)= q. Then e < e, and for any h such that e + h lies in (e, c), """°" ^/(e) > 0. Thus for any t in J, e, lies in @. As ^ ranges over S, let e« range over gj < @. To each point e of gj corresponds but one point t of Si. For 0=g(e,f)-g(ie,t') = {t-t')ie-a). Hence . ,, t = t' , as e > a. '^^'^^^ Card X = Card gj < Card @, which is absurd. 2. Letf(x) he continuous in 21[ = (a< J), iei 6 denote the points of 21 where one of the derivates has one sign. If S exists. Card S = c, the cardinal number of the continuum. The proof is entirely similar to that in 1. For let c be a point of g. Then there exists a, d > e such that f(:d}-fCc-)=p>o. We now introduce the function g(x,f)=f{x)-fCc}-t(x-c} , f>0, and reason on this as we did on the corresponding g in 1, using here the interval (c, c?) instead of (a, by. We get Card gi = Card S=c. 3. Letf(x'), g(3>) he continuous in the interval 21. Let a pair of corresponding derivates as Rf\ Rg' he finite and equal, except pos- sibly at an enumerable set g. Then f = g-\-G, in 21(, where is a constant. 518 DERIVATES, EXTREMES, VARIATION For let Then in ^ = 21 - g, M(f>' > Rf-Rg' = , R^|t' > 0. But if Rij}' < at one point in 21, it is < at a set of points 93 whose cardinal number is c. But 93 lies in S. Hence Ii(f) is never < 0, in 21. The same holds for ■^. Hence, by 508, cf) and ■\jr are both monotone increasing. This is impossible unless 0. If ^ exists, it cannot be a null set. DERIVATES 519 For let c be a j^int of S, then there exists a point d > c such that f(:d)-f(c-) = p>0. (1 Let 6„ denote the points of S where n-l0. Hence ^^^^ _ ^^^^ - IQ^c) - QCd)l >0, or using 1), 3) p-q<0, which is not so, as ^ is > q. 2. (Lebesgue.) Let f(x'), g(_x) be continuous in the interval 21, and have a pair of corresponding derivates as Rf', Rg' which are finite at each point of 21, and also equal, the equality holding except possibly at a null set. Thenf{x) — g(x) = constant in 21. The proof is entirely similar to that of 515, 3, the enumerable set (S being here replaced bj'' a null set. We then make use of 1. 518. Letf'(x') be continuous in some interval A = (m — S, m + 8). Letf"(x') exist, finite or infinite, in A, but be finite at the point x = u. Then f(,u~) = \imQf (1 where Qf^^^^f(u+h)+f(u-h)-2f(u) ^ ^^^ Let us _^rs( suppose that/"(M) = 0. We have for 0 0, for any X in some V(a^, and 2°, /(a) —f(x) > for some x in any V{a'). If the sign > can be replaced by > in 1°, we will say / has a proper maximum at a, when we wish to emphasize this fact ; and when > cannot be replaced by >, we will say / has an improper maximum. A similar extension of the old definition holds for the minimum. A common term for maximum and minimum is extreme. 2. If /(re) is a constant in some segment S3, lying in the inter- val 21, SB is called a segment of invariability, or a constant segment of /in 21. Example. Let/(a;) be continuous in 21 = (0, 1*). Let /-I x= ■ a-^a^ag •■• (1 be the expression of a point of 21 in the normal form in the dyadic system. Let ^= . a^o^^a^- (2 be expressed in the triadic system, where «„=«„, when a„ = 0, and =2 when a„ = l. The points S={f| form a Cantor set, I, 272. Let j3f„| be the adjoint set of intervals. We associate 522 DERIVATES, EXTREMES, VARIATION now the point 1) with the point 2), which we indicate as usual by x~^. We define now a function g(x) as follows : 5'(l) =/(a;) , when a; - f This defines g for all the points of S. In the interval 3„, let g have a constant value. Obviously g is continuous, and has a pantactic set of intervals in each of which g is constant. 3. We have given criteria for maxima and minima in I, 468 seq., to which we may add the following : Let f{x) he continuous in (a— S, a + 8). If Rf'Qa)>Q and Lf'{d)< 0, finite or infinite, /(x") has a minimum at x = a. If Rf'(cL)<. and Lf'(ji)>(^, finite or infinite, f (x) has a maxi- mum at x = a. For on the 1° hypothesis, let us take a such that ^ — a > 0. Then there exists a S' > such that A Hence ^^^ _|_ ^^ >y(a) , a + A in (a*, a + S') • Similarly if /3 is chosen so that i + /3 < 0, there exists a 8" > 0, such that J, TV /.^ ^ /(a-A)-/(a) ^2 ^ ^ — h Hence ^^^ _ j^-^ -^/(^a) , a + A in (a - S", a*). 520. Example 1. Let /(a;) oscillate between the a;-axis and the two lines y = x and y = — x, similar to y = ■ IT a; Sin- ai In any interval about the origin, y oscillates infinitely often, hav- ing an infinite number of proper maxima and minima. At the point a; = 0, / has an improper minimum. Example 2. Let us take two parabolas Pj , Pj defined hy y = x^, y = lQ^. Through the points x-=±\, ±\"- let us erect ordi- nates, and join the points of intersection with Pj, Pg, alternately by straight lines, getting a broken line oscillating between the MAXIMA AND MINIMA 523 parabolas Pj , Pj* The resulting graph defines a continuous func- tion fix) which has proper extremes at the points S = | ± - [ • However, unlike Ex. 1, the limit point a; = of these extremes is also a point at which /(a;) has a proper extreme. Example 3. Let {8 j be a set of intervals which determine a Harnack set § lying in 31 = (0, 1). Over each interval S = (a, (8) belonging to the n'" stage, let us erect a curve, like a segment of a sine curve, of height A„ = 0, as w = oo, and having horizontal tangents at a, /S, and. at 7, the middle point of the interval S. At the points \^\ of 21 not in any interval S, let/(a;) = 0. The func- tion/ is now defined in 21 and is obviously continuous. At the points 1 7 J,/ has a proper maximum; at points of the type a, /3, ^, /has an improper minimum. These latter points form the set § whose cardinal number is c. The function is increasing in each interval (a, 7), and decreasing in each (7, /3). It oscillates in- finitely often in the vicinity of any point of §. We note that while the points where / has a proper extreme form an enumerable set, the points of improper extreme may form a set whose cardinal number is c. Example 4- We use the same set of intervals fSj but change the curve over S, so that it has a constant segment tj = (X, fi) in its middle portion. As before /= 0, at the points | not in the intervals B. The function / (a;) has now no proper extremes. At the points of §, / has an improper minimum ; at the points of the type \, /*, it has an improper maximum. Example 5. Weierstrass' Function. Let S denote the points in an interval 21 of the type X = ~ , r, s, positive integers. h' For such an x we have, using the notation of 502, I'^x = t„ + f m = h'^~'r. Hence f m = , for»w>s. Thus e„=(-l)'"'^' = (-l)'-^^- 524 DERIVATES, EXTREMES, VARIATION A IT Hence sgn -— = sgn ^ = sgn e„rj„ = sgn ( - lyh. "^^"^ BgnRf'(x) = + \ , sgni/'(a;) = -l, if r is even, and reversed if r is odd. Thus at the points (J, the curve has a vertical cusp. By 619, 3, F has a maximum at the points S, when r is odd, and a minimum when r is even. The points S are pantactic in 21. Weierstrass' function has no constant segment S, for then f'(^x) = in S. But F' does not exist at any point. 521. 1. Let f(^x-y ••• x^) he continuous in the limited or unlimited set 21. Let S denote the points of 2t where f has a proper extreme. Then (£ is enumerable. Let us first suppose that 21 is limited. Let S > be a fixed positive number. There can be but a finite number of points « in 21 such that /(«) >/(^) , in Fj^Ca). (1 For if there were an infinity of such points, let ;S be a limiting point and t; < | S. Then in F^(/3) there exist points a', a" such that F5(a'), Fj(a") overlap. Thus in one case and in the. other which contradicts the first. Let now B^> S^> ••• =0. There are but a finite number of points a for which 1) holds for S = 8j, only a finite number for 8= §25 etc. Hence (S is enumerable. The case that 21 is unlim- ited follows now easily. 2. We have seen that Weierstrass' function has a pantactic set of proper extremes. However, according to 1, they must be enumerable. In Ex. 3, the function has a minimum at each point of the non-enumerable set §; but these minima are improper. On the other hand, the function has a proper maximum at the points J7}, but these form an enumerable set. MAXIMA AND MINIMA 525 522. 1. Leffix) be continuous in the interval 21. Letf have a proper maximum atx = a, and x = ^ in 21. Then there is a point y between a, ^ where f has a minimum, which need not however be a proper minimum. For say « < yS. In the vicinity of «, /(a;) is 0, by > 0. 4. Let fix) be continuous in the interval 21, and let f{x) have no constant segments in 21. The points fi^) •, for any x in b. Let /t = Min/(a;), in b. Then there is a point x where / takes on this minimum value. The point e divides the interval b into two intervals. Let t be that one of these intervals which contains x, the other interval we denote by ttt- Within m let us take a point Cj of c. Then in I there is a point c[ such that Ac,) =/(ci). The point e^ determines an interval bj , just as c determined b. Obviously bjl,y is an increasing function, increas- ing still faster than before. MAXIMA AND MINIMA 529 All this can be^verified by analysis. For setting y = sinx+ ax , a > 1, we get , ° y' =a + cos X, and , ^ Thus ^ is a lineo-oscillating function in any interval. Example 2. /(a;) = a^sin- , x=f=0 X = , x=0. l(x') = ax+b , y =f(x) +l(x). Then ^ ^ «/' = 2 a; sin cos - + a , a; #: a; iB = a , »= 0. Hence, if a > 1 + 2 tt, ?/ is an increasing function in 31 = ( — tt, tt). The function /oscillates iniinitely often in 2t, but is a lineo-oscil- lating function. Example S. f{p) = a; sin - , x4=^ X = , x = Q. l(x') = ax+h , y=f{x) + l(x'). Here /■111, ^n y' = sni cos — \- a , x=^0. XX X For x—0,y' does not exist, finitely or infinitely. Obviously, however great a is taken, y has an infinity of oscilla- tions in any interval about x=0. Hence /is not a lineo-oscillat- ing function in such an interval. 2. If one of the four derivates of the continuous function /(a;) is limited in the interval 21, f(x) is at most lineo-oscillating in 21. For say ^/' > - « in 21. LetO<«=f(x-) + ax be monotone. Then Uf'=Ug'-a, and this case is reduced to the preceding. Remark. This shows that no continuous function whose graph has a vertical cusp can be lineo-oscillating. All its vertical tan- gents correspond to points of inflection, as in y = x^. Variation 526. 1. Letf(x) he continuous in the interval 21, and have limited variation. Let D he a division ofUof norm d. Then using the no- tation of 443, lim F^/=Ff , limP^/=P/ , limiV^/=i^. (1 For there exists a division A such that where for brevity we have dropped / after the symbol V. Let now A divide 21 into v segments whose minimum length call \. Let D be a division of 21 of norm d < M^d, < Mn. 2. LetfCx) he limited and R-integrahle iw 21= (a< 5). Then F (x) = I fdx , a^. ^= ^ | ^(a,+j) - ir'(aO | = 2 | f'^/cZs; | 1/ da; k, are finite in number. For suppose they were not. Then however large Gr is taken, we may take n so large that nk > Gr. There exists a division D VARIATION 533 of 21, such that'^here are at least n intervals, each containing a point of ^ within it. Thus for the division D, ^ Osc f>nk> a. Thus the variation of / is large at pleasure, and therefore is not limited. 2. If f has limited variation in the interval 21, its points of dis- continuity form an enumerable set. This follows at once from 1. 530. 1. Let 2)j, D^ ■■■be a sequence of superposed divisions, of norms d^ = 0, of the interval 21. Let fi^^ be the sum of the oscilla- tions of f in the intervals of D„ . If Max Il^Jn *'® finite, f (x) has limited variation in 21. For suppose / does not have limited variation in 21. Then there exists a sequence of divisions E^, U^ ••• such that if il^^ is the sum of the oscillations of/ in the intervals of U„, then ^E^<^£^< ■■• = +00- (1 Let us take v so large that no interval of L^ contains more than one interval of E„ or at most parts of two Il„ intervals. Let F„= U„-\- Dy. Then an interval S of D^ is split up into at most two intervals S', 8" in F„. Let w, to', co" denote the oscillation of /in S, B', 8". Then the term » in D, goes over into in n^^. Hence if Max ilj,^ = M, which contradicts 1). 2. Let F^„ = 2 |/(aO — /(flt+i) h t^ie summation extended over the intervals (a., a.+j) of the division D„. If Max F^„ is n finite with respect to a sequence of superposed divisions \I)^\, we cannot say that/ has limited variation. Example. For let /(a;) = 0, at the rational points in the inter- val 21 = (0, 1), and = 1, at the irrational points. Let D„ be 534 DEKIVATES, EXTREMES, VARIATION obtained by interpolating the points — -^t_ in 21. Then / = A at the end points a^, a^+i of the intervals of i>„. Hence Fb„ = 0. On the other hand, f{x) has not limited variation in 31 as is obvious. 531. Let FQc) = lim/(a;, t}, t finite or infinite, for x in the interval 21. Let Var/(2;, i)7>0 , in 21 thenf/g has limited variation. Let us show, for example, that h=fg has limited variation. For let Min/=m , Min ^ = n in the interval d^. Osc/= CD , Osc ^ = T VARIATION 535 Then • . , a ■ ■, 0h < 4 ^2o) + 2 ^2t, < some Cr, and h has limited variation. 533. 1. Let us see what change will be introduced if we replace the finite divisions D employed up to the present by- divisions -ff, which divide the interval 21 = (a < S) into an infinite enumerable set of intervals (a,, a.