IN t c l£RPO'TfgHH| ■ '' K 1^^ n JQjBtOPPlloBoBBMr. : ■>Sm^^^^^Knisn'X'!' •- III III ■iJflF-- . >»vi^i««Oa0(:Mti;sSC(9ML;^tri. .Ft, ..,.-.s zo^^J:f %ATH. OEPT. LECTURES ON THE THEORY OF FUNCTIONS OF REAL VARIABLES A^OLUME I BY JAMES PIEEPONT Professor of Mathematics in Yale University '^i^^... Math, dbpt GINN AND COMPANY BOSTON • NEW YORK • CHICAGO • LONDON ATLANTA • DALLAS ■ COLUMBUS • SAN FRANCISCO Entered at STAxioitEKS' Hall, COPi'KIGHT, 1905 By JAMES PIERPONT ALL RIGHTS RESERVED PRINTED IN THE UNITED STATES OP AMERICA 529.5 GINN A^ D COMPANY • PRO- PRIETORS ■ BOSTON • U.S.A. PREFACE The present work is based on lectures which the author is accustomed to give at Yale University on advanced calculus and the theory of functions of real variables. It falls in two volumes, and the following remarks apply only to the first. The student of mathematics, on entering the graduate school of American universities, often has no inconsiderable knowledge of the methods and processes of the calculus. He knows how to differentiate and integrate complicated expressions, to evaluate indeterminate forms, to find maxima and minima, to differentiate a definite integral with respect to a parameter, etc. But no em- phasis has been placed on the conditions under which these pro- cesses are valid. Great is his surprise to learn that they do not always lead to correct results. Numerous simple examples, how- ever, readily convince him that such is nevertheless the case. The problem therefore arises to examine more carefully the conditions under which the theorems and processes of the calculus are correct, and to extend as far as possible or useful the limits of their applicability. In doing this it soon becomes manifest that the style of reason- ing which the student has heretofore employed must be abandoned. Examples of curves without tangents, of curves completely filling areas, and other strange configurations so familiar to the analyst of to-day, make it clear that the rough and ready reasoning which rests on geometric intuition must give way to a finer and more delicate analysis. It is necessary for him to learn to think in the e, 8 forms of Cauchy and Weierstrass. We have here the beginnings of the theory of functions of real variables, and the twofold problem just sketched characterizes sufticiently well the subject-matter and form of treatment of the present volume. iv PREFACE To obtain a foundation, the author has begun by developing the real number system after the manner of Cantor and Dedekind, postulating the theory of positive integers. To obtain sufficient generality, he has employed from the start the more simple prop- erties of point aggregates. No attempt, however, has been made to state every theorem with all possible generality. The author has allowed himself a wide liberty in this respect. Some theorems are stated under very broad conditions, while others are enunciated under extremely narrow ones. Some of these latter will be taken up later on. Two features of this volume may be mentioned here. In the first place, the Euclidean form of exposition has been adopted. Each theorem with its appropriate conditions is stated and then proved. Without doubt this makes the book less attractive to read, but on the other hand it increases its usefulness as a book of reference. One is thus often saved the labor of running through a complicated piece of reasoning to pick up sundry conditions which have been introduced, sometimes without any explicit mention, in the course of the demonstration. Secondly, numerous examples of incorrect forms of reasoning currently found in standard works on the calculus have been scattered through the earlier part of the volume. It is the author's experience that nothing stimulates the student's critical sense so powerfully as to ask him to detect the flaws in a piece of reasoning which at an earlier stage of his training he considered correct. A few new terms and symbols have been introduced, but only after long deliberation. It is hoped that their employment suffi- ciently facilitates the reasoning, and the enunciation of certain theorems, to justify their introduction. It may be well to note here the author's use of the word " any " in the sense of any one at pleasure, and not in the sense of some one. The words " each," "every," "some," "any," are often used in an indiscriminate manner, and to this is due a part of the difficulty the beginner experiences in modern rigorous analysis. No attempt has been made to attribute the various results here given to their respective authors. That has been rendered un- necessary by the very full bibliographies of the Encyclopddie der PREFACE V Mathematischen Wissenschaften. The author feels it his pleasant duty, however, to acknowledge his large indebtedness to the writ- ings of Jordan, Stolz, and Vallee-Poussin. He hopes, howevei-, that it will be found that he has not used them servilely, but in an individual and independent manner. Finally, he wishes to express his hearty thanks to his friend Professor M. B. Porter, and to his former pupil Dr. E. L. Dodd, for the unflagging interest they have shown during the composi- tion of this volume and for their many and valuable suggestions. JAMES PIERPONT. New Haven, Conn., August, 1905. Note A list of some of the mathematical terms and symbols employed in this woik will be found at the end of the volume. CONTENTS CHAPTER I RATIONAL NUMBERS ARTICLES PAGE ]. Historical Introduction 1 2-19. Fractions 5 20-30. Negative Numbers 12 31-;>5. Some Properties of the System B 19 36-39. Some Inequalities ^ . . 22 40-51. Rational Limits 24 CHAPTER II IRRATIONAL NUMBERS 62-53. Insufficiency of JS 31 54-80. Cantor's Theory ' 32 81-85. Some Properties of 9{ 52 86-96. Numerical Values and Inequalities 64 97-111. Limits ' 61 112-122. The Measurement of Rectilinear Segments. Distance ... 72 123-127. Correspondence between 91 and the Points of a Right Line . = 78 128-131. Dedekind's Partitions ~r 82 132-143. Infinite Limits 85 144-145. Different Systems for Expressing Numbers . . , • . 91 CHAPTER III EXPONENTIALS AND LOGARITHMS 146-159. Rational Exponents . 95 160-172. Irrational Exponents . 101 173-179. Logarithms . 109 180-184. Some Theorems on Limits .... . 112 185. Examples . 117 Vlll CONTENTS CHAPTEE IV THE ELEMENTARY FUNCTIONS. NOTION OF A FUNCTION IN GENERAL 186-193. 194-195. 196. 197-199. 200-208. 209-210. 211-215. 216. 217-220. 221-224. 225-229. 230-237. 238-239. 240-241. Fdnctions of One Variable PAGE Definitions * 118 Integral Rational Functions 121 Rational Functions 123 Algebraic Functions 123 Circular Functions 125 Exponential Functions 131 One-valued Inverse Functions 131 Logarithmic Functions 134 Many-valued Inverse Functions ■ . 135 Inverse Circular Functions 137 Functions of Several Variables Rational and Algebraic Functions 139 Functions of Several Variables in General 143 Composite Functions 145 Limited Functions 147 CHAPTER V FIRST NOTIONS CONCERNING POINT AGGREGATES 242-253. Preliminary Definitions 148 254-255. Limiting Points 157 256-261. Limiting Points connected with Certain Functions .... 158 262-269. Derivatives of Point Aggregates 162 270-272. Various Classes of Point Aggregates 167 CHAPTER VI LIMITS OF FUNCTIONS Functions of One Variable 273-277. Definitions and Elementary Theorems 278-284. Second Definition of a Limit . 285-294. Graphical Representation of Limits 295-305. Examples of Limits of Functions 306-312. The Limit e and Related Limits 171 175 180 184 190 CONTENTS IX Functions of Several Variables ARTICLES PAGE 313—317. Definitions and Elementary Tlieorems 193 318-321. A Method for Determining tlie Non-existence of a Limit . . 196 322-324. Iterated Limits 198 325-328. Uniform Convergence 199 329-335. Remarks on Diriclilet's Definition of a Function .... 202 336-338. Upper and Lower Limits 206 CHAPTER VII CONTINUITY AND DISCONTINUITY OF FUNCTIONS 339-342. Definitions and Elementary Theorems 208 343-346. Continuity of the Elementary Functions 210 347. Discontinuity 211 348. Finite Discontinuities 212 349. Infinite Discontinuities 212 350-358. Some Properties of Continuous Functions 214 369-361. The Branches of Many-valued Functions 219 362. Notion of a Curve 220 CHAPTER VIII DIFFERENTIA TION 363-364. 365-366. 367-371. 372-389. 390-392. 393-404. 405-408. 409-413. 414-417. 418-422. 423-430. 431-433. 434^435. Functions of One Variable Definitions 222 Geometric Interpretations 223 Non-existence of the Differential Coefficient 225 Fundamental Formulae of Differentiation 229 Differentials and Infinitesimals 244 The Law of the Mean 246 Derivatives of Higher Order 252 Taylor's Development in Finite Form 266 Functions of Several Variables Partial Differentiation 259 Change in the Order of Differentiating 262 Totally Differentiable Functions 268 Properties of Differentials 276 Taylor's Development in Finite Form 279 CONTENTS CHAPTER IX IMPLICIT FUNCTIONS AKTIOLBS 436. Definitions 437-438. Existence Tiieorems ; one independent and one dependent variable 439. Extension of tlie Domain of Existence 440-443. Existence Theorems ; several Variables 282 284 290 291 CHAPTER X INDETERMINATE FORMS 444-449. Application of Taylor's Development in Finite Form 450-451. The Form 0/0 452-153. The Form 00 /qo ..... 454. Other Forms 455-459. Criticisms 460-464. Scales of Infinitesimals and Infinities 465. Order of Infinitesimals and Infinities 298 301 305 307 308 312 316 CHAPTER XI MAXIMA AND MINIMA One Variable 466-467. Definitions. Geometric Orientation 317 468-473. Criteria for an Extreme . . . 318 474-475. Criticism 321 Several Variables 476-478. Definite and Indefinite Forms 322 479-480. Semidefinite Forms . . . .326 481-483. Criticism 327 484-486. Relative Extremes 329 CHAPTER XII INTEGRATION 487-488. Geometric Orientation 333 489. Analytical Definition of an Integral 335 490-492. Upper and Lower Integrals ........ 337 493-498. Criteria for Integrability 340 499-503. Classes of Limited Integrable Functions 344 CONTENTS XI ARTICLES PAGE 504-508. Properties of Integrable Functions 346 509-513. Functions with Limited Variation 349 514-518. Content of Point Aggregates 352 619-522. Generalized Definition of an Integral , 356 523-530. 631-535. 536-638. 539. 640-544. 545-546. 647-549. 550-551. 652-554. 565-559. 560. 661-566. 567. 568. 569-570. CHAPTER XIII PROPER INTEGRALS First Properties , . 361 First Tiieorem of the Mean 366 The Integral considered as a Function of its Upper Limit . . 368 Criticism ... ...... . . .371 Change of Variable 371 Second Theorem of the Mean 377 Indefinite Integrals Primitive Functions 380 Methods of Integration 383 Integration by Parts. 384 Change of Variable 386 Integrals Depending on a Parameter Definitions 387 Continuity 388 Differentiation 392 Integration . 394 Inversion of the Order of Integration 395 671-577. 578-590. 591-605. 606-607. 608. 609-615. 616-619. 620-621. 622-628. 629-6-31, CHAPTER XIV IMPROPER INTEGRALS. INTEGRAND INFINITE Preliminary Definitions . . .* 399 Criteria for Convergence 405 Properties of Improper Integrals 412 Change of Variable 420 Second Theorem of the Mean 421 Integrals Depending on a Parameter Uniform Convergence 424 Continuity 429 Integration 432 Inversion 435 Differentiation 441 Xll CONTENTS CHAPTER XV IMPROPER INTEGRALS. INTERVAL OF INTEGRATION INFINITE ARTICLES 632-634. 635-646. 647-652. 653-654. 655-657. 658-665. 666-670. 671-682. 683-691. 692, PAGE Definitions 445 Tests of Convergence 450 Properties of Integrals 456 Theorems of the Mean . , . 459 Change of Variable 462 Integrals Depending on a Parameter Uniform Convergence 464 Continuity 474 Integration and Inversion 479 Differentiation 493 Elementary Properties of B(?<, u), r(M) . . . . . .501 CHAPTER XVI MULTIPLE PROPER INTEGRALS 693. Notation 506 694-701. Upper and Lower Integrals 507 702-703. Content of Point Aggregates 512 704. Frontier Points 514 705-710. Discrete Aggregates 515 711-716. Properties of Content 519 717-718. Plane and Rectilinear Sections of an Aggregate .... 