+j). ^'^' FE = i|/(0-/(««+i)|, (1 ^°^ TF=MaxF^, for the class of finite or infinite enumerable divisions {S}- Obviously W> V; hence if TTis finite, so is V. We show that if V is finite, so is W. For suppose W were infinite. Then for any 6f > 0, there exists a division U, and an n, such that the sum of the first n terms in 1) is > Cr, or w^,„>a. (2 Let now D be the finite division determined by the points a^ , '^2 '" *n+i which figure in 2). Then jr^^a, hence 7"= 00, which is contrary to our hypothesis. 536 DERIVATES, EXTREMES, VARIATION We show now that V and W are equal, when finite. For let H be so chosen that W-%We^^, and hence Hence W~ Vj, < £. We may therefore state the theorem : 2. If f has limited variation in the interval 21 with respect to the class of finite divisions D, it has with respect to the class of enumer- able divisions E, and conversely. Moreover Max Vo = Max V^ • 534. Let us show that Weierstrass" function JF, considered in 502, does not have limited variation in any interval 21 = (a < /8) when ah > 1. Since F is periodic, we may suppose « > 0. Let h^ k+ 1 ^ + /Lt be the fractions of denominator 6™ which lie in 21. These points effect a division 2)„ of 21, and ^^.= ^(^)-^(«) If I is the minimum of the terms Fj under the S sign, Vn>t^l- Now Hence h-\ < a. A: + At+1 >/8. < Jm /X > J-21 - 2. (1 (2 NON-INTUITIONAL CURVES 537 On the other h«nd, using the notation and results of 502, and also h > a"'b'" -^i)- <-' Let us now take Hence from 3), ^ ^ /2 tt \ ^^^"^ F,^>«-(|--^)(5'"i-2) , byl),2). As a < 1, and ab > 1, we see that Fi„ = + CO, as m = CO. Non-intuitional Curves 535. 1. Let f(x) be continuous in the interval 31. The graph of / is a continuous curve 0. If / has only a finite number of os- cillations in 21, and has a tangent at each point, we would call an ordinary or intuitional curve. It might even have a finite num- ber of angle points, i.e. points where the right-hand tangent is different from the left-hand one [cf. I, 366]. But if there were an infinity of such points, or an infinity of points in the vicinity of each of which / oscillates infinitely often, the curve grows less and less clear to the intuition as these singularities increase in number and complexity. Just where the dividing point lies be- tween curves whose peculiarities can be clearly seen by the intui- tion, and those which cannot, is hard to say. Probably different persons would set this point at different places. For example, one might ask : Is it possible for a continuous curve to have tangents at a pantactic set of points, and no tangent at another pantactic set? If one were asked to picture such a curve to the imagination, it would probably prove an impossibility. 538 DERIVATES, EXTREMES, VARIATION Yet such curves exist, as Ex. 3 in 601 shows. Such curves might properly be called non-intuitional. Again we might ask of our intuition : Is it possible for a con- tinuous curve to have a tangent at every point of an interval 21, which moreover turns abruptly at a pantactic set of points ? Again the answer would not be forthcoming. Such curves exist, how- ever, as was shown in Ex. 2 in 501. We wish now to give other examples of non-intuitional curves. Since their singularity depends on their derivatives or the nature of their oscillations, they may be considered in this chapter. Let us first show how to define curves, which, like Weierstrass' curve, have a pantactic set of cusps. To effect this we will extend the theorem of 500, 2, .so as to allow g(^z') to have a cusp at z=Q. 536. Let @ = {e„j denote the rational points in the interval S(=(— a, a). Let gQc) he continuous in 93= (—2a, 2 a), and = 0, at x = Q. Let 33* denote the interval 93 after removing the point a; = 0. Let g have a derivative in 93*, such that M l^'(^)l<: «> 0. (1 Then -^(^)=5j-^T;^K«-«»)=2«n5'(«-e„) , /3> 6.F is a continuous function in 21, and — — behaves at x = e^ essentially as —2 does at the origin.* Ax To simplify matters, let us suppose that (S does not contain the origin. Having established this case, it is easy to dispose of the general case. We begin by ordering the e„ as in 233. Then obviously if e„ = ^ , q > , p positive or negative, we have ^ n> q. Let «»» = p_r q s >1>J-. qs mn (2 * Cf. Dini, Theorie der Functionen, etc., p. 192 seq. Leipzig, 1892. NON-INTUITIONAL CUKVES 539 Let E{x) be tjje J' series after deleting the m'" term. Then We show that E has a differential coefficient at a; = e„, obtained by differentiating E termwise. To this end we show that as A = 0, ill converges to fi. = 2a„^'(..„) , m^n. (4 That is, we show e>0 , 9j>0 , \D(}i)-a\K.|-1»1>2^>2^ . forr<«<. 2 ms Thus by 1), I g\e^^ + A') I < 2" ilfm°w° < ilfiW , i^^i a constant. 540 DERIVATES, EXTREMES, VARIATION Beno. a fortion, , ^, (,^^^ , < ^^„.. ^9 Now the sum ^ converges if /* > 0. Hence J3j,^ ^ and ^ may be made as small as we choose, by taking p sufficiently large. Let us note that by 91, S^<--. (10 Thus a fi= Min (a, ^S), JB<\D„\ + \a„\<2X^,= ^^iSr,. I < ^^ Hence ^ » ,t ht < '* 1 s+l ** \h\ l + a + /3 gl+a+/3 < ^ 1 1 if -m , on using 10). Taking « still larger if necessary, we can make Thus ^<-3- NON-INTUITIONAL CURVES 541 The reader noig sees why we broke the sum 6) into three parts. As A = 0, the middle term contains an increasing number of terms. But whatever given value Ji has, s has a finite value. Thus as A, B, are each < e/3, the relation 5) is established. Hence U has a differential coefficient at a; = e„, and as AF_ A(0) , AH our theorem is established. 537. Example 1. Let i-— 2 1 Then for x ^ 0, g' (x) = - -j^. Here a = J. " Va; Forx=0, Eg'Cx-y^ + oo , Lg'{x) = -oo. Thus 3/7 r^ is a continuous function, and at the rational points e^ in the in- terval 21, BF' (x-)=+cc , iJ"(x) = -oo. Hence the graph of F has a pantactic set of cuspidal tangents in 21. The curve is not monotone in any interval of 21, however small. Example 2. Let -^ g (x) = a; sin - , a; ^ X = , a;=0. Then 111 a a(x')=sin cos- , x=^0. Here « = 1. For a; = 0, ^'(x)=+\ , l(x) I. 542 DERIVATES, EXTREMES, VARIATION Then ^ . 1 a; — «„ is a continuous function in 21, and at the rational point e„, where U is the series obtained from F by deleting the m*^ term. 538. Pompeiu Curves.* Let us now show the existence of curves which have a tangent at each point, and a pantactic set of vertical inflectional tangents. We first prove the theorem (^BoreV) : Let Bix-)=V^^ = V'^ , a„>0, ^x - e„ ^r„ where @ = {e„f is an enumerable set in the interval 21, and A = 2Va„ is convergent. Then B converges absolutely and uniformly in a set 33 < 21, and S is as near 21 as we choose. The points © where adjoint B is divergent form a null set. For let us enclose each point e„ in an interval S„ of length — -^^ with e„ as center. The sum of these intervals is ■.2-Va^^2A k k — ^ k k ' iov k> sufficiently large. Let now k be fixed. A point a; of 21 will not lie in any S„ 1f ^n=h-e„|>^». Then at such a point, k Adjoint B < '2a„—p=^ = klVcT^ = kA. * Math. Annalen, v. 03 (1907), p. 326. NON-INTUITIONAL CURVES 543 As S > 21 — e, the points S) where B does not converge ab- solutely form a null set. 539. 1. We now consider the function Fix) = I a„(a; - O^ = 2/„(^) (1 where @ = {e„} is an enumerable pantactic set in an interval SI, and A = 2a„ (2 is a convergent positive term series. Then F is a continuous function of x in 21. For ] a; — e„ |^ is < some Mm 21. Let us note that each /„(a;) is an increasing function and the curve corresponding to it has a vertical inflectional tangent at the point a; = e„ . We next show that Fix) is an increasing function in 21. For let x' < a;". Then /.(^') ^ the corresponding term in D„. Thus Q„.(K)>M , 0<|^|<,,'. Since each term of ^ is > 0, as 7) shows, Q(K) > M. Hence 8) is established. 5. Let us finally consider the points a; = e„ . If denotes the series obtained from F by deleting the m'" term, we have AZ=^ + M , iovx = e„. AcB ^1 Aa; As F is increasing, the last term is > 0. ^'(a:)=+oo , in Hence r7f/„\_ As a result we see the curve F has at each point a tangent. At an enumerahle pantactic set V, it has points of inflection with vertical tangents. 7. Let us now consider the inverse of the function F, which vs^e denote by x=a(t~). (9 As X in 1) ranges over the interval %t=F(x') will range over an interval 93, and by I, 381, the inverse function 9) is a one- valued continuous function of « in SB which has a tangent at each 546 DERIVATES, EXTREMES, VARIATION point of S3. If TTare the points in 35 which correspond to the points F" in SI, then the tangent is parallel to the ^axis at the points W, or Gr' (t) = 0, at these points. The points W are pan- tactic in 35. Let Z denote the points of S3 at which (r'(t) = 0. We show that Z is of the 2° category, and therefore Card Z=c. For Gr'(() being of class < 1 in 33) its points of discontinuity S form a set of the 1° category, by 486, 2. On the other hand, the points of continuity of G' form precisely the set Z, since the points W are pantactic in i8 and Gr' =0 in W. In passing let us note that the points Z in 58 correspond 1-1 to a set of points S at which the series 3) diverges. For at these points the tangent to F is vertical. But at any point of convergence of 3), we saw in 2 that the tangent is not vertical. Fiially we observe that 3) shows that MmD(x}>l • — 2a„ , in 21. 3 gf Hence „ jjz Summing up, we have this result : 8. Let the positive term series 2Va„ converge. Let @ = je„J be an enumerable pantactic set in the interval 21. Tlie Pompeiu curves defined by F(x-)='E.a„(_x-cJ have a tangent at each point in 21, whose slope is given by ^'i^}=lX when this series is convergent, i.e. for all x in ^ except a null set. At a point set 3 of the 2° category which embraces <§., the tangents are vertical. The ordinates of the curve F increase with x. 540. 1. Faber Curves.* Let F(x') be continuous in the interval 21 =(0, 1). Its graph we denote by F. For simplicity let • Math. Annalen, v. 66 (1908), p. 81. NON-INTUITIONAL CURVES 547 P(0) = 0, -F(l) » Iq . We proceed to construct a sequence of broken lines or polygons, Lq, ij, ij ... (1 which converge to the curve F as follows : As first line Lq we take the segment joining the end points of F. Let us now divide 21 into n^ equal intervals Sji, 8i2— 8i,„, (2 of length . 1 Oj = — 1 and having „„„... (3 as end points. As second line L^ we take the broken line or polygon joining the points on J' whose abscissae are the points 3). We now divide each of the intervals 2) into n^ equal intervals, getting the n^n^ intervals ^21' ^22' ^23'" (* of length ^= ' and having 2 rijn^ *21' "^22' "^23 (5 as end points. In this way we proceed on indefinitely. Let us call the points terminal points. The number of intervals in the r*'' division is v^ = n^ .Mj... w,. If i„(a;) denote the one-valued continuous function in 21 whose value is the ordinate of a point on L^, we have since the vertices of L^ lie on the curve F. 2. For each x in 21, Urn i„(:c) = Fix). (7 77t=ico For if a; is a terminal point, 7) is true by 6). 548 DERIVATES, EXTREMES, VARIATION If X is not a terminal point, it lies in a sequence of intervals belonging to the 1°, 2° ••• division of 21. Since -F(2;) is continuous, there exists an 8, such that |l'(a;)-J'(a^,„)|<|, m>s (8 for any a; in 8„. As LJ^x) is monotone in S„, I J^m(«) - i^mCamn) 1 < | ^J^a^^ - L,^{a^,n+i) I < I -^mC^mn) — -^m(«m, n+l) I <| ' by 8). '^^'^^ li„(a.)-J?'„(a„„)|<|. (9 Hence from 8), 9), \F{x')-L^(x')\s which is 7). 3. We can write 7) as a telescopic series. For ii = Xo + (ij - io) A = ^1 + (A - -^i) = A + ( A - A) + (J^2 - A) etc. Hence F(x) = lim LXx) = L,(x) + 1 }i„(a;) - i„_,(a;) | . If we set f,(x) = L^ix-) , Ux) = LXx~)-L„_,(x), (10 we have ^(a;) = tfjx) , (11 ^"•^ J'„(a;) = i/.(rr) = i;„(a;). (12 The function /„(a;), as 10) shows, is the difference between the ordinates of two successive polygons -Z/„_i, i„ at the point x. It may be positive or negative. In any case its graph is a polygon NON-INTUITIONAL CURVES 549 /„ which has a'*Vertex on the a;-axis at the end pomt of each interval S„_j. Let 4s be the value of /„(a;) at the point x = a^, that is, at a point corresponding to one of the vertices of /„. We call Ina the vertex differences of the polygon L^- Let p„ = Min I Z„J , g„ = Max \l^,\. s s Then |/„(a;)|<5„ , in 21. (13 In the foregoing we have supposed -F(a;) given. Obviously if the vertex differences were given, the polygons 1) could be con- structed successively. We now show : ^^ ^qn (14 is convergent, J'(^) = 2/„(:.) is uniformly convergent in 21, and is a continuous function in 21. For by 13), 14), F converges uniformly in 21. As each f„(x) is continuous, F is continuous in 21. The functions so defined may be called Faber functions. 541. 1. We now investigate the derivatives of Faber' s functions, and begin by proving the theorem : If ^n^---n,q, = 'Lv,q, (1 converge, the unilateral derivatives of F(x^ exist in 21 = (0, 1) . More- over they are equal, except possibly at the terminal points A= Ja„„j. For let a; be a point not in A. Let x', x" lie in V= V*(x) ; letx'-x=h', x" -x=h". Tpf Fix'-, -Fix-) FCx")-F(x) Let V- ^, ^„ Then F' (x) exists at x, if e>0 , 7]>Q , |^|<€ , for any a;', a;" ill F. (2 550 DEBIVATES, EXTREMES, VARIATION Now Q\< h' h" F„a , m=l,2..; Gr being large at pleasure. Hence at least one of the difference quotients 5) belonging to this sequence of divisions is numerically large at pleasure. ^- -^^ ^ = 2?™ (1 is absolutely convergent, the functions FQx) have limited variation in 31. For/„(a;) is monotone in each interval S^,. Hence in 8,^, Var/„ = I Z™ - C.+1 1 < I C I + I C+i I- Hence in 21, Var /„(:.)