524 719-721. Classes of Integrable Functions 526 722-723. Generalized Definition of Multiple Integrals 528 724-731. Properties of Integrals 531 732-737. Reduction of Multiple Integrals to Iterated Integrals . . • 537 738-740. Application to Inversion ^ 543 741-746. Transformation of the Variable 547 FUI^OTION THEORY OF REAL VARIABLES CHAPTER I RATIONAL NUMBERS Historical Introduction 1. The reader is familiar with the classification of real numbers into rational and irrational numbers. The rational numbers are subdivided into integers and fractions. Besides the real numbers there is another class of numbers currently employed in modern analysis, viz. complex or imaginary numbers. In this work we shall deal almost exclusively with real numbers. Historically, the first numbers to be considered were the posi- tive integers 1, 2, 3, 4, 5, 6, . . . (3^ We shall denote this system of numbers by 3- It is not our intention to develop the theory of these numbers ; instead, we shall merely call attention to some of their funda- mental properties.* In the first place, we observe that the elements of ^ are ar- ranged in a certain fixed order ; that is, if a, h are two different numbers, then one of them, say a, precedes the other h. This we express by saying that a is less than 6, or that h is greater than a. In symbols a<6, h>a. * For an extended treatment of this subject we refer to the excellent work of O Stolz and J. A. Gmeiner, Theoretische Arithmetik, Leipzig, 1900. 1 2 RATIONAL NUMBERS We say the system ^ is ordered. Furthermore, if a= b, 5 = c, then a> c. Also if a > 5, b>c, then a>c. Secondly, we observe that the system 3' i^ infinite ; after each element a follows another element, and so on without end. On the elements of 3^ we perform four operations, viz. addition, subtraction, multiplication, and division. They are called the four rational operations. Of these operations, two may be re- garded as direct, viz. addition and multiplication. The other two are their inverses, viz. subtraction, the inverse of addition ; and division, the inverse of multiplication. The formal laws governing addition are : the associative law, expressed by the formula a-\-(b + e) = (a + b}-\-c; and the commutative law, expressed by a + b = b + a. As regards the position of a + 6 in the system Q, relative to a or 6, we have , , a + > a or 0. We have also the relation a-{-b>a' + b, if a>a'. The formal laws governing multiplication are the three fol- lowing : The associative law, expressed by a • be = ab • c. The distributive law, expressed by Qa-}-b)c = ac -\-bc, a(b -\- c") = ab + ao. The commutative law, expressed by ab = ba. We have also the relation, with respect to order, ab>a'b if a>a'. HISTORICAL INTRODUCTION 3 « Another important property is this : If ac = he, then a = h. The result of subtracting h from a is defined to be the number X in ^, satisfying the relation a = h -\-x. But when a^h, no such number exists in 3^. Similarly, the result of dividing a by 6 is defined to be the number x in 3, satisfying the relation a = hx. If, however, a is not a multiple of 6, no such number exists in 3. Thus when we limit ourselves to the number system ^, the two operations of subtraction and division cannot always be per- formed. In order that they may be, we enlarge our number system by introducing new elements, viz. fractions and negative numbers. Tlie introduction of fractions into arithmetic was comparatively easy ; on the contrary, the negative numbers caused a great deal of trouble. For a time negative numbers were called absurd or fictitious. That the product of two of these fictitious numbers, — a and —5, could give a real number, -{-ah, was long a stumbling block for many good minds. The introduction of irrational numbers, i.e. numbers like V2, ^5, 7r = 3.14159..., e = 2.7182..., never excited much comment. In actual calculations one used approximate rational values, and it was perfectly natural to sub- ject them to the same laws as rational numbers. It is true that the Greeks of the time of Euclid were perfectly aware of the difficulties which beset a rigorous theory of incommensurable magnitudes ; witness the fifth and tenth book of Euclid's Ele- ments. But these subtle speculations found little attention during the Renaissance of mathematics in the seventeenth and eighteenth centuries. The contemporaries and successors of Newton and Leibnitz were too much absorbed in developing and applying the infinitesimal calculus to think* much about its foundations. 4 RATIONAL NUMBERS At the close of the eighteenth and the beginning of the nine- teenth centuries a cliange of attitude is observed. Gauss, La- grange, Cauchy, and Abel called for a return to the rigor of the ancient Greek geometers. Certain paradoxes and even results obviously false had been obtained by methods in good repute. It became evident that the foundations of the calculus required a critical revision. Abel in a letter to Hansteen in 1826 writes : * "I mean to devote all my strength to spread light in the immense obscurity which prevails to-day in anal3^sis. It is so devoid of all plan and system that one may well be astonished that so many occupy themselves with it, — what is worse, it is absolutely devoid of rigor. In the higher analysis there exist very few propositions which have been demonstrated with complete rigor. Every- where one observes the unfortunate habit of generalizing, without demonstration, from special cases ; it is indeed marvelous that such methods lead so rarely to so-called paradoxes." In another place he writes : f " I believe you could show me but few theorems in infinite series to whose demonstration I could not urge well-founded objections. The binomial theorem itself has never been rigorously demonstrated. . . . Taylor's expan- sion, the foundation of the whole calculus, has not fared better." The critical movement inaugurated by the above-mentioned mathematicians found its greatest exponent in Weierstrass. It is no doubt largely due to his teachings that we may boast to-day that the great structure of modern analysis is built on the securest foundations known ; that its methods have attained, if not sur- passed, the justly famed rigor of the ancient Greek geometers. The saying of D'Alembert, " Allez en avant, la foi vous viendra," has lost its force. To-day, it is not faith that is required, but a little patience and maturity of mind. As Weierstrass has shown, it is necessary, in order to place analysis on a satisfactory basis, to go to the very root of the