< 2 2^. Var#„(a;)<2isZ„. = 2\ , in 21. We apply now 531. 552 DERIVATES, EXTREMES, VARIATION Tio. 1 542. Faber Functions without Finite or Infinite Derivatives. To simplify matters let us consider the following example. The method employed admits easy generalization and gives a class of functions of this type. We use the notation of the preceding sections. Let /o(a;) have as graph Fig. 1. We next divide 21 =(0, 1) into 2^' equal parts Su, Sjj and take /i(a;) as in Fig. 2. We now divide 21 into 2^' equal parts S^j, Sjj, Sjg, \^ and take/2(a;) as in Fig. 3. The height of the peaks is 1^ = ——. In the m'" division 21 falls into 2"*' equal parts one of which may be denoted by h = -. Fia. 2 K = («™ l)- '^=ro' Its length may be denoted by the same letter, FiQ. 3 thus S„ = - i)mi In Fig. 4, B^ is an interval of the m- division. J8t I AAAA The maximum ordinate of /m(2;) is Z„ = — — = - 5w Fig. 4 1 The part of the curve whose points have an ordinate <^lm have been marked more heavily. The x of such points, form class 1. The other z's make up class 2. With each x in class 1, we associate the points «„ < /3„ corresponding to the peaks of /„ adjacent to x. Thus a„|C. « — e„ 1 2"-' 410'" On the other hand, for any x^x' in S„, x' —X <24 Hence setting «' = e„, and letting w >m, |A(e„)-/™(aj)|<|^|e„-x| logg 10 , for n sufficiently large. H®'^^® (w-l)!(w-l)>log2l0''. Thus 20-1)!'' "" ' "* 2 and this establishes 4). Let us now extend the definition of the functions /„ (a;) by giv- ing them the period 1. The corresponding Faber function F(x) defined b)-^ 540, 12) will admit 1 as period. We have now Fie J - F(x) = i/„(0 -Ux^\ + !^n-i(«n) - F^^.ix-)] From 2) we have tt ~-. i 7 -^1 jrl 2 *"• 554 DERIVATES, EXTREMES, VARIATION As to T^, we have, using 4) and taking n sufficiently large, l^l.lv^l 11 2 ' ^ 10» A lO™ 10" Y 10*" 9 10" m=l ■«■ Similarly <§?, S'n- m=n+l »n=n+l »+i 9 10" Ui-|-|) >TV^n. ^^ \T,\>\T,\ + \T,\ e„ - a; e„ - a; Thus F(eJ-FCx^ e„-x 18 S„ 1 2"' >3610» = °°- As e„ may be at pleasure «„ or /3„, and as the signs of 1) are opposite, we see that F'(a;)=+ , F'(x)=-ao; and FQc) has neither a finite nor an infinite differential coefficient at any point. CHAPTER XVI SUB- AND INFRA-UNIFORM CONVERGENCE Continuity 543. In many places in the preceding pages we have seen how important the notion of uniform convergence is when dealing with iterated limits. We wish in this chapter to treat a kind of uniform convergence first introduced by Arzeld, and which we will call subuniform. By its aid we shall be able to give condi- tions for integrating and differentiating series termwise much more general than those in Chapter V. We refer the reader to Arzela's two papers, '■'■ Sulle Serie di Funzioni," R. Accad. di Bologna, ser. V, vol. 8 (1899). Also to a fundamental paper by Osgood, Am. Journ. of Math., vol. 19 (1897), and to another by Mobson, Proc. Lond. Math. Soc, ser. 2, vol. 1 (1904). 544. 1. Let/(a;j ••• x„, t^--- tn)=f(x, i) be a function of two sets of variables. Let x = (xi--- x^) range over 3£ in an jw-way space, and « = (*j ••• <„) range over X in an w-way space. As x ranges over S and t over Z, the point {x^-.-- 1^ ••■') = (x, () will range over a set 21 lying in a space 9tj,, ^ = m + w. Let T, finite or infinite, be a limiting point of X. Let i=r lim/(a;i ••• x„, t^ ••• «„) = (^i - ^m) in X- Let the point x range over S30, 77>0: 535 556 SUB- AND INFRA-UNIFORM CONVERGENCE 1° There exists a finite number of layers S, whose ordinates t lie in V,i*(^t'). 2° Each point x of 3E is associated with one or more of these layers. Moreover if a; = a belongs to the layer 8(, all the points X in some T^(a) also belong to 8j. 3° \fix,t)-4,(ix)\ 0, and for each v there exists a finite set of layers of ordinates > v, call them 8i, 22- (2 such that each point a; of 21 belongs to one or more of them, and if x= a belongs to ?„, then all the points of 21 near a also belong to 8„. 2° \F„Cx^.-x^-)\ to he continuous at the point x= a in 3£, it is necessary that for each e > 0, there exists an ?? > 0, and a dt for each t in Fi,*(T) such that \f(x,t)-.l>Cx)\ I , for any x in some Vs,(_a)i and hence in particular. |/(a,^)-<^(a)|<|. Also since /(a;, /8) is continuous in x, \f(_x, 0')—f(a, /8)| <^ , for any x in some Vi„(a). Thus if S < 8', S", these unequalities hold simultaneously. Add- ing them we get I ^(a;) — ^(a)| is continuous at a; = a. 2. As a corollary we get : J^et J'(a;) = 2/.,....„(a;i-a;„) converge in SI, each term being continuous in 21. For F(x') to he con- tinuous at the point x = a in 21, it is necessary that for each e > 0, and for any cell M^ > some M),^ there exists a S^ such that I ^m(^)I < e , /"»• any x in Fs^(a). It is sufficient if there exists an It>, and a S>0 such that I F^(^x')\ < e , for any x in Fi(a). 547. 1. Let Urn f(x^---x„, subuniformly in 36, ^ is continuous in 3c. 2° If Ti is complete, and ^ is continuous in 2i,f=(f) subuniformly in I. To prove 1°. Let a; = a be a point of J. Let e > be taken at pleasure and fixed. Then there is a layer S^ to which the point a belongs and such that \fQx,t)-(x-)\ \, liml\(a;)= 0. Then F converges subuniformly in 21. For let 2) be a cubical division of norm d of the space 9J„ in which 21 lies. We may take d so small that SSo is small at pleasure. Let Bj, denote the cells of D containing points of 21 but none of 93. Then by hypothesis # converges uniformly in Bj). Thus there exists a /*(, such that for any ^ > /*o' I J; (x) I < e , for any a; of 21 in Bj,. 560 SUB- AND INFRA-UNIFORM CONVERGENCE At a point b of 95, there exists by hypothesis a Fi(6) and a \q such that for each X > Xq |l\(a;)|<€ , for any a: in Vs(b). Let C^^)^ be a cube lying in i>s(J), having 6 as center. Since 93 is complete there exists a finite number of these cubes such that each point of 53 lies within one of them. Moreover , ^ . -. , for any a; of SI lying in the «"' cube of 1). As Bj, embraces but a finite number of cubes, and as the same is true of 1), there is a finite set of layers S such that I Fy(x) I < e , in each 8. The convergence is thus subuniform, as \, fi are arbitrarily large, 2. The reasoning of the preceding section gives us also the theorem : lim/(a;i--.a;„, ij ••.<„) = ^(a^j ••• a;„) in I, T finite or infinite. Let the convergence he uniform in 3£ except possibly for the points of a complete discrete set ^ = \e\. For each point e, let there exist an rj such that setting e(x, f) =f(x,ty — (}> (x), lim e(a;, <) = , for any t in V.^*(j). Thenf= ^ subuniformly in 3E. 3. As a special case of 1 we have the theorem : ^'* FCx-)=f,cx-)+f,az-)+.- converge in 21, and converge uniformly in 21, except at x = a^,--- x=a,. At X = a^ let there exist a v^ such that limJ;Xa;)=0 , n,>v, , t = 1, 2 .■• «. Then F converges subuniformly in 21. CONTINUITY 561 4. When • t ^r ^\ ^ /• n lim/(a;, t} = , /x > 0. The convergence is subuniform in 21. For a; = is the only- possible point of non-uniform convergence, and for any m, I e (a;, wi~) 1 = =0 , as a; = 0. Example 2. f(x, n) = = ^ (x) = , as w = oo, a; in 21 = (0 < a) , a, /3, \, /u > , /x > X , c> 0. The convergence is uniform in 53 = (e < «), where e > 0. For e(x, u) < , in 58 < e , for n > some m. Thus the convergence is uniform in 21, except possibly at a; = 0. The convergence is subuniform in 21. For obviously for a given n lim/(a;, w) = 0. 550. 1. Let liin/(a;i ••• x„ t-^ ••• t„) = (f)(x^ ■•• a;„) in X, t finite t=r or infinite. Let the convergence be uniform in 3E except at the points 562 SUB- AND INFRA-UNIFORM CONVERGENCE For the convergence to he mh-uniform in X, it is necessary that for each b in 58, and for each e > 0, there exists a t= ^ near t, such that Hm |e(a;, 01 >e. (1 For if the convergence is subuniform, there exists for each e and t; > a finite set of layers 8j, t in F,*(t) such that I e(x, I < ^ 5 xin^t. Now the point x=b lies in one of these layers, say in S^. Then I €(a;, ;8) I < € , for all x in some F'*(J). But then 1) holds. 2. Hxample. Let ^'(a;) = L^Cl -a;). This is the series considered in 140, Ex. 2. F converges uniformly in 21 =( — 1, 1), except at a; = 1. we see that Hence F is not subuniformly convergent in 31, limF^Cx-) = -l. Integrability 551. 1. Infra-uniform Convergence. It often happens that f(xi ••• x^t^ ■■• «„) = 0(a;i ••• a;„) subuniformly in X except possibly at certain points (g= jej form- ing a discrete set. To be more specific, let A be a cubical divi- sion of $R„ in which X lies, of norm S. Let Xa denote those cells containing points of X, but none of @. Since (S is discrete, X^ = Ti. Suppose now/=^ subuniformly in any X^; we shall say the convergence is infra-uniform in 3E. When there are no exceptional points, infra-uniform convergence goes over into sub- uniform convergence. INTEGRABILITY 563 This kind of convergence Arzela calls uniform convergence by segments, in general. 2. We can make the above definition independent of the set @, and this is desirable at times. Let J = (X, y) be an unmixed division of 3E such that j may be taken small at pleasure. If /=^ subuniformly in each X, we say the convergence is infra-uniform in 36. 3. Then to each e, ?; >0, and a given X, there exists a set of layers Ij, Ij "■ •« ^ ^^ ^i*(''')' such that the residual function e(a;, f) is numerically < e for each of these layers. As the projections of these layers I do not in general embrace all the points of % we call them deleted layers. 4. The points y we shall call the residual points. 5. Example 1. F = t il +n.^)(f+ (n+l^a^^) This series was studied in 150. We saw that it converges uni- formly in 21= (0, 1), except at a; = 0. As _ -t 1 + nx and as this = 1 as a; = for an arbitrary but fixed n, F does not converge subuniformly in 31, by 550. The series converges infra- uniformly in 21, obviously. 6. Example 2. J'=Sa;"(l-a;). This series was considered in 550, 2. Although it does not converge subuniformly in an interval containing the point a; = 1, the convergence is obviously infra-uniform. 552. 1. Let lim f {x^-- x^t^---t„) = 4>{^i--- ^m) ^e limited in 1, T finite or infinite. For each t near r,letfbe limited and R-integrahle in H. For ^ to he B-integrahle in 3E, it is sufficient thatf = ^ infra- uniformly in H. IfTi is complete, this condition is necessary. 564 SUB- AND INFRA-UNIFORM CONVERGENCE It is sufficient. We show that for each e, «» > there exists a division I) of 3J„ such that the cells in which Osc > a) (1 have a volume < o-. For setting as usual we have in any point set, Osc < Osc/4- Osc e. Using the notation of 551, \e(x,t)\<^ in the finite set of deleted laj'ers Ij, Ij ••• corresponding to t=ti, t^--- For each of these ordinates t^,f(x, t^) is integrable in I. There exists, therefore, a rectangular division D of 9?„, such that those cells in which Osc/(2;, 0>| have a content < -, whichever ordinate t, is used. Let -2^ be a division of 9J„ such that the cells containing points of the residual set y have a content < o-/2. Let F = D + E. Then those cells of F in which Osc /(a;, «0 >|, or Osc | e(x, *0 I ^| t = l, 2 ••■ have a content < a-. Hence those cells in which 1) holds have a content < cr. It is necessary, if I is complete. For let ^1' ^2 "■ ^ ''^' Since <^ a,ndf(x, i„) are integrable, the points of discontinuity of (^x) and of /(a;, <„) are null sets by 462, 6. Hence if S, S« denote the points of continuity of m. (2 But f lying in 35, it lies in S and ©^ . Thus |<^(^)-'^(D1<|, \f(^,t^)-.fao\■ ' ^1' ^2 ■■■ ^»' such that each point of I lies within one of them. But corre- sponding to the r's, are layers such that in each of them I e(x, I < e- Thus/= cj) subuniformly in X = (T^, F^--- T,). Let y be the residual set. Obviously f < cr. Thus the convergence is infra- uniform. 2. As a corollary we have : Let Fix) = ^f,^...^(x,...x„) converge in 21. Let F be limited, and eachf^ he limited and Jt-in- tegrable in 21. For F to be R-integrable in 21, it is sufficient that F converges infra-uniformli/ in 21. if 21 is complete, this condition is necessary. 553. Infinite Peaks. 1. Let lim/(a;, •■■a:„ is limited in 3£. For X being taken at pleasure in 3E and fixed, ^(x) is a limit point of the points / (a;, i) as * = t. But all these points lie in some interval (— 6r, Gr^ independent of x. Hence (x) to hold, it is sufficient thatf=4> infra- uniformly in I. If 1 is for each t complete, this condition is necessary. For by 552, iufra-uniformly, and when I is complete, this condition is necessary. By 424, 4, each /(a;, t) is measurable. Thus we may apply 381, 2 and 413, 2. 2. As a corollary we have the theorem : Let -f'(^) = 2/,....>i-^-) converge in the complete or metric field 21. Let the partial sums F^ be uniformly limited in 21. Let each termf be limited and B-integrable in 21. Then for the relation fF = -E ff 568 SUB- AND INFRA-UNIFORM CONVERGENCE to hold it is sufficient that F is infra -uniformly convergent in 21. If SI is complete, this condition is necessary. 555. Example 1. Let us reconsider the example of 150, We saw that we may integrate termwise in 21 = (0, 1), al- though F does not converge uniformly in 21. The only point of non-uniform convergence is x •-= 0. In 551, 5, we saw that it con- verges, however, infra-uniformly in 21. As I Fn(^x) I < 1 , for any x in 21, and for every w, all the conditions of 554 are satisfied and we can integrate the .series termwise, in accordance with the result already obtained in 150. Example^. Let ^(.) = |; { i?