matter and create a theory of irrational numbers with the same care and rigor as contemplated by Euclid, in his theory of incom- mensurable magnitudes, only on a f?r grander scale. It is too * Abel, (Euvres, 2° ed., Vol. 2, p. 263. t Abel, I.e., p. 257. FRACTIONS 5 early to make the reader see the necessity of this step, but it will appear over and over again in the course of this work. Fractions 2. Before taking up the theory of irrational numbers, we wish to develop in some detail the modern theory of fractions and negative numbers. We shall rest our treatment of these numbers on the properties of the positive integers 3^, which we therefore suppose given. One of these properties, on account of its impor- tUnce, deserves especial mention, viz.: If the product ah is divisible by e, and if a and c are relatively prime, then b is divisible by c. 3. Let us begin with the positive fractions. As we saw, divi- sion of a by b, where a, b are two numbers in Q, is not possible unless a is a multiple of b. Our object is therefore to form a new system of numbers, call it %, formed of the numbers of ^ and cer- tain other numbers, in which division shall be always possible. We start by forming all possible pairs of numbers in 3- These pairs we represent by the notation a = (a, a'), y8 = (5, 5') — In any one of these pairs, as a = (a, a'), we call a the first con- stituent and a' the second constituent of a. The system g consists of the totality of these pairs a, /3, ••• The elements of g we have represented by the symbol (a, a'). Any other symbol would do. The customary ones are a/b and a: b. We have purposely avoided these symbols, so familiar to the reader, in order that his attention shall be more closely fixed on the logical processes employed. 4. The objects of 5 have as yet no properties ; we proceed to assign them one arithmetic property after another, taking care that no property shall contradict preceding ones. We begin by setting 5 in relation to ^. We say : (a, a') shall be a number c, in 3^? when a = a'c. Thus, any element of ^ whose first constituent is a multiple of its second, is an element of ^, i.e. a positive integer. 6 RATIONAL NUMBERS From this follows that every number a of -3^ lies in ^. For, (a, 1) lies in g. On the other hand (a, 1) = a. Hence a lies in ^. 5. We define next the terms, equals greater than, less than. Let «=(«, a'), /3=(^', b'). We say : a = /3 according as ah' = a'h. We observe that to decide the equality or inequality of two elements in 5, the operations required are on the elements of Q. 6. We deduce now some of the consequences of the above defi- nition of equality and inequality. In the first place, suppose a, /3 both lie in 3^ ; i.e. let a = (aa', a') = a, y8 = (hh\ J') = J, by 4. Now, according to the definition in 5, according as that is, according as «|^ aa'b' = a'bb'; a^b. Thus, when a considered as a number of -3^, equals /3 considered as a number of Q, the two are equal, considered as numbers in ^, and conversely. 7. If « = 7? y8 = 7i the7i a = 0. For, let a = (a, a'), /3 = (J, b'), 7 = (c, c'). Since « = 7, ac' = a'c, by 5. (1 Since /8=7, bc' = b'c. (2 Multiply 1) by 6', and 2) by a' and subtract. Then ah'c' = a'Jc'. .-. ah' = a'h. .-. a= 8, by 5. FRACTIONS 7 8. The two numbers {ma, ma'^ and (a, a'') are equal. This follows at once from the definition in 5. From this fact we conclude : We can multiply the first and second constituent of a number without changing its value. Conversely : If the first and second constituents of a number have a common factor, it can be removed without changing the value of the 7iumber, 9. Let a, a' be relative prime. For a = (a, a') and /3 = (5, 6') to be equal, it is necessary and sufficient that b = ta, b' = ta'. (1 Obviously if 1) holds, a = j3. The condition 1) is thus sufficient. It is necessary. For, from « = /S, we have ab' = a'b. (2 We apply now the property mentioned in 2. Since a'b is divisible by a, by virtue of 2) ; and since a, a' are relative prime, b must be divisible by a. Say b = ta. (3 Similarly, since ab' is divisible by a', and a, a' are relative prime, b' must be divisible by a'. Say b' = sa'. (4 Putting 3), 4) in 2), we get s = t. Hence 3), 4) give now 1). 10. Our next step is to define the four rational operations on the elements of g. We begin by defining the two direct operations. Let « = (a, a'), /3 = (5, J'). We define addition by the equation, a + l3=(ab' + a'b, a'b''); (1 and multiplication, by a^ = (^ab, a'b'y. (2 8 RATIONAL NUMBERS It can be shown that the operations just defined enjoy the same properties as those of ordinary fractions. Without stopping to show this in detail, we demonstrate a few of these properties, by way of illustration. 11. Let a, ^ lie in ^ ; and say a = {aa\ a') = a, /3 = {bb\ 5') = h. Then a + y8, as defined in 10, 1), should give a + 6 ; and ayS, as defined in 10, 2), should give ab. This is indeed so. For a + ^ = (iaa'b' + a'bb', a'b'), by 10, 1) = (« + J, 1), by8 = a + 5, by 4. Similarly, a-l3 = (aa'bb', a'b'), by 10, 2) = (a5, 1), by8 = ab, by 4. 12. From ay = fiy, follows a = /S. For, let 7 = (e, c') ; we have : «7 = (a, a')(c, e') = (ac, «'c'), by 10, 2) ^y=(b,b'Xc,c'} = (bc,b'c'}. Since by hypothesis ay = ^y, we have acb'c' = a'c'bc, by 5. .-. ab' = a'b. .'. «= A by 5. 13. We establish now the following relations: 1) a + /3>«. 2) If yS>7, then a-{-/3>a + y. 3) If a + /3 = a + 7, then ^ = y. 4) If a>y3 and /3>7, then «>7. FRACTIONS 9 To prove 1): « + ;8 = (a, a') + Q>. ^') = (ab' + a'b, a'b'), by 10, 1). But a'(ab' + a'b)>aa'b'. .'. a + /3 > «, by 5. To prove 2): a + ^ = (ab' + a'b, a'V) = (ab'c' + a'bc\ a'b'c'), by 8. (4 Similarly, « + ry = (ac' + a'c, a'c') = {ab'c' + a'b'c, a'b'c'y (5 By hypothesis yS > 7 ; hence, by 5, Jc' > b'c. (6 Comparing 4), 5), we see the second constituents are equal, while the first constituent in 4) is greater than the first constituent in 5), by virtue of 6). From this follows, by 5, that a + yS>a + 7, which is 2). To prove 3): Suppose the contrary ; then since /3 9^ 7, either /3 > 7 or y8 < 7. If /3>7, then « + /8>« + 7, by 2). (7 If /3<7, then a + 7>« + yg, by 2). (8 But both 7), 8) contradict the hypothesis that a + yS = a + 7. (9 (10 To prove 4) : Since «>/3, ab' >a'b. Since /3>7, be' '>b'c. From 9), 10) we have abb'c' >a'bb'c; whence ac' >a'c. Hence ', by 5, a :>y. 10 RATIONAL NUMBERS 14. As an illustration of the demonstration for the formal laws governing addition and multiplication, let us show that the dis- tributive law holds in g. We wish to prove that 1) a(/S + 7) = «/3 + ay. Now /S + 7 = (be' + b'c, b'c% by 10, 1). a{l3 + 7) = (a, a') • (be' + b'c, h'c') 2) = {abc' + aJ'c, a'b'c% by 10, 2). Also a^ = (ab^ a'b'); ay = Qac, a'c'^. .'