£ - (^L^ | = 0. Then ^nW=5. We considered this series in 152, i. We saw there that this series cannot be integrated termwise in 21 = (0 < a). It is, how- ever, subunifornily convergent in 21 as we saw in 549, Ex. 1. We cannot apply 554, however, as F^ is not uniformly limited. In fact we saw in 152, i, that a; = is a point with an infinite peak. oo Example 3. F(x) = 2a;"(l - 2;). We saw in 551, 6, that F converges infra-uniformly in 21 = (0, 1). ^^'■® ^„(a;) I = 1 1 - a;" I < some M, for any a; in 21 = (0 < u), m < 1, and any n. Thus the F^ are uniformly limited in 21. We may therefore integrate termwise by 554, 2. We may verify this at once. For F{x')=\ , 0^2: <1 = , a; = 0. INTEGRABILITY 669 Hence * (""FC^xydx = u. (1 On the other hand, F„ax = u = M , as w = 00. (2 n + 1 From 1), 2) we have ■^ [n + 1 n + 2 I 556. 1. l/l°/(a;i ••• a;„ *! ■••«„) = (^(kj ••• a;„) infra-uniformly in the metric or complete field '£, as t = t, r finite or infinite ; 2° fix, f) is uniformly limited in 26 and R-integrahle for each t near t; Then lim ) fix, t) = frf), (=T "^H ^% uniformly with respect to the set of measurable fields 31 in ?£.• If J is complete, condition 1° may be replaced by 3° 4>(x^ is R-integrable in 3£. For by 552, 1, when 3° holds, 1° holds ; and when 1° holds, ^ is i2-integrable in 3£. Now the points (Sj where \e(x,tj\ >e are such that ^ lim d, = , by 412. Let3e = g, + 3e,. Then Ceix, t) = r e(a:, + f <^^ t), Xdi <*'©« '^x, Ka I Xx e I < 2 Md, + el ^^^ lim I, = 0, which establishes the theorem. 2. As a corollary we have : If 1° jF'(2;)= 2/.,....„(a;i ••• a;„) converges infra-uniformly, and each of its terms f is R-integrable in the metric or complete field 21; 670 SUB- AND INFRA-UNIFORM CONVERGENCE 2° F^,(x) is uniformly limited in 21; Then X^(^)=sX^" and the series on the right converges uniformly with respect to all measurable 33 <. 21. 3. If 1° \\mf(x, ^j ••• t„') = (x')dx = (a;), uniformly in 21, and $(a;) is continuous in 21. 4- ^f^° F{x-)=^f^^...^ix-) and also each termf^ are R-integrable in the interval 2l = (a< 5); 2° FxCx) is uniformly limited in 21; ^^*"* a(ix~)=l.rf^(x)dx , xinU is continuous. For G' is a uniformly convergent series in 21, each of whose terms is a continuous function of x. Differentiahility 557. 1. If Vlim f(ix,t^ — t^)= ^Qx) in % = {a Kb), T finite or infinite ; 2° fKjx, t^ is R-integrable for each t near t, and uniformly limited in 21; 3° fl(x, i)= ■ylr(^x') infra-uniformly in %, as t= t; Then at a point x of continuity of yfr in % . , '(x•)=^}r(ix), (1 or what is the same -f lim/(a:, = lim ffCx, t). (2 w «=rr t=^ ax DIFFERENTIABILITY 571 For by 554, . = lin)[/(.-r, 0-/(a, 0] , by I, 538 = 4>(x~)-,^ia) , byl°. Now by I, 537, at a point of continuity of yjr, —j^f(x)dx==^^ix). (4 From 3), 4), we have 1), or what is the same 2), 2. In the interval 21, if 1° F(x')=1,f^...^^(x') converges; (1 2° Uachf[(x) is limited and R-integrahle ; 3° Fl(x) is uniformly limited ; 4° G-(x)= 1,f[ is infra- uniformly convergent ; Then at a point of continuity of Gf(x^ in 21, we may differentiate the series 1) termwise, or F'(x')= Gijc). 3. In the interval 21, if 1° f (xi ij ••■ ^„) = ^(a;) as t = T, t finite or infinite ; 2° f(x, t~) is uniformly limited, and a continuous function of x ; 3° ■<^{x)= lim/j(a;, () is continuous ; or what is the same ^lunf(x,t) = \imff{x,f). (2 ax i=T t=T dx For by 547, l, condition 3° requires that /' = i/r subuniformly in 21. But then the conditions of 1 are satisfied and 1) and 2) hold. 4. In the interval 21 let us suppose that 1° F(x') = 'Lf^...,^(x) converges; (1 572 SUB- AND INFRA-UNIFORM CONVERGENCE 2° Each termf, is continuous; 3° FKx) is uniformly limited; 4° (r(aj) = 2/'(a;) is continuous ; Then we may differentiate 1) termwise, or F'(x)= GrQe). 558. Example 1. We saw in 655, Ex. 3 that ^(^x)=x=% n + 1 «-|- 2 . , in 21 = (0,1). (1 The series got by differentiating termwise is a{x) = 2a;''(l -x}=l , 0(x) = lim I f(x, t)dx = t(^\ (1 or ^^ ^^ lim I f{x, t')dx= I lim/(a;, t)dx, and the set @ is complete and discrete. For ® is discrete by 553, 3. Let a be a point of -4 = 21 — @. Then in an interval a about a, l/(a;, f) I < some M , a; in a, any t near t. (2 Now by 556, 3, taking e > small at pleasure, there exists an 77 > such that for any x in a, and t in V^*(^t'). If we set a; = a + h, we have Also by 556, 3, we have fy(x, f)dx = J'^ijiCx^dx + e" , | e" | < e for any x in a, and < in V^*{t). Thus h-^a h h Ax n From 3), 4) we have Ax h Ax h -^ = -— , tor X in A. ax ax DIFFERENTIABILITY 575 Now e may be«iade small at pleasure, and that independent of k. Thus the last relation gives — — = — — , tor X in A. Ax Ax As this holds however small h = Axis taken, we have Hence by 515, 3, -<^(a;)= ^(a;)+ const , in 21. For x = a, ■v|r(a) = OQa) = ; and thus fix-)=e{x') , in 21. 2. As a corollary we have : If 1° F(x)= 2/n...^(a;) is limited and R integrahle in the inter- val "&= (aKh'); 2° Fx{x) is limited and each termf^ is R-integrahle ; 3° Gr(x')='^\ f^ is continuous; 4° The points (g in 21 in whose vicinity the peaks of F^ix) are in- finitely high form an enumerable set; Then ^^ ^^ or we may integrate the F series termwise. 560. 1. If 1° lim/(a;, t^ ■■■ t^)=^(x) in 2[ = (a < 6), r finite or t=T infinite; 2° fl(x, f) is limited and R-integrable for each t near t; 3° The points d of ^ in whose vicinity fK^x^ ^«« infinite peaks as t = T form an enumerable set; 4° ^(a;) is continuous at the points @; 5° -^(x) = limf^(x, t) is limited and R-integrahle in 21; 576 SUB- AND INFRA-UNIFORM CONVERGENCE Then at a point of continuity of ■v/r(a;) in 21 ,/.'(x) = V^Ca;), (1 or what is the same |-lim/(z, = lim//(x, 0- ax t=T i=T ax For let S = (a < yS) be au interval in 21 containing no point of S. Then for any a; in S CyXx,t)dz=fCx,t)-fCa,f) , by 2°. lim f /i(a;, t)dx = limlf(x, t) -f(a, t)l t=T ♦^a l=T Hence = (x) - ^{a-) , byl°. (2 By 556, 3, (x) is continuous in S. Thus <^(a;) is continuous at any point not in ©. Hence by 4° it is continuous in 21. We may thus apply 559, l, replacing therein /(a;, f) by/j(a;, <). We get flQc, fydx = I \imfl(x, t')dx = I ■^(x)dx. (3 Since 2) obviously holds when we replace a by a, this relation with 3) gives 1 ■ylr(x)dx = ^(a;) — ^(a).. At a point of continuity, this gives 1) on differentiating. 2. If 1° -F(2;) = 2/ij ... i„(a;) converges in the interval 21; 2° G-{x) = S/[(a;) awe? eacA of its terms are limited and R- integrahle in 21; 3° The points of 21 in whose vicinity G-k(x) has infinite peaks as X = (xi,form an enumerable set at which F(jc) is continuous; Then at a point of continuity of GQc) we have F'ix) = acx-), or what is the same A^fCx) = X^^- dx dx DIFFERENTIABILITY 577 561. Examplel^ Hence Fix-) = % F'^X) = 22: 2 3? (1 The series obtained by differentiating F termwise is Here V e..,w.?,'-?^-] Hence is a continuous function of x. (2 The convergence of the G- series is not uniform at a; == 0. For set a„ = l/w. Then (it} , y = ylr(t^ (1 be one- valued continuous functions of t in the interval 21= (a< J). As t ranges over 21 the point x, y will describe a curve or an arc of a curve C. We might agree to call such curves analytic, in distinction to those given by our intuition. The interval 21 is the interval corresponding to C. Let 2) be a finite division of 21 of norm d, defined by a|A2;|. Hence r. ^ v i a i P^ > S I Ax I . But the sum on the right is the variation of for the division D. If now (^ does not have limited variation in 21, the limit 3) does not exist. The same holds for i/r. Hence limited variation is a necessary condition. The condition is sufficient. For P^< S I Aa; I + 2 1 Aw I = Var rf> + Var i/r, D J) As (f>, yjr have limited variation, this shows that Po = Max Pj, D is finite. We show now that limPa = Po. (4 For there exists a division A such that P,-\Pd or P^. 584 GEOMETRIC NOTIONS Suppose that the point t^ of A falls in the interval (t^, «,+j) of 2). Then the chord (i, t + 1) in P^ is replaced by the two chords (t, «), (k, t + 1) in Pg. Hence where ^^ ^ ^^^ ^^ + («,, + !)_ (,, , + i) . Obviously as , yjr are continuous we may take cIq so small. that each 2 effect a division of 21, and since s = lim Pe (8 with respect to the class of all divisions of 21, the limit 8) is the . same when E is restricted to range over divisions of the type of D. Now Pj) = Pjh "^ Pd,- Passing to the limit, we get 7). PLANE CURVES 585 The preceding reasoning also shows that if C^ , C^ are rectifiahle curves, then, C is, and 7) holds again. 5. If 1) define a rectifiahle curve, its length s is a continuous func- tion s(f) of t. For 0 so small that Sc^j, 8<^2' ^''/^i' ^''1^2 "^ ^Z"^' if ^< '?• Hence As = s(« + A) — s(«) < Max Pj < e , ifO7;. Thus s is continuous. 6. The length s of the rectifiahle arc corresponding to the inter- val (a < is a monotone increasing function oft. This follows from 4. 7. If X, y do not have simultaneous intervals of invariability, sQ) is an increasing function of t. The inverse function is one-valued and increasing and the coordinates x, y are one-valued functions of s. That the inverse function t (s) is one-valued follows from 1, 214. We can thus express t in terms of s, and so eliminate t in 1). 570. 1. if <^', -v^' are continuous in the interval 21, «= ri«V(^'2 + i/r'2. (1 For i = limSVA(^2-|- AA|r2. (2 0 but not all =0. m m = 1, 2 •■• w. (4 For Hence (Vfa + V6f + - )(Va2+ ... +V6f+ -.) = (af+...+a2)-(Jf+... +62) = (a2_52)+...+(«2_J2) Vaf + ... _ V6f + - = ;g(a„- 6.) But ™=i Va2 + ... + vjfir::: (5 ~m__i ^m Vaf+ ■•• +VJf + This in 5) gives 4). <1. PLANE CURVES 587 Let us appl3fc4) to prove the folio wing theorem, more general than 1. 3. (^Baire.) If <^' ■, '^' are limited and M-integrable, then s = rV^'2 + f 2 dt. (1 For by 4), + lt'(0-V^'a)|; where j;,' , ?;" are numerically < 1. Thus I 2S.'^+ f 2 cZt. rf=o •''a Thus passing to the limit in 6), we have lim lAt^ ^WtI7+W0^= r This with 2), 3) gives 1) at once. 571. Vblterra's Curve. It is interesting to note that there are rectifiable curves for which (f)' (f), ■^' (t) are not both M-integrable. Such a curve is Volterra's curve, discussed in 503. Let its equa- tion be y=f{x). Then /'(a;) behaves as I X sin cos - X X in the vicinity of a non null set in 21 = (0, 1). Hence /'(a;) is not iZ-integrable in 21. But then it is easy to show that does not exist. For suppose that ^ = VH-/'Ca;)2 588 GEOMETRIC NOTIONS were ^-integrable. Then g^=l+f'(xy^ is i2-integrable, and hence /'(a;)^ also. But the points of discontinuity of /'^ in 21 do not form a null set. Hence /'^ is not i2-integrable. On the other hand, Volterra's curve is rectifiable by 569, 2, and 528, 1. 572. Taking the definition of length given in 569, 1, we saw that the coordinates must have limited variation for the curve to be rectifiable. But we have had many examples of functions not having limited variation in an interval 21. Thus the curve defined by y =x sin - , a; ^ X (4 =0 , x=0 does not have a length in 21 = (— 1, 1) ; while y = x^ sin - , a: =?!= ^ X ' (5 =0 , x=0 does. It certainly astonishes the naive intuition to learn that the curve 4) has no length in any interval S about the origin how- ever small, or if we like, that this length is infinite, however small h is taken. For the same reason we see that No arc of Weientrass' curve has a length {or its length is infinite') however near the end points are taken to each other, when a6 > 1. 573. 1. 6° Property. Space-filling Curves. We wish now to exhibit a curve which passes through every point of a square, i.e. which completely fills a square. Having seen how to define one such curve, it is easy to construct such curves in great variety, not only for the plane but for space. The first to show how this may be done was Peano in 1890. The curve we wish now to define is due to Hubert. We start with a unit interval 21 = (0, 1) over which t ranges, and a unit square 58 over which the point x, y ranges. We define x^^ifi, , y = ^{t~) (1 PLANE CURVES 689 as one-valued continuous functions of < in 21 so that xy ranges over S3 as < ranges over 21. The analytic curve Q defined by 1) thus completely fills the square 33. We do this as follows. We effect a division of 21 into four equal segments SJ, S^, S^, h\, and of S3 into equal squares 77J, 7;^, 17^, i)\, as in Fig. 1. We call this the first division or By The corre- spondence between 21 and S3 is given in first approximation by saying that to each point P in h[ shall correspond some point Q iwq'^. We now effect a second division D^ by dividing each interval and square of B^ into four equal parts. We number them as in Fig. 2, 2 3 1 4 12 3^ Fia. 1. v'i V'le 6 7 10 11 f 5^8 9 12 ■ 4 . 