. a^ -\- ay = {aa'bc' -\- aa'b'e, a'Wc'') 3) = (abc' + ab'c, a'b'c'}, by 8. The comparison of 2), 3) gives 1). 15. We turn now to the inverse operations, subtraction and division ; considering first division. We define the quotient of a by /8 to be the element or elements, I, if any exist, of ^ which satisfy the relation « = /3|. (1 Set f = Qx, a;'). Since | must satisfy 1), we iiave (a, a') = (^ b')(ix, x')=::(bx, b'x'). The first and third members give, by 5, ab'x' = a'bx, (2 which a:, x' must satisfy. A solution of 2) is obviously x=ab', x' = a'b. Thus, is a solution of 1). This is the only solution of 1). For, sup- pose ?; is a solution. Then by definition a = ^7). (3 Then 1), 3) give, by 7, which gives, by 12, FRACTIONS 11 16. The quotient of « by /3, we shall now represent b)'' a//3. All the numbers of ^ may be regarded as quotients of numbers in Q. For, let a=(a, a') be any number of g. It evidently satisfies the equation ^ t^ which, as we have just seen, admits only one root, viz. the quotient of a by a'. Hence a = (a, a') = a/a'. Thus the elements or numbers in ^ are ordinary positive frac- tions. 17. We have now this result. In the system ^, division is always possible and unique. In the old system Q, this is not true ; the division of a by J being only possible when a is a multiple of b. We see, then, that, on properly enlarging our number system by introducing new elements, we obtain a system 1^ which has this advantage over 3, that the quotient of any two numbers in ^ exists and is unique. 18. We treat now subtraction. We define the result of subtracting /3 from a to be the element or elements, call them |, in ^, which satisfy the relation « = yS + ^ (1 7f /3 ^ a, there exists no number ^ in ^ which satisfies 1). For, '^" ^ " ''' /S + I = « + I > «, by 13, 1). If /3 > «, /3 + I > « + I, by 13, 2). Also a-\- ^> a, by 13, 1). .-. /S + I > «, by 13, 4). Thus when /3 ^ «, /S -f- f > «, and hence /3 + | =^ a. Suppose then, that /3 < «. Then ab' > a'b. (2 From 1), we have, setting ^ = (x, x') ; ia,a') = (6, b') + (ix,x') = Cbx' + b'x,b'x'),hylO,f), 12 RATIONAL NUMBERS Hence by 5, observing 2), a'b'x = x'(ab' -a'b^. (3 A solution of 3) is evidently X = ah' — a'h, x' = a'h'. Hence I = (a5' _ a'h, a'h') is a solution of 1). This is the only solution; for if ?; is a solution, w§ have, by definition, ^ , ,, a = y8 + 7?. (4 The comparison of 1), 4) gives yS + I = /S + .;. Hence, by 13, 3), 19. We have thus this result : In the system 5, the subtraction of yS from a is possible and unique, when a > /8 ; when a ^ /3, it is impossible. That is, there is no number | in % which satisfies 18, 1). When subtraction is possible, we represent the result of sub- tracting y8 from a by a — yS. Negative Numbers 20. In the system of positive fractions %, subtraction is only possible when the minuend is greater than the subtrahend. To remove this restriction, we propose to form a new number system _B, which contains all the numbers of ^ ; and in which subtraction of a greater from a less shall be possible. Since the method of forming M is identical with that employed for g, we shall be more brief now. The numbers in ^ we now denote hy a, b, c, . . . , while the Greek letters a, /3, y, . . . shall denote numbers in the new system M. 21. 1. We begin by taking the elements of ^ in pairs, to form new objects, which we de'note by the new symbol {a, b\. The totality of all such pairs forms the system R. Next, we place H in relation to g. Let a = \a, bl. In case a > b, we say a shall be the number a — b, which obviously lies in NEGATIVE NUMBERS 13 5. Thus every number in g lies in R. For, let a be any number in g, and let h be any other number in g. Then \a -\- h^h\ = (^a + b') — b = a; that is, a lies in jR. 2. We next order the system i2. We say \a,a'i>\b,b'l, according as a + b'fa' + b. (1 3. Addition is defined by the relation a + 0= \a, a' I + {b, h'\ = ja + J, «' + b'\. (2 Multiplication is defined by «./3 = ^a5 +a'J', «J' + a'i|. (3 4. As a consequence of 1), we have la, a'l = {a + b,a' + b\, (4 where b is any number in ^. In words, 4) states : We can add the same number b to both constituents of a = \a,a'\ without changing the value of a; and if a, a' are both > b, we can subtract b from both constituents, without altering the value of a. 5. It is easy now to prove results analogous to those in 6, 7, 11, 14 ; in particular the associative, commutative, and distributive laws. 22. According to our definition of equality, all the elements of R whose first and second constituents are the same, i.e. all elements of the type \a, a\, are equal. We set fa, a|=0, and call this number zero. Then, if in a = | a, a' | , a> a', a > ; if a' > a, a < 0. 14 RATIONAL NUMBERS Numbers in R which are > 0, are called positive ; those < 0, are called negative. The number is neither positive nor negative. From this, it follows that the positive numbers of R are simply the numbers of g; while and all negative numbers do not lie in g. 23. 1. We observe that «+0 = «=0 + «, (1 a . = = • a. (2 To prove 1). Let a= \a, a'l, 0= lb,bl. Then a+0 ^ \a + b, a' + b\, by 21, d; = la, a'|,by21, 4; = a. To prove 2). a • = \ab -\- a'b, ab + a'b\ = 0, by 22. 2. We also note the relations + = 0, 0-0 = 0. 24. We can prove now easily the Rule of signs. TJie product of tzvo positive or two negative numbers is positive. The product of a positive and a negative number is negative. Let «= \a, a'\, ^= \b, b'\. 1°. «, /8 > 0, then a^ > 0. For here a> a', b>b', hy 22. .: a/3= \ab + a'b\ ob' + a'b\ = {b(a ~ a')-\- a'b\ ab'\, by 21, 4 = lb(a- a'), 6'(a - a') I, by 21, 4 >0, since b{a ^ a'^>b'(a — a'). 2°. «, /3<0, then «/3>0. Here a'>a, b' >b, by 22. NEGATIVE NUMBERS 15 /. a^=\ah^a'(h' -h-), ab' U by 21, 4; = la'(6'-6), aCb'-b^l, by 21, 4; >0, since a'(b' -b)>a(b' -h^. 3°. a>0, y8<0, then «/3<0. Here a>a\ b'>b. .-. a/8=fa6 + a'(5'-5), ab'\ = \a'(b'-b~), a(b'-by\ ■ <0, since a' (b' -b) 7, then a + /8 > « + 7. 2) From a + /3 = « + 7, follows /S = 7. 8) If a =5^ and a/3 = «7, then yS = 7. To prove 1): a-f./3= \a + b, a' + b'\, by 21, 3 ; = {« + ^ + c', a' + 5' + c'|, by 21, 4. (4 Similarly, a + 7= fa + c, a' + c'( = fa + 6'+c, «' + 5' + c?'|. (5 Since /3> 7, J + ,' > 6' + ., by 21, 2. .-. a + J + c' > a + 6' + c. (6 16 RATIONAL NUMBERS If we now apply the definition for greater than given in 21, 2 to « + /3 and a 4- 7, the relations 4), 5), 6) show that « + yS > a + 7. To prove 2): Suppose the contrary, i.e. suppose /3 > 7 or /3 < 7. If /3>7, « + ;S>« + 7, by 1). If7>/3, « + 7>« + /3, by 1). Thus in both cases, a + /3 ^t a + 7, which is contrary to hypothe- sis. Hence /S = 7. To prove 3) : From a/S = 057 we have a(/3-7)=0. Applying 25, we have /3 = 7. 27. We turn now to subtraction. This we define as in 5, viz. : the result of ^btracting /3 from « is the element or elements |, of R. which satisfy , t- /-1 This equation gives, setting |= fa;, x'|, {a, a'|= 16, 6'f + fa;, a;'^ = f5 + a:, h' + x'\^hy21, 3. Hence by 21, 2, a + 6' + a;'=a' +5 +a;. This equation is evidently satisfied by x = a + h\ x' = a' + h. Hence <.,■,,.-,. ^=\a + h\ a' + h\ (2 is a solution of 1). This is the only solution. For, let ?/ be a solution. Then by definition, a , ^o The comparison of 1}, 3) gives Hence by 26, 2), ^ NEGATIVE NUMBERS 17 28. We have thus this result : in the system R subtraction is always possible and unique. The result of subtracting /3 from a, we represent by a — /3; it is a number in R. Then any number «= {a, a'\ in R, is the result of subtracting a' from a, or For, Similarly, Hence a= a — a. a = \a + h, h\, by 21, 4. a' = \a' + h, h\. a — a' = \a + h,h\ — \a' -\-h,h\ = {a + 25, a' + 26|,by27, 2) = fa, a'|,by21, 4. 29. 1. Let a = fa, a'\ be any number of R. The number fa', a\ is called minus «, and we write fa', a| = — «. — (— «)= — fa', a| = fa, a'j = a. ^^^''' « + (-«)= fa, a'| + fa', a| = f a + a', a + «' I = = «— a. If a is positive, — a is negative ; and conversely, if a is nega- tive, — a is positive. 2. The number —a may be defined as the number |, such that « + | = 0. For, f = — a satisfies this equation ; and, as we saw in 27, this equation admits but one solution. This shows that the numbers in i2, =f= 0, may be grouped in pairs, such that their sum is zero. 3. If - a = - /3, then a = /3. For, multiplpng both sides of — a = — yS by — 1, we get a = yS. 4. Every number a, of i?, different from zero, can be written in the form « = a, or a— — a^) where a is a number in 5. 18 RATIONAL NUMBERS For if a > 0, we already know by 22 that a is a number in ^. If a < 0, then — « is positive, so that — « = a, a number in ^. Multiplying this equation by — 1, we get oc = — a. 30. 1. We treat now division. We say : the result of dividing « by /3 is the number or num- bers P, of M, such that «=I/S. (1 Suppose /3^0; then there is one and only one number f ; i.e. in this case^ division is possible and unique. There can be at most one. For, if t] satisfies 1), we should have a = v/3. (2 Comparing 1), 2), we have 1/3 = 7;^; whence by 26, 3), To show that there is always one solution of 1), we have the following cases. Let a, /3 > ; then a = a, /3 = 6, by 29, 4 ; and 1) becomes a = |5. (3 But by 15 the solution of 3) is I = (a, 6) = a/b. Let a, /S < ; then a = - a, y8 = - 6, by 29, 4. Then 1) becomes - a = -b^', or by 29, 3, Hence as before, Let a > 0, yS < ; then a =a, ^ = — b, and 1) becomes a = -bl (4 SOME PROPERTIES OF THE SYSTEM K lU Set -1=7?; then 4)g: ives a = : hrj. Hence, id V = 1 = a/b ■ - a/b. lfa<0,and^>0 , we get again ^ = ■■ - a/b. Finally, let a = 0; then 1) becomes Hence by 25, = :0. 2. We consider now the case that /3 = 0. The equation 1) admits now no solution, unless a = also. For, when yS = 0, /3| = 0, whatever f may be. If now a = 0, the equation 1) is satisfied for every number f in H. We have thus this result : When the divisor is zero, division is either impossible or entirely indeterminate. For this reason, division by zero is excluded in modern mathe- matics. The admission of division by zero by the older mathe- maticians, Euler for example, has caused untold confusion. We shall see it is entirely superfluous. Some Properties of the System R 31. The system B,, which we have just formed, is made up of the totality of positive and negative integers and fractions, and also zero. It is called the system of rational numbers ; any element in it being called a rational number. The elementary arithmetical properties of these numbers having been established, there is no further occasion to employ the special notations {a, 6) and \a, b\\ we shall, instead, employ the customary ones. Furthermore, we shall represent for the rest of this chapter the numbers in B, indifferently by Greek and Latin letters a, b, c, ••• a, y8, 7, ••• For the sake of completeness we now proceed to deduce a few properties of R, although the reader is probably familiar with them. 20 > RATIONAL NUMBERS 32. The system R is invariant with respect to the four rational operations. This simply means that the addition, subtraction, multiplica- tion, and division of any two elements of i2, division by of course excluded, always leads to an element in R. We saw this is not true for the systems ^ and %. 33. 1. The system R is dense. This term, taken from the theory of aggregates, which we shall take up later, simply means that between any two numbers a, h in R., exists a third and hence an infinity of numbers. For, let and say a>h. Then d = a^Jg ~ ^2^1 is an integer > 1. Let e be a positive integer. By taking it large enough, we can make ed>n, where n is an arbitrarily large positive integer. Let h be any integer, such that ea^-^ a^b. na-i a-,aj) ^ 7-^7 na = — 1 > -1-2- > a^ > 0. 