3 14 13 1 ■ 2 15' t IG I'lO. 2. As to the numbering of the 97's we observe the following two principles : 1° we may pass over the squares 1 to 16 continuously without passing the same square twice, and 2° in doing this we pass over the squares of B^ in the same order as in Fig. 1. The correspondence between 21 and 58 is given in second approximation by saying that to each point P in 8[' shall correspond some point Q in rj['. In this way we continue indefinitely. To find the point Q in 93 corresponding to P in 21 we observe that P lies in a sequence of intervals (2 8' > S" > 8'" > - = 0, to which correspond uniquely a sequence of squares V' > v" > V'" > - = 0. (3 The sequence 3) determines uniquely a point whose coordinates are one-valued functions of t, viz. the functions given in 1). The functions 1) are continuous in 21. For let t' be a point near t ; it either lies in the same interval as t in Bn or in the adjacent interval. Thus the point Q' corre- 590 GEOMETRIC NOTIONS spending to <' either lies in the same square of i)„ as the point Q corresponding to t, or in an adjacent square. But the diagonal of the squares = 0, as n = oo. Thus Dist(^'^)=0 , asw = ao. both = 0, as t' = t. As t ranges over 21, the point x, y ranges over every point in the square 58. For let ^ be a given point of S3. It lies in a sequence "of squares as 3). If ^ lies on a side or at a vertex of one of the t} squares, there is more than one such sequence. But having taken such a sequence, the corresponding sequence 2) is uniquelj^ de- termined. Thus to each Q corresponds at least one P. A more careful analysis shows that to a given Q never more than four points P can correspond. 2. The method we have used here may obviously be extended to space. By passing median planes through a unit cube we divide it into 2^ equal cubes. Thus to get our correspondence each division D„ should divide each interval and cube of the pre- ceding division 2?„_i into 2^ equal parts. The cubes of each divi- sion should be numbered according to the 1° and 2° principles of enumeration mentioned in 1. By this process we define x = i(f) , y = 2(t} , Z=(f>s(t') as one-valued continuous functions of t such that as t ranges over the unit interval (0, 1), the point x, y, z ranges over the unit cube. 574. 1. Hubert's Curve. We wish now to study in detail the correspondence between the unit interval 21 and the unit square S3 afforded by Hilbert's curve defined in 573. A number of inter- esting facts will reward our labor. We begin by seeking the points P in 21 which correspond to a given Q in S3. To this end let us note how P enters and leaves an r} square. Let 5 be a square of D„. In the next division B falls into four PLANE CURVES 591 squares B^ ■■• S^ and in the n + 2^ division in 16 squares Bj. Of these last, four lie at the vertices of ^ ; we call them vertex squares. The other 12 are median squares. A simple considera- tion shows that the t] squares of I)„+^ are so numbered that we always enter a square B belonging to i)„, and also leave it by a vertex square. Since this is true of every division, we see on passing to the limit that the point Q enters and leaves any rj square at the ver- tices of r). We call this the vertex law. Let us now classify the points P, Q. If P is an end point of some division 2)„ > we call it a terminal point, otherwise an inner point, because it lies within a sequence of S intervals S' > 8" > ••■ = 0. The points Q we divide into four classes : 1° vertex^ points, when § is a vertex of some division. 2° inner points, when Q lies within a sequence of squares V>7?">-=0. 3° lateral points, when Q lies on a side of some t) square but never at a vertex. 4° points lying on the edge of the original square S3. Points of this class also lie in 1°, 3°. We now seek the points B corresponding to a ^ lying in one of these four classes. Class 1°. Q a Vertex Point. Let i)„ be the first division such that § is at a vertex. Then Q lies in four squares rj,, rjj, ??«, vi of A- There are 5 cases :• a) ij kl ai-e consecutive. )S) ij k are consecutive, but not I. 7) ij are consecutive, but not k I. ' S) ij, also k I, are consecutive. e) no two are consecutive. A simple analysis shows that a), 0) are not permanent in the following divisions ; 7), 8) may or may not be permanent ; e) is permanent. 592 GEOMETRIC NOTIONS Now, whenever a case is permanent, we can enclose § in a se- quence of 7/ squares whose sides = 0. To this sequence corre- sponds uniquely a sequence of B intervals of lengths = 0. Thus to two consecutive squares will correspond two consecutive inter- vals which converge to a single point P in 21. If the squares are not consecutive, the corresponding intervals converge to two dis- tinct points in 21. Thus we see that when 7) is permanent, to Q correspond three points P- When S) is permanent, to Q corre- spond two points P. While when Q belongs to e), four points P correspond to it. Class 2°. Q an Inner Point. Obviously to each Q corresponds one point P and only one. Glass 3°. Q a Lateral Point. To fix the ideas let Q lie on a ver- tical side of one of the j;'s. Let it lie between?;,, ly,- of i)„. There are two cases : «) y = t + 1. We see easily that a) is not permanent, while of course /S) is. Thus to each Q in class 3°, there correspond two points P- Class 4°. Q lies on the edge of 93. If ^ is a vertex point, to it may correspond one or two points P. If Q is not a vertex point, only one point P corresponds to it. To sum up we may say : To each inner point Q corresponds one inner point P- To each lateral point Q correspond two points P. To each edge point Q correspond one or two points P. To each vertex point Q, correspond two, three, or four points P. 2. As a result of the preceding investigation we show easily that: To the points on a line parallel to one of the sides of i8 correspond in 21 an apantactic perfect set. 3. Let us now consider the tangents to Hilbert's curve which we denote by H. PLANE CURVES 593 Let Q he a veMex point. We saw there were three permanent cases 7), S), e). In cases 7), 8) we saw that to two consecutive S intervals cor- respond permanently two contiguous ver- tical or horizontal squares. Thus as t ranges over <—: 1 — ^ — ' S, , S^+i, the point x, y ranges over these squares, and the secant line \ \ V.+i '7.+1 Q joining Q and this variable point x, y oscillates through 180". There is thus no tangent at Q. In case e) we see similarly that the secant line ranges through 90°. Again there is no tangent at q. In the same way we may treat the three other classes. We find that the secant line never converges to a fixed position, and may oscillate through 360°, viz. when Q is an inner point. As a result we see that Hilberfs curve has at no point a tangent, nor even a unilateral tangent. 4. Associated with Hilbert's curve ^are two other curves, X =(!>(€) , and «/ = ->|r(«). The functions <^, i/r being one-valued and continuous in 21, these curves are continuous and they do not have a multiple point. A very simple consideration shows that they do not have even a unilateral tangent at a pantactie set of points in 21. 575. Property 7°. liquations of a Curve. As already remarked, it is commonly thought that the equation of a curve may be written in any one of the three forms y=/(^), (1 ^(x,y)=0, (2 x = <^(t-) , y = ^(f), (3 and if these functions are continuous, these equations define con- tinuous curves. Let us look at the Hilbert curve H. We saw its equation could be expressed in the form 8). ^cuts an ordinate at every point of it for which < «/ < 1. Thus if we tried to define H by 594 GEOMETRIC NOTIONS an equation of the type 1), /(a;) would have to take on every value between and 1 for each value of a; in 31 = (0, 1). No such functions are considered in analysis. Again, we saw that to any value a; = a in 21 corresponds a perfect apantactic set of values \ta\ having the cardinal number c- Thus the inverse function of a; = (<) is a many-valued function of x whose different values form a set whose cardinal number is c. Such functions have not yet been studied in analysis. How is it possible in the light of such facts to say that we may pass from 3) to 1) or 2) by eliminating t from 3). And if we cannot, how can we say a curve can be represented equally well by any of the above three equations, or if the curve is given by one of these three equations, we may suppose it replaced by one of the other two whenever convenient. Yet this is often done. In this connection we may call attention to the loose way elimination is treated. Suppose we have a set of equations fn+i(^i---Xmh •••«„)=0. We often see it stated that one can eliminate ij ••• t„ and obtain a relation involving the a:'s alone. Any reasoning based on such a procedure must be regarded as highly unsatisfactory, in view of what we have just seen, until this elimination process has been established. 576. Property 8°. Closed Curves. A circle, a rectangle, an ellipse are examples of closed curves. Our intuition tells us that it is impossible to pass from the inside to the outside without crossing the curve itself. If we adopt the definition of a closed curve without multiple point given in I, 362, we find it no easy matter to establish this property which is so obvious for the simple closed curves of our daily experience. The first to effect the demonstration was Jordan in 1892. We give here * a proof due to de la VallSe-Poussin.-f Let us call for brevity a continuous curve without double point * The reader is referred to a second proof due to Brouwer and given in 598 seq. t Cours d' Analyse, Paris, 1903, Vol. 1, p. 307. PLANE CURVES 595 a Jordan curve. • A continuous closed curve without double point will then be a closed Jordan curve. Cf . I, 362. 577. Let G he a closed Jordan curve. However small (r> is taken, there exists a polygonal ring B containing G and such that 1° Each point of R is at a distance < or from G. 2° Each point of G is at a distance < a from the edges of R. For let X = ^(f) , y = f{t} (1 be continuous one-valued functions of tin T={a< 6) defining G. Let D = (a, a^, a^ — J) be a division of T of norm d. Let a, ttj, ttj — be points of G corresponding to a, a-^ ••• If c? is suffi- ciently small, the distance between two points on the arc Ci= («!_!, aj is •■■ =0. Let ^j, i?2 "■ ^® ^^^® corresponding rings, and let be their inner and outer edges. A point P of the plane not on C which lies inside some G-^ we call an interior or inner point of C. HP lies outside some G^, we call it an exterior or outer point of G. Each point P not on C must belong to one of these two classes. For let p = Dist (P, C); then p is > some o-„. It therefore lies within (7["' or without G'"\ and is thus an inner or an outer point. Obviously this definition is independent of the sequence of rings j^nl employed. The points of the curve Care interior to each G'p^ and exterior to each Gl^K Inner points must exist, since the inner polygons exist as al- ready observed. Let us denote the inner points by 3 and the outei' points by O. Then the frontiers of 3f and O are the curve 0. 2. We show now that 1° Two inner points can ie joined by a broken line L, lying in 3f- 2° Two outer points can be joined by a broken line L^ lying in O- 3° Any continuous curve R joining an inner point i and an outer point e has a point in common mth C. To prove 3°, let ^=/(0 , y = git') be the equations of S, the variable t ranging over an interval T= (^a<^'), t = a corresponding to i and t= fi to e. Let t' be PLANE CURVES 597 such that a so small that no square of A contains a point lying on non-consecutive arcs 0^ of C. Let us also take A so that the point a corresponding to t = a lies within a square, call it S^, of A. As t increases from t = a, there is a last point Pj on where the curve leaves S^. The point Pj lies in another square of A, call it aSj, containing other points of 0. Let Pj be the last point of in dSj . In this way we may continue, getting a sequence 1). There exists at least one point of 1) on each arc C. . For other- wise a square of A would contain points lying on non-consecutive arcs C, . The polj^gon determined by 1) cannot have a double 598 GEOMETRIC NOTIONS point, since each side of it lies in one square. The sides are < e, provided we take S V2 < e, since the diagonal is the longest line we can draw in a square of side 8. 580. Existence of Inner Points. To show that the links form a ring with inner points, Schonfliess * has given a proof which may be rendered as follows : Let us take the number of links to be even, and call them Xj, L^, ••• Xjn- Then L^, L^, L^--- lie entirely outside each other. Since ij, i/^ overlap, let P be an inner common point. Simi- larly let Q be an inner common point of Zi^, Lg. Then P, Q lying within L^ may be joined by a finite broken line b lying within ij- Let b^ be that part of it lying between the last point of leaving L^ and the following point of meeting L^. In this way the pairs of links define finite broken lines No two of these can have a common point, since they lie in non-consecutive links. The union of the points in the sets Li , b^ , Lg , &4 ••• ijB-l ' *2n we call a ring, and denote it by SR. Tlie points of the plane not in 3J fall into two parts, separated by SR. Let X denote the part which is limited, together with its frontier. We call 2; the inte- rior of 5R. That X has inner points is regarded as obvious since it is defined by the links which pairwise have no point in common, and by the broken lines each of which latter lies entirely within a link. Let S2™ = ^HA». 