2. Let a he an arbitrarily large number of R; there exists a posi- tive integer n, such that a/71 < b, where b is arbitrarily small. For, by 1, there exists a positive n, such that nb>a. Hence a/n0, h<0. Let h = -bQ, 5o>0. Then a + h = a — b^, a — b = a + Bq. li a>bQ, \a-{-b\^\a — bQ\=a — bQ0, and \a\m. In symbols we write , ,. ■^ 1 = lim a^ ; n=oo we also use the shorter forms Z = lim a^, or a^ = I, when no confusion can arise. We shall also employ at times the symbol I = lim a„. A When I is the limit of A, we say A is a convergent sequence, and that a„ converges to Z as a limit. 2. Notation. We shall find it extremely convenient to employ the following abbreviation : €>0, m, |Z— a„|w (2 to mean that, for each positive rational e there exists an index m, such that I Z — a„| < efor every n'p-m. The reader should therefore repeat the italics often enough to himself to be able to read the line of symbols 2) without hesitation. 26 RATIONAL NUMBERS 3. The reader should observe that from 2) we can conclude also that for each positive M there exists an m\ such that i'-«.i<^' If, therefore, {a„| has the rational limit Z, we can write €>0, m, K-«»|<-^^ n>m. (3 We have, of course, changed the notation slightly in 3) by dropping the accent of m'. 43. The graphical interpretation of this definition will prove most helpful in our subsequent reasoning. Ui Let us lay off the points on our axis, corresponding to the num- bers «„, also the point corresponding to I. On either side of I lay off the points I — e^ I -\- e. These determine an interval, marked heavy in the figure, which we shall call the e-interval. If now I is the limit of the sequence A, there must exist for each little e-interval, an index m, such that the images of all the numbers a^+i? ^m+21 "• f^'H within the e-interval. See 39. In general, as e is taken smaller and smaller, the index w increases. The definition, however, only requires that for each given e there exists some corresponding m such that 42, 1) holds for every n greater than this m. 44. Another useful graphical interpretation of the definition of a limit is the following. H 'o.* RATIONAL LIMITS 27 We take two axes a:, y as in analytic geometry. On the a;-axis mark off points 1, 2, 3, ••• at equal distances apart. Lay off the numbers a^, a^^ a^, ••• as ordinates on lines through the points 1, 2, 3, ••• parallel to the ?/-axis. (See Fig.) These points we may consider as the images of the numbers a„. On either side of the line y = I, draw parallel lines at a distance e from it. We get then a band, shaded in the figure, which we shall call the e-hand. Then, if I is the limit of A, there exists for each e an index m, such that the images of all the numbers a,„+i, «m+2? ••• fall within the-corresponding e-band. 45. EXAMPLES 1. ^ = -[-]■; lim n!„ = lim- = 0. L 71 J n 2. M^-D-'H'-i) 3. ^ = 1, -i, +i - 1, ••• «„= (-l)»+i-; lima„ = 0. n The reader will find it helpful to construct the graphs, ex- plained in 43, 44, for each of these sequences. 46. If it is known of tivo rational numbers p^ q^ that \p — q\< €, however small e > may he taken, then p = q- For, if p 4^ q, say p > q, then p — q is a definite positive rational number ; call it d. Then \p — q\ is not < d, and this contradicts the hypothesis. Hence p = q- 47. A rational sequence A = \a„l cannot have two rational limits Z, V . For, since a^ = I, we have by definition, e > 0, Wj, \l — a„\<€, n> m^. (1 Also, since a„ = T, we have €>0, m^, I?' — a„|m^. (2 Let m>mp m^; then from 1), 2) follows e>0, ?w, |Z— a„|m. (3 e>0, w, 1^' — a„|m. (4 28 RATIONAL NUMBERS The inequalities 3), 4), holding now for the same m, we can add them, and get, by 38, 3), \l-l'\<2e. (5 But since e is small at pleasure, so is 2 e. The inequality 5) gives, by 46, ^^^,^ 48. If the rational sequence ]aj<, has a rational limit I, there exists an index m, such that bm, (1 where h is any rational number < Z, and c any rational number > I. For, since a„ = Z, e>0, m, |Z — a„|m. .-. l-€b, l + e0, m', |a— a„lm'. (2 Since 5„ = 6, we have e>0, m", \b-b„\<€/2. n>m". (3 By choosing m so large that m>m', m'\ we can suppose 2), 3) hold for the same m.* *When a>h, a'^c, a^d- vf^ shaW oit&n set vaoTQ sh.ovtlY a>6, c, d"« Similarly a^O, b^O, c^O-- may be written more shortly a, b, c, ••• ^ RATIONAL LIMITS 29 Then 1) becomes, using 2), 3), I (a + 5) - (a„ + 6„) I < I + 1 = e. n>m. This states that lira (a„ + h„^=a + h. Similarly, we prove the other half of our theorem. 50. If the tivo sequences ja^j, \hj<^ have the rational limits a, h respectively, then lim a„5„ = ah. {X For, I ^^ = a5 - a„6„ = a(5 - 6„) + 5„(a - a„). .-. \d,\<\a\\b-hr,\^\br\\a-a^\, (2 by 37, 3), 5). Since h^ = 6, we have, by 48, \K\ m'. Also, by 42, 3, i^-*"|<2i«r W>7?i". Since a„ = a, we have, by 42, 3, k ^n\<^^' n > m'". Evidently by taking m large enough, we can use the same m in these three inequalities. Then they give in 2) which proves 1). 51. Let the two sequences \a„}, {5„|, have the rational limits a, 5, iespeetively . Let h and b„^0. Then lim^- = ?. (1 o„ b 30 RATIONAL NUMBERS For, d ^ ^ «n ^ «^» - ^^« — (^^» — ab^ + {ah - a„b') ^ a(b„-b) ^ a-a„ ^ bb^ b^ Since Jt^ 0, |J| >0. Let J5 be a rational number, such that Then, by 48, there exists an m, such that \b^\>B. 71 >m. (3 Also, by 42, 3, e > 0, w, 1 ^ — ^„ I < -pr-' — n>m. (4 e>0, m, |a— a„|< — . n>m. (5 By taking w large enough, we can use the same m in these inequalities. Putting 3) in 2), we get <| + | = ^' by 4), 5), 2 2 which proves 1). CHAPTER II IRRATIONAL NUMBERS Insufficiency of R 52. Although the system of rational numbers R is dense, and so apparently complete^ it is easy to show that it is quite insufficient for the needs of even elementary mathematics. Consider, for example, the length h of the diagonal of a unit square. This length is defined by the equation 83 = 2. (1 We can show there is no number in R which satisfies 1). For, suppose where a, b are two positive integers, which we can take without loss of generality, relatively prime. Then 1) gives a2=2 52. Let jt) be any prime factor of h. It is then a divisor of a\ and so of a. Thus a and b are both divisible by p. They are thus not relatively prime, unless p = l. Thus 5 = 1; and 8 is an integer. But obviously there is no integer whose square is 2. 53. 