3:) ' m=l, 2, ... » Die Entwickelung der Lehre von den Punktmannigfaltigkeiten. Leipzig, 1908, Part 2, p. 170. PLANE CURVES 699 Then these 8 kave pairwise no point in common since the L^„ have not. Let 2 = 82 + 8^+... +8^^ + ^. Then S > 0. For let us adjoin L^ to 9t, getting a ring SRg whose interior call 2^2- That Stj bas inner points follows from the fact that it contains 8^, 8g •■• Let us continue adjoining the links -£4, Lq ••• Finally we reach L^^, to which corresponds the ring 9?2„, whose interior, if it exists, is 2lj„. If Z^^ does not exist, 2^2m-2 contains only ggn- This is not so, for on the edge of L^ bounding St, is a point P, such that some Dp{P') contains points of no L except L^. In fact there is a point P on the edge of L^ not in either L^ or L^^, as otherwise these would have a point in common. Nov/, if however small p > is taken, Dp(^P} contains points of some L other than ij , the point P must lie in i, which is absurd, since ij has only points in common with Zi^, L^^, and P is not in either of these. Thus the adjunction of L^, L^, ... Xg^ produces a ring ^^ whose interior 2^2„ S,j>S,j^ ••• to which correspond intervals in which the corresponding ^'s lie. But only one point t is thus determined. TJie functions 4) are continuous. This is obvious for points yS lying within the black intervals of T. It is true for the points \. For X lies within a sequence of white intervals, and while t ranges over one of these, the point on O ranges in a square. But these squares shut down to a point as the intervals do. Thus S„-> •■■ which converge to a point of £). Now On the other hand, B contains a null set of points of £), viz. the black segments. Thus = S= A , and hence O = 1 — A and 5) is established. Thus Osgood's curve is continuous, has no double point, and its upper content is 1 — A. 3. To get a continuous closed curve C without double point we have merely to join the two end points a, /3 of Osgood's curve by a broken line which does not cut itself or have a point in com- mon with the square S except of course the end points a, /8. Then bounds a figure % whose frontier is not discrete, and jj does not have an area. Let us call such curves closed Osgood curves. Thus we see that there exist regions bounded by Jordan curves which do not have area in the sense current since the Greek geometers down to the present day. Suppose, however, we discard this traditional definition, and employ as definition of area its measure. Then we can say : A figure % formed of a closed Jordan curve J and its interior Q has an area, viz. Meas g. For Front fj = t7. Hence g is complete, and is therefore meas- ureable. We note that ^ = ^j- ^ We have seen there are Jordan curves such that />0. DETACHED AJSID CONNECTED SETS 603 We now have a^definition of area which is in accordance with the promptings of our geometric intuition. It must be remembered, however, that this definition has been only recently discovered, and that the definition which for centuries has been accepted leads to results which flatly contradict our intuition, which leads us to say that a figure bounded by a continuous closed curve has an area. 583. At this point we will break off our discussion of the relation between our intuitional notion of a curve, and the con- figuration determined by the equations where , yjr are one-valued continuous functions of t in an interval T. Let us look back at the list of properties of an intuitional curve drawn up in 563. We have seen that the analytic curve 1) does not need to have tangents at a pautactic set of points on it ; no arc on it needs have a finite length ; it may completely fill the interior of a square ; its equations cannot always be brought in the forms y—f{pS) or F(xy^=Q, if we restrict ourselves to functions /or F employed in analysis up to the present; it does not need to have an area as that term is ordinarily understood. On the other hand, it is continuous, and when closed and with- out double point it forms the complete boundary of a region. Enough in any case has been said to justify the thesis that geometric reasoning in analysis must be used with the greatest circumspection. Detached and Connected Sets 584. In the foregoing sections we have studied in detail some of the properties of curves defined by the equations a; =0(0 , y = t(0- Now the notion of a curve, like many other geometric notions, is independent of an analytic rejiresentation. We wish in the fol- lowing sections to consider some of these notions from this point of view. 604 GEOMETRIC NOTIONS 585. 1. Let 21, 33 be point sets in m-way space SJ™. If Dist(2l, «8)>0, we say 21, S3 are detached. If 21 cannot be split up into two parts 58, S such that they are detached, we say 21 has no detached parts. If 21 = S3 + S and Dist (S3, £)> 0, we say S3, E are detached parts of 21. Let the set of points, finite or infinite, a, a-i, a^, •■• b (1 be such that the distance between two successive ones is < e. We call 1) an e-sequence between a, b; or a sequence with segments (a., a.+i) of length <€. We suppose the segments ordered so that we can pass continuously from a to S over the segments without retracing. If 1) is a finite set, the sequence is finite, otherwise infinite. 2. Let 21 have no detached parts. Let a, b be two of its points. For each e > 0, there exists a finite e-sequence between a, b, and lying inn. For about a describe a sphere of radius e. About each point of 21 in this sphere describe a sphere of radius e. About each point of 21 in each of these spheres describe a sphere of radius e. Let this process be repeated indefinitely. Let S3 denote the points of 21 made use of in this procedure. If S3 < 21, let g = 21 - S3. Then Dist (S3, E)>e, and 21 has detached parts, which is contrary to hypothesis. Thus there are sets of e-spheres in 21 joining a and b. Among these sets there are finite ones. For let % denote the set of points in 21 which may be joined to a by finite sequences ; let @ = 21 - g. Then Dist (g, ®) > e. For if < e, there is a point / in g, and a point ^ in ® whose distance is < e. Then a and ff can be joined by a finite e-sequence, which is contrary to hypothesis. 3. 7/21 has no detached parts, it is dense. For if not dense, it must have at least one isolated point a. But then a, and 21 — a are detached parts of 21, which contradicts the hypothesis. 4. Let 21, S3, S be complete and 21 = (S3, 6). 7/21 has no de- tached parts, S3, E have at least one common point. IMAGES 605 For if SB, S Jiave no common point, S = Dist (33, E) is > 0. But 8 cannot > 0, since S3, S would then be detached parts of 21. Since B = and since 93, E are complete, they have a point in common. 5. If 31 is such that any two of its points may be Joined hy an e-Bequenee lying in 21, where e > is small at pleasure, 21 has no detached parts. For if 21 had 93, S as detached parts, let Dist (93, E) = 8. Then S > 0. Hence there is no sequence joining a point of 93 with a point of g with segments < S. 6. If 21 is complete and has no detached parts, it is said to be connected. We also call 21 a connex. As a special case, a point may be regarded as a connex. 7. If^is connected, it is perfect. For by 3 it is dense, and by definition it is complete. %. If ^ is a rectilinear connex, it has a first point a and a last point /8, and contains every point in the interval (a, yS). For being limited and complete its minimum and maximum lie in 21 and these are respectively a and /3. Let now ei>e2>-" =0. There exists an e^-sequence Cj between a, yS. Each segment has an ej-sequence C^. Each segment of C^ has an eg-sequence Cj, etc. Let be the union of all these sequences. It is pantactic in (a, ff). As 21 is complete, 21 = («, ^). Images 586. Let ari=/iOi-U - a;„ =/„( 1, it is manifold. If r = 1, 21 is a simple image of J, other- wise it is a multiple image. If the functions 1) are one-valued and continuous in J, we say 21 is a continuous image of X. 587. Transformations of the Plane. Example 1. Let u-=x sin y , v =x cos y. (1 We have in the first place This shows that the image of a line x = a, a^O, parallel to the y-axis is a circle whose center is the origin in the u, v plane, and whose radius is a. To the «/-axis in the x, y plane corresponds the origin in the u, v plane. From 1) we have, secondly, u . - = tan y. This shows that the image of a line y = b, is a, line through the origin in the u, v plane. From 1) we have finally that u, v are periodic in y, having the period 2 tt. Thus as x, y ranges in the band B, formed by the two parallels y=±'ir, or — tt < ^ < tt, the point u, v ranges over the entire m, v plane once and once only. The correspondence between B and the w, v plane is unifold, except, as is obvious, to the origin in the u, v plane corresponds the points on the ?/-axis. Let us apply the theorem of I, 441, on implicit functions. The determinant A is here 9(m, v') _ sin y, cos y _ _ d(x,y) xcosy, —xsiny As this is =#= when x, y is not on the y-Axis, we see that the correspondence between the domain of any such point and its image is 1 to 1. This accords with what we have found above. IMAGES 607 It is, however, % much more restricted result than we have found ; for we have seen that the correspondence between any limited point set 21 in B which does not contain a point of the «/-axis and its image is unifold. 588. Example 2. Let u = — " , V = Va;^ + y\ ■ (1 the radical having the positive sign. Let us find the image of the first quadrant Q in the x, y plane. From 1) we have at once 00. Hence the image of § is a band B parallel to the v-axis. From 1) we get secondly Hence .., , ..,_„2 y = uv , x = 1) Vl — w^. (2 x2 + y2 Thus the image of a circle in Q whose center is the origin and whose radius is a is a segment of a right line v = a. When x = y=0, the equations 1) do not define the correspond- ing point in the m, v plane. If we use 2) to define the corre- spondence, we may say that to the line d = in 5 corresponds the origin in the x, y plane. With this exception the correspondence between Q and B is uniform, as 1), 2) show. The determinant A of 1) is, setting r = Vx^ + y'' 1 9(m, v~) _ d(x, y) — xy x^ X y r r — X x^^y' for any point a;, y different from the origin. 589. Example S. Reciprocal Radii. Let be the origin in the X, y plane and fi the origin in the m, v plane. To any point P = (x, y) in the x, y plane different from the origin shall cor- respond a point Q = (m, v") in the u, v plane such that D.Q has M2 + t;2 ' " m2 + 1,2 X y x^ + y^ ' " " ^2 + /• 608 GEOMETRIC NOTIONS the same direction as OP and such that OP • ilQ = 1. Analyti- cally we have x=\y , u=\v , \ > 0, and (m2 + v^)(3^ + f) = 1. From these equations we get ^ = .c . ..,. ' ^ = 7^-7-^ O and also M = The correspondence between the two planes is obviously unifold except that no point in either plane corresponds to the origin in the other plane. We find for any point x, y different from the origin that -.^ ^ ■, d{x,y) Cx^ + y^ Obviously from the definition, to a line through the origin in the X, y plane corresponds a similar line in the m, v plane. As xy moves toward the origin, u, v moves toward infinity. Let x, y move on the line x = a ^ 0. Then 1) shows that u, v moves along the circle a (m^ -)- 1,2) _ m = which passes through the origin. A similar remark holds when X, y moves along the line y = b ^ 0. 590. Such relations between two point sets 21, S3 as defined in 586 may be formulated independently of the functions /. In fact with each point a of 21 we may associate one or more points 6j, Jj •■• of S3 according to some law. Then S3 may be regarded as the image of 21. We may now define the terms simple, manifold, etc., as in 686. When h corresponds to a we may write h '^ a. We shall call 83 a continuous image of 21 when the following con- ditions are satisfied. 1° To each a in 21 shall correspond but one h in S3, that is, S3 is a simple image of 21. 2° Let 6 ~ a, let a^, a^ ••■ be any sequence of points in 21 which = a. Let 6„ ~ a„. Then 5„ must = h however the sequence \a„\ is chosen. IMAGES 609 When SB is a simple image of 21, the law which determines which 6 of 95 is associated with a point a of 21 determines obviously th one-valued functions as in 586, 1), where e. 2. Let f{t.y ... <„) be one-valued and continuous in the connex 21, then the image of 21 is an interval including its end points. This follows from the above and from 585, 8. 3. Let the correspondence between 21, 95 he unifold. If 95 is a continuous image of 21, then 21 is a continuous image of 95. For let |5„| be a set of points in 95 which = b. Let a„ ~ 5„, a ~b. We have only to show that a„ = a. For suppose that it does not, suppose in fact that there is a sequence a,j, a^^ ••■ which 610 • GEOMETRIC NOTIONS = a^ a. Let ^ ~ a. Then i^^, 6^, ••• = )3. But any partial se- quence of |6„S must = 6. Thus b = ^, hence a = a, hence a„ = a. 4. A Jordan curve J is a wnifold continuous image of an interval T. Conversely if J is a unifold continuous image of an interval T, there exist two one-valued continuous functions ^ = 'l>(0 > y - -fif) such that as t ranges over T, the point x, y ranges over J. In case J is closed it may be regarded as the image of a circle T. All but the last part of the theorem has been already established. To prove the last sentence we have only to remark that if we set x = r cos t , y = r sin t we have a unifold continuous correspondence between the points of the interval (0, 2 tt*) and the points of a circle. 5. The first part of 4 may be regarded as a geometrical definition of a Jordan curve. The image of a segment of the interval T or of the circle F, will be called an arc of J. 592. Side Lights on Jordan Curves. These curves have been defined by means of the equations x=it), y = ^{f). (1 As t ranges over the interval T = (a< b), the point P —(x, y~) ranges over the curve J. This curve is a certain point set in the x, y plane. We may now propose this problem : We have given a point set S in the a;, y plane ; may it be regarded as a Jordan curve ? That is, do there exist two continuous one-valued func- tions 1) such that as t ranges over some interval T, the point P ranges over the given set S without returning on itself, except possibly for f = a, t = b, when the curve would be closed? Let us look at a number of point sets from this point of view. 593. Example 1. «/ = sin - , a; in the interval 21 = (— 1, 1), but ^ X = , for a; = 0. IMAGES 611 is this point set S a Jordan curve ? The answer is, No. For a Jordan curve is a continuous image of an interval 21. By 591, l, it is complete. But S is not complete, as all the points on the y axis, — 1 < 3/ < 1 are limiting points of S, and only one of them belongs to S, viz. the origin. 2. Let us modify E by adjoining to it all these missing limiting points, and call the resulting point set Q. Is C a Jordan curve ? The answer is again, No. For if it were, we can divide the inter- val T into intervals S so small that the oscillation of ^, i/r- in any one of them is < a. To the intervals S, will correspond arcs (7, on the curve, and two non-consecutive arcs 0^ are distant from each other by an amount > some e, small at pleasure. This shows that one of these arcs, say (7,, must contain the segment on the ^-axis — 1 ^ y :^ !• But then Osc ■^ = 2 as f ranges over the correspond- ing S, interval. Thus the oscillation of ■^ cannot be made < e, however small 8, is taken. 3. Let us return to the set S defined in 1. Let A, B be the two end points corresponding to a; = —1, 2; = 1. Let us join them by an ordinary curve, a polygon if we please, which does not cut itself or 6. The resulting point set S divides all the other points of the plane into two parts which cannot be joined by a contin- uous curve without crossing ^. For this point of view ^ must be regarded as a closed configuration. Yet ® is obviously not complete. On the other hand, let us look at the curve formed by removing the points on a circle between two given points a, b on it. The remaining arc 8 including the end points a, J is a complete set, but as it does not divide the other points of the plane into two sepa- rated parts, we cannot say 8 is a closed configuration. We mention this circumstance because many English writers use the term closed set where we have used the term complete. Cantor, who first introduced this notion, called such sets ahge- schlossen, which is quite different from geschlossen = closed. _i 594. Example 2. Let p = e *, for ^ in the interval 21 = (0, 1) except ^ = 0, where /> = 0. These polar coordinates may easily be replaced by Cartesian coordinates -1 -1 x = ^(0}=e *cos^ , y = e 'sin9 , in 21, 612 GEOMETRIC NOTIONS except ^ = 0, when x, y both = 0. Tlie curve thus defined is a Jordan curve. Let us take a second Jordan curve with p = for 6 = 0. If we join the two end points on these curves corresponding to ^ = 1 by a straight line, we get a closed Jordan curve J, which has an interior 3» and an exterior O. The peculiarity of this curve J is the fact that one point of it, viz. the origin x = y = 0, cannot be joined to an arbitrary point of 3 by a finite broken line lying entirely in ^ ; nor can it be joined to an arbitrary point in O by such a line lying in £>. 595. 1. It will be convenient to introduce the following terms. Let 21 be a limited or unlimited point set in the plane. A set of distinct points in 21 determine a broken line. In case 1) is an infinite sequence, let a„ converge to a fixed point. If this line has no double point, we call it a chain, and the segments of the line links. In case not only the points 1) but also the links lie in 21, we call the chain a path. If the chain or path has but a finite number of links, it is called finite. Let us call a, precinct a region, i.e. a set all of whose points are inner points, limited or unlimited, such than any two of its points may be joined by a finite path. 2. Using the results of 578, we may say that, — A closed Jordan curve J divides the other points of the plane into two precincts, an inner ^ and an outer O- Moreover, they have a common frontier which is J. 3. The closed Jordan curve considered in 594 shows that not every point of such a closed Jordan curve can always be joined to an arbitrary point of 3f or D by a finite path. Obviously it can by an infinite path. For about this point, call it P, we can describe a sequence of circles of radii r = 0. Between any two of these circles there lie points of ^ and of O, if r is suf- IMAGES 613 ficiently small. ^In this way we may get a sequence of points in 3, viz. Jj, Jj ••• = P. Any two of these i^, I^^^ may be joined by a path which does not cut the path joining ij to I„. For if a loop were formed, it could be omitted. 4. Any arc ^ of a closed Jordan curve J can be joined hy a path to an arbitrary point of the interior or exterior., which call 21. For let J'= ^ + 8. Let A be a point of ^ not an end point. Let S = Dist (k, 8), let a be a point of 21 such that Dist (a, A) <1S. Then , = Dist(a,?)>lS. Hence the link I = (a, K) has no point in common with 8. Let b be the first point of I in common with S. Then the link m = (a, J) lies in 21. If now a is any point of 21, it may be joined to a by a path p. Then p + m is a path in 21 joining the arbi- trary point a to a point b on the arc ^. 596. Example 3. For ^ in 21 = (0*, 1) let p = a(l + e~~% p = a(l + e ^ ''). These equations in polar coordinates define two non-intersecting spirals S-^, S^ which coil about p = a as an asymptotic circle F. Let us join the end points of the spirals corresponding to ^ = 1 by a straight line L. Let S denote the figure formed by the spirals S-^^, S^., the segment L and the asymptotic circle F. Is S a closed Jordan curve ? The answer is. No. This may be seen in many ways. For example, S does not divide the other points into two precincts, but into three, one of which is formed of points within F. Another way is to employ the reasoning of 593, 2. Here the circle F takes the place of the segment on the y-axis which figures there. Still another way is to observe that no point on F can be joined to a point within S by a path. 597. Example 4- Let S be formed of the edge (£ of a unit square, together with the ordinates o erected at the points 614 GEOMETRIC NOTIONS x= —, of length — , w = 1, 2 ••• Although S divides the other points of the plane into two precincts 3 and £), we can say that S is not a closed Jordan curve. For if it were, 3 and O would have to have S as a common frontier. But the frontier of O is @, while that of 3 is S. That E is not a Jordan curve is seen in other ways. For example, let 7 be an inner segment of one of the ordinates 0. Obviously it cannot be reached by a path in O. Brouwer's Proof of Jordan's Theorem 598. We have already given one proof of this theorem in 677 seq., based on the fact that the coordinates of the closed curve are expressed as one-valued continuous functions Brouwer's proof* is entirely geometrical in nature and rests on the definition of a closed Jordan curve as the unifold continu- ous image of a circle, cf. 591, 5. If 21, 93, ■■■ are point sets in the plane, it will be convenient to denote their frontiers by %,^, gjg ... so that ^51= Front 21 , etc. We admit that any closed polygon ^ having a finite number of sides, without double point, divides the other points of the plane into an inner and an outer precinct ^^ , ^<, respectively. In the following sections we shall call such a polygon simple, and usu- ally denote it bj' ^. We shall denote the whole plane by (g. "^^^^ 6 = ^-1-^.4-^,. Let 21 be complete. The complementary set A is formed, as we saw in 328, of an enumerable set of precincts, say A— Jj1„|. • Math. Annalen, vol. 69 (1910), p. 169. BROUWER'S PROOF OF JORDAN'S THEOREM 615 599. 1. If mpreeinot 21 and its complement* A each contain a point of the connex S, then ^^ contains a point of S. For in the contrary case c = I)v(^, S) is complete. In fact S3 = 21 + i^a ^® complete. As S is complete, Dv(SS, S) is com- plete. But if 55j( does not contain a point of S, c = Dv(SB, 6). Thus on this hypothesis, c is complete. Now c = Dv(^A, S) is complete in any case. Thus 6 = c + c, which contradicts 585, 4. 2. // '!P„ i|}j, the interior and exterior of a simple polygon $ each contain a point of a connex S, then ^ contains a point of E. 3. Let ^ be complete and not connected. There exists a simple polygon '^ such that no point of ^ lies on ^, while a part of ^ lies in ^, and another part in ^^ . For let £i, ^2 be two non-connected parts of ^ whose distance from each other is /o > 0. Let A be a quadrate division of the plane of norm S, so small that no cell contains a point of ^j and ^2- Let Aj denote the cells of A containing points of ^j. It is complete, and the complementary set Ag = @ — Aj is formed of one or more precincts. No point of ^j lies in A^ or on its frontier. Let Pj, Pj be points in Sj, ^^ respectively. Let I) be that precinct containing P^. Then ^J^ embraces a simple polygon ^ which separates P^ and P2 • 4. Let Sj, ^2 ^* ^*<"' detached connexes. There exists a simple polygon ^ which separates them.. One of them is in ^,, the other in ^j, and no point of either connex lies on 'iJJ. For the previous theorem shows that there is a simple polygon ^ which separates a point Pj in Sj from a point P^ in ^2 '^^^ ^o point of Sj or ^2 lies on ^. Call this fact F. Let now Pj lie in '^, . Then every point of ^j lies in % . For otherwise ^. and ^^ each contain a point of the connex ^j . Then 2 shows that a point of , 0. Let P^ be a point of gSi such that some domain b of Pj contains only points of ^j and of I. Let Q-^ be a point of f in b. Join P^, Q^ by a right line, let it cut 5% fi''st at the point P'. In a similar way we may reason on ^2) obtaining the points P", Q^. Then P'Q^Q^P" is the path in question. If we denote it by v, we may let v* denote this path after removing its two end points. 8. Let ^j, ^2 ^^ *""' detached connexes. A path v Joining Sj, ^2 and lying ini= Inter (Sj, Sg)' ^'^'^ points excepted, determines one and only one precinct in I. BROUWER'S PROOF OF JORDAN'S THEOREM 617 For from an arbitrary point P in t, let us draw all possible paths to V. Those paths ending on the same side (left or right) of V certainly lie in one and the same precinct f^ or \ iii ^ Then since one end point of v is inside, the other end point outside $, there must be a part of "^ which is not met by v and which joins the right and left sides of v. We take this as an evident property of finite broken lines and polygons without double points. Thus fj and f, are not detached ; they are parts of one precinct. .9. Two paths Vj, v^ without common point, lying in f and joining ^j, ^2' ^P^** ^ *'*^'' two precincts. Let i = ! — i;j ; this we have just seen is a precinct. From any point of it let us draw paths to v^. Those paths ending on the same side of v^ determine precincts tj, i, which may be identical. Suppose they are. Then the two sides of v^ can be joined by a path lying in f, which does not touch v^ (end points excepted), has no point in common with i)^, and together with a segment of Wj forms a simple polygon '^ which has one end point of v-y in ip^, the other end point in ip^. Thus by 2, ^ contains a point of tlie connex v-y. This is contrary to hypothesis.. Similar reasoning shows that 10, The n paths v^ ••• v„pairwise without common point, lying in t, and joining the connexes ^j, ^2 ^i^^*'* ^ *'*^^'' n precincts. Let us finally note that the reasoning of 595, 4, being independ- ent of an analytic representation of a Jordan curve, enables us to use the geometric definition of 591, 6, and we have therefore the theorem 11. Let %he a precinct whose frontier '^ is a Jordan curve. Then there exists a path in 21 joining an arbitrary point of 21 with any arc of%- Having established these preliminary theorems, we may now take up the body of the proof. 600. 1. Let St Je a precinct determined by a closed Jordan curve J. Then JJ = Front 21 is identical with J. If J determines but one precinct 21 which is pantactic in (g, we have obviously % = J. 618 GEOMETRIC NOTIONS Suppose then that SI is a precinct, not pantactic in @. Let 33 be a precinct ^t 31 determined by %. Let b = Front 93. Then b<^}^<^J. Suppose now b m. Let us effect a reticulation M of 3I„ of norm p. If S > is taken sufficiently small A = D^s(^a') lies in 21. Let Ij=Ds(^a') ; if /o is taken sufficiently small, the cells ^1' ^2 '■■ ^« (■'■ of R which contain points of E, lie in A. Let the image of E be @, and that of the cells 1) be Si, Sj-S.. (2 These are complete. Each point of (g lies in one of the sets 2). Hence by 2, they contain a point /3 which lies in n + 1 of them. Then a~/8 lies in n + 1 of the cells 1). But these, being part of the reticulation R, are such that no point lies in more than m+1 of them. Hence the contradiction. 602. 1. Sehonfliess' Theorem. Let be one- valued and continuous in a unit square A whose center is the origin. These equations define a transformation T. If T is regular, we have seen in I, 742, that the domain D^(P') of a point P — (x, y) within A goes over into a set H such that if ^ ~ P then -Z)<,(§) lies in ^, if o- >0 is sufficiently small. 622 GEOMETRIC JSOTIONS These conditions on /, g which make T regular are sufficient, but they are much more than necessary as the following theorem due to Schonfliess * shows. 2. Let A = B + c be a unit square in the x, y plane, whose center is the origin and whose frontier is c. u =f{x, y-) , v = g(x, y) be one-valued continuous functions in A. As (x, y) ranges over A, let (m, v) range over 21 = 33 + c where c ~ c. Let the correspondence between A and 21 be uniform. Then c is a closed Jordan curve and the interior c. of c is identical with 33. That c is a closed Jordan curve follows from 576 seq., or 598 seq. Obviously if one point of S lies in c,, all do. For if /S„ /S^ are points of 93, one within c and the other without, let 6, ~yS,, 6„~/8j. Then 6., b^ lying in B can be joined by a path in B which has no point in common with c. The image of this path is a continuous curve which has no point in common with c, which contradicts 578, 2. be the equation of c in polar coordinates. li (e)sine\ = F{fi,0), v=g\ !=(?(/., ^). These c^ curves have now the following property : If a point (p, q) is exterior (interior^ to t^^, it is exterior {in- terior) to c^,/or all fi such that I A* — Mo I ^ some e > 0. For let p^ be the distance of ( p, q) from a point (m, v) on c^ . Then , *6oeUingen Naehrichten, 1899. The demonstration here given is due to Osgood, Ooett. Naehr., 1900. AREA OF CURVED SURFACES 623 is a continuou^f unction of 6, fx. wliich does not vanish for fi = fi^, when < < 2 TT. But being continuous, it is uniformly con- tinuous. It therefore does not vanish in the rectangle -e + iiQ of norm d of the m, v plane. The rectangles fall into triangles t^ on drawing the diagonals. Such a division of the plane we call quaii rectangular. Po=(Mo, fo) , Pi=(mo + S, «;) , Pj = («„,«„ + 7;) be the vertices of t^. To these points in the m, v plane corre- spond three points ^. = (a;., y,, aj, t= 1, 2, 3, of *S' which form the vertices of one of the triangular faces t, of the inscribed polyhe- dron n^) corresponding to the division D. Here, as in the follow- ing sections, we consider only triangles lying in 21. We may do this since 21 is metric. Let X,, P",, Z^ be the projections of t, on the coordinate planes. Then, as is shown in analytic geometry, where T\ = Xl+ Y\ + Z\ 2X.= ^0 20 1 ^2 H 1 = = and similar expressions for y, , Z,. Thus the area of IT^) i s = 2vxj-t-r?+^2 » A"y A'z A"3 the summation extending over all the triangles t^ lying in the set 21. Let a;, y, z have continuous first derivatives in 21. Then ^'x = x^-x,A^cJl■, A"x = x,-x, = f^v + <^'% AREA OF CURVED SURFACES 625 with similar exoressions for the other increments. Let dg Bz Bx Bz Bx By A = Bu Bu By Bz , B = Bu Bu Bx Bz , 0= Bu Bu Bx By 3v 3i Bv Bv Bv Bv (2 Then where «, /S, 7, are uniformly evanescent with d in 3)[. Thus if A, B, do not simultaneously vanish at any point of 21, we have as area of the surface S lim Sj)= f-y/A^ + J^+ CUudv. d=0 »/2l (3 2. An objection which at once arises to this definition lies in the fact that we have taken the faces of our inscribed polyhedra in a very restricted manner. We cannot help asking, Would we get the same area for *S' if we had chosen a different system of polyhedra ? To lessen the force of this objection we observe that by replac- ing the parameters u, v hj two new parameters u', v' we may replace the above quasi rectangular divisions which correspond to the family of right lines m = constant, «; = constant by the infinitely richer system of divisions corresponding to the family of curves u' = constant, v' = constant. In fact, by subjecting u', v' to cer- tain very general conditions, we may transform the integral 3) to the new variables u', v' without altering its value. But even this does not exhaust all possible ways of dividing SI into a system of triangles with evanescent sides. Let us there- fore take at pleasure a system of points in the u, v plane having no limiting points, and join them in such a way as to cover the plane without overlapping with a set of triangles t^. If each triangle lies in a square of side d, we may call this a triangular division of norm d. We may now inquire if Sf) still converges to the limit 3), as d = 0, for this more general system of divisions. It was generally believed that such was the case, and standard treatises even contained demonstrations to this effect. These demonstrations are wrong ; for Schwarz * has shown that by • Werke, vol. 2, p. 309. 626 GEOMETRIC NOTIONS properly choosing the triangular divisions D, it is possible to make Sj) converge to a value large at pleasure, for an extensive class of simple surfaces. 604. 1. Schwarz's Example. Let C be a right circular cj'lin- der of radius 1 and height 1. A set of planes parallel to the base at a distance - apart cuts out a system of circles Fj, Fj ••• Let us divide each of these circles into m equal arcs, in such a way that the end points of the arcs on Fj, Fj, Fg ••• lie on the same vertical generators, while the end points of Fg, F^, Fg ••■ lie on generators halfway between those of the first set. We now inscribe a polyhedron so that the base of one of the triangular facets lies on one circle while the vertex lies on the next circle above or below, as in the figure. The area t of one of these facets is < = ^6A 6 = 2 sin — m ^^4. + 1 cos- my Thus t = sin • 1 ' -M m 'n^ 2 m There are 2 m such triangles in each layer, and there are n layers. Hence the area of the polyhedron corresponding to this triangular division D is S^ = 2;e.= 2 mn sin -a/^ + 4 Sin' m "W 2m' Since the integers m, n are independent of each other, let us consider various relations which may be placed on them. Case 1°. Let n = \m. Then Sj)=:2 m^\ sin - TT^/ 1 m 'V« + 4sin*-^ \%2 2 m. = 2'mh. IT m IT sin — m 'X%2 + 4 — 9 , m as m = 00. tt" Sin 2ni. 7r 2*m* 2m AREA OF CURVED SURFACES 627 Case 2°. Let% = \m^. Then m m TT m = 2^yj. \2to* + 4 TT* 2m 2*m* TT 2m 1 + — - X^ , as m = 00. Case 3°. Let n = Xm^. Then >S'^ = 2^ sin^ TT m '1 + Jm2x2 sm ■ TT 2m = 4-00 as m ; ; 00. 2. Thus only in the first case does S^, converge to 2 tt, which is the area of the cylinder C as universally understood. In the 2° and 3° cases the ratio h/b = 0. As equations of C we may take X = cos u ■,' y = sin M , z = v. Then to a triangular facet of the inscribed polyhedron will cor- respond a triangle in the w, v plane. In cases 2° and 3° this tri- angle has an angle which con verges, to tt as m = oo. This is not so in case 1°. Triangular divisions of this latter type are of great importance. Let us call then a triangular division of the m, v plane such that no angle of any of its triangles is greater than TT — e, where e > is small at pleasure but fixed, positive triangu- lar divisions. We employ this term since the sine of one of the angles is > some fixed positive number. 605. The Outer Definition. Having seen one of the serious diffi- culties which arise from the inner definition, let us consider briefly the outer definition. We begin with the simplest case in which the equation of the surface S is s =f(x, y). (1 / being one-valued and having continuous first derivatives. Let us effect a metric division A of the a;, y plane of norm S, and on 628 GEOMETRIC NOTIONS each cell d, as base, Ave erect a right cylinder C, which cuts out an element of surface S^ from S. Let ^^ be an arbitrary point of S, and J, the tangent plane at this point. The cylinder O cuts out of 2, an element A*S', . Let v, be the angle that the normal to £, makes with the s-axis. Then 1 cos V, = VWIHI. and ^^ _ d^ cos Vic The area of S is now defined to be lim 2A»S, (2 {=0 when this limit exists. The derivatives being continuous, we have at once that this limit is x-w>Hiy-(sy which agrees with the result obtained by the inner definition in 603, 3). The advantages of this form of definition are obvious. In the first place, the nature of the divisions A is quite arbitrary ; however they are chosen, one and the same limit exists. Secondly, the most general type of division is as easy to treat as the most narrow, viz. when the cells d„ are squares. Let us look at its disadvantages. In the first place, the elements AiS, do not form a circumscribing polyhedron of *S'. On the con- trary, they are little patches attached to 8 At the points ^,, and having in general no contact with one another. Secondly, let us suppose that S has tangent planes parallel to the g-axis. The de- rivatives which enter the integral 603, 3) are no longer continuous, and the reasoning employed to establish the existence of the limit 2) breaks down. Thirdly, we have the case that s is not one- valued, or that the tangent planes to S do not turn continuously, or do not even exist at certain points. AREA OF CURVED SURFACES 629 To get rid M these disadvantages various other forms of outer definitions liave been proposed. One of these is given by Q-oursat in liis Qours d'Analyse. Instead of projecting an arbitrary element of surface on a fixed plane, the xy plane, it is projected on one of the tangent planes belonging to that element. Hereby the more general type of surfaces defined by 603, 1) instead of those defined by 1) above is considered. The restriction is, however, made that the normals to the tangent planes cut the elements of surface but once, also the first derivatives of the coordinates are assumed to be continuous in 21. Under these conditions we get the same value for the area as that given in 603, 3). When the first derivatives of x, y, z are not continuous or do not exist, this definition breaks down. To obviate this difficulty de la VallSe-Poussin has proposed a third form of definition in his Cours d'Analyse, vol. 2, p. 30 seq. Instead of projecting the element of surface on a tangent plane, let us project it on a plane for which the projection is a maximum. In case that S has a con- tinuously turning tangent plane nowhere parallel to the z-axis, de la Vallee-Poussin shows that this definition leads to the same value of the area of ;S' as before. He does not consider other cases in detail. Before leaving this section let us note that Jordan in his Cours employs the form of outer definition first noted, using the paramet- ric form of the equations of S. In the preface to this treatise the author avows that the notion of area is still somewhat obscure, and that he has not been able "S, definir d'une mani^re satisfaisante I'aire-d'une surface gauche que dans le cas ou la surface a un plan tangent variant suivant une loi continue." 606. 1. Regular Surfaces. Let us return to the inner definition considered in 603. We have seen in 604 that not every system of triangular divisions can be employed. Let us see, however, if we cannot employ divisions much more general than the quasi rec- tangular. We suppose the given surface is defined by X = 0(m, v} , y = ■f(u, v) , z = x(". *>) (1 the functions ^, yjr, x being one-valued, totally differentiable func- tions of the parameters u, v which latter range over the complete 630 GEOMETRIC NOTIONS metric set 21. Surfaces characterized by these conditions we shall call regular. Let ^O^K'^'o) ' Pi=(«o+S', I'o+'j') , P^ = (u^+h",v^ + r,") be the vertices of one of the triangles <,, of a triangular division D of norm d of 21. As before let ^q, ^j, ^g be the corresponding points on the surface S. Then A':r = rri - x„ = g S' + g V + «iS' + ;SiV, A"a: = a;, .zo=^8" + ^v'+«;'s+/8;v', 5m 5d and similar expressions hold for the other increments. Also 2X = du ''JLS+^rf dv du 8v du ■ dv du dv + 2X^, where XJ, denotes the sum of several determinants, involving the infinitesimals Similar expressions hold for I^,, Z,. We get thus X, = A J, + XI , F. = A«. + YI , Z. = (7<«. + Zl where A, B, are the determinants 2) in 603. Then the area of the inscribed polyhedron corresponding to this division D is Let us suppose that A^ + B^+0^>q , q>0 (2 as u, V ranges over 21. Also let us assume that (3 Y' AREA OF CURVED SURFACES 631 remain numeri4j^,lly < e for any division D of norm dA"z- A'3A"^/| < I {I A't/| . I A"z| + 1 A'2| • | A"z/|| < M^P^Pj • PJ^^ = 2 A I cosec e, I AREA OF CURVED SURFACES 633 where Q^ is the^ngle made by the sides P^P\i PqPz- As i) is a positive division, one of the angles of t^ is such that cosec 6,^ is numerically less than some positive number M. Thus \x^\<2,xmt,, vfhere /u, M are independent of k and d. Similar relations hold for I F«|, \Z^\. Hence Sjy<^Q /uW- <« = 6 f>?MiM. + 7/) where 77 > is small at pleasure, for cZq sufficiently small. 2. Let 21 and. a;, y, z he as in 606, 3, except at certain points form- ing a discrete set a, the first partial derivatives do not exist. Let their total difference quotients he limited in 91. Then \\mSo= C^A^ + B^+O^dudv, where D denotes a positive trianyular division of norm d. Let us first show that the limit on the left exists. We may choose a metric part 53 of 21 such that (S = 21 — S3 is complete and exterior to 21 and such that S3 is as small as we please. Let S(^ denote the area of the surface corresponding to S. The triangles <, fall into two groups : Gti containing points of S3 ; Gr^ containing only points of S. Then aS^ = 2 VZI + 72 + 22 = S + 2. But 93 may be chosen so small that the first sum is < e/-! for any dKd^. Moreover by taking c?,, still smaller if necessary, we have |2-ASg|, = Osc z in the square s,. If the triangle t^ lies in the squares s,^, ■■■ s,^. Thus, n denoting a sufficiently large constant, 2 I X, I < fi.'^d^w.^ + ••• m is of constant occurrence, and is to be read : for each e > 0, there exists an index m, such that | « — a„ j < t, for every n > m Similarly a line of symbols as ; t>0, 8>0, |/(a:)-«l<£, lin Vs*(a) is to be read : for each e > 0, there exists a 8 > 0, such that \f(x) -«|