1. A similar reasoning shows that ■\/a does not lie in R, unless a is the nth. power of a rational number. 31 32 IRRATIONAL NUMBERS The numbers 6 = 2.71828-, 7r = 3.14159. can be shown to be irrational ; the numbers log a;, e*, sin a;, tana; are in general not rational. 2. Let us show that , , r I = log 5, the base being 10, does not lie in R. If I were rational, we should have 1 = a "V where a, h are integers. Then a 10* = 5. . •. : 10" = 5^ (1 Obviously I cannot be negative ; we can thus suppose a, 5 > 0. Now any integral positive power of 10 is an integer ending in ; while any integral positive power of 5 ends in 5. Thus 1) requires that a number ending in should equal a number ending in 5, which is absurd. Hence I is not rational. Cantor's TJieory 54. 1. The preceding remarks show clearly the necessity of forming a more comprehensive system of numbers than R. How this may be done in various ways has been shown by Weierstrass, Cantor, Dedekind, Hilbert, and others. We adduce now certain considerations which lead up to Cantor's theory. We have seen no rational number exists which satisfies the equation a;2 = 2. (1 It is, however, possible to determine an infinite sequence of rational numbers such that ,. or, lim a J = 2. CANTOR'S THEORY 33 The method we now give for finding such a sequence A has no practical value ; it has, however, theoretical importance. For aj, we take the greatest integer, such that In the present case, a^ = l. From the numbers we take for a^ the number whose square is < 2, while the next number of 2) gives a square > 2. Suppose ^ Then 1^ S'<2<(a, + A)2. -brom the numbers we take for a^ the number whose square is < 2, while the next number of 3) gives a square > 2. Suppose ^^ 3 2^102 Then / i \2 2^9^ I n _1__L ) . i02y «3'<2

0, we can find an m, such that 3 , 1 ^ Then 1 2 — a„2 1 < e. n>m. Hence lim a„2 = 2. 55. 1. The method given in 54 for forming the sequence a^, ag, ag, ••• admits a simple graphical interpretation. 103= i-« 1 1,41.5 2 2 We first divide the indefinite right line L into unit segments ; flfj is end point of one of these segments. In the present case a^ = 1. We next divide the segment 1, 2 into 10 equal parts ; a^ is the end point of one of these segments. In the present case a^ = lA. We next divide the segment 1.4, 1.5 into 10 equal parts ; a^ is the end point of one of these segments. In this way, we continue subdividing each successive little interval or segment into 10 smaller parts, without end. CANTOR'S THEORY 35 We observe that each little segment is contained in the im- mediately preceding one, and therefore in all preceding ones. Also, that the lengths of these segments form a sequence whose limit is zero. 1 1 J- J- ' 10' 102' 103' 56. 1. The method of 54 may be used to find an infinite sequence of rational numbers which more and more nearly satisfy the equation 10^^ = 5, which defines log 5. We find : ftj = 0, a2= .6, a^= .69, a^= .69S, ••• 2. The same method may evidently be applied to any problem which defines an irrational number. In each case it leads to a sequence of rational numbers fli, (Zo* ^31 • * * -^ such that 1°. Each number a„ satisfies more nearly than the preceding ones the conditions of the problem. 2°. For each positive rational e, arbitrarily small, there exists an index m, such that la„-ajm. 57. Regular Sequences. 1. It is this second property of the sequences A, that '^antor seizes on to construct the elements of his number system. We lay down now the following definitions. Any infinite sequence of rational numbers ^\i ^2^ ^^3' *" which ha"" property 2° in 56 is called regular. As in 42, 2, we shall indicate this property by the abbreviated notation : „ , , e > 0, m, a„ — a^ <€, n,v>m. (1 36 IRRATIONAL NUMBERS 2. Every regular sequence defines a number^ which we represent by the symbol « = («!, «2' ^3' ••■)• The totality of such numbers forms a number system^ called the system of real numbers. We shall denote it by 9fJ, which may be read German R. For the convenience of the reader, we shall denote in this chap- ter the new numbers, i.e. the numbers in QfJ, by the Greek letters a, /3, 7, •••; while the Latin letters a, b, c, ••• denote numbers in the old system M. To see if a given sequence is regular, we must see if the in- equalities 1) are satisfied. For this reason we shall speak of these inequalities as the e,w test. 3. The €,m test is equivalent to the following : e > 0, wi, I a^ — a„, I < e, w > m. (2 The difference between 1), 2) being that in |a„ — a„, |, only one index, n, varies. For, when 1) holds, 2) is satisfied. For we pass from 1) to 2) by setting v = m in 1). Conversely, if 2) holds, 1) is satisfied. For, since e in 2) is small at pleasure, let us take p > m. Adding the inequalities, we get, by 38, 3), l«« — «vl<°"» n,v>m, which is 1). -1 Then 2) gives |««- 1 ^ o" Also \a,- -<«»1<2 CANTOR'S THEORY 37 4. We observe finally that we may replace n, v>m in 1) by w, v> m. For, if 11^ /Q for every w, i' ^ w, it is true for every n^ v> m. Conversely, if 3) is true for every n, i^ > m, it is true for every n, v ^ m + 1. We would therefore in 1) replace m hy m + 1. 58. EXAMPLES 1. That the sequeuces A, defined in 54, are regular, is readily shown. We have , Ui , , a„-i a„ = ai + — + ••• H — ^ • 10 10"-i 0,^ = a\ + —-+•■• -\ 10 10"-! For simplicity, suppose v > « ; then „,_„, = ii. + i|^+...+i=l, <-!,,. (1 as the considerations of 55 show. If we choose m so large that i0^i<^' then, by 1), dv — ««<«• n, v'>m. The €,m test is therefore satisfied. 2. Consider the sequence Here If we take now then 1 _1 1 _1 1' 2' 3' 4' "* |a„-a,| = |^±-b-+-. (2 \n v\ ^ 71 V 2 m> 1111 - + -< — n, j'>m. n V m m Hence 2) gives 3. Consider the sequence |««-«^,|<|+| = e. 1, 1, 1, 1, 1, Here «^^ _ «^ ^ 1 _ 1 = 0, and this sequence evidently satisfies the e,ni test, and is therefore regular. 38 IRRATIONAL NUMBERS 4. Consider ^ i i i . 1, —1, 1, —1, ••• ^^^® |an-a^,| = or 2. Evidently no m exists, such that I «„ — a^ I < e. n, v>-m. The sequence is thus not regular. 5. Consider 1 2 3 4 ••• and the e,m test is obviously not satisfied. The sequence is therefore not regular. 59. For any regular sequence of rational numbers a^, ag, ••• there exists a positive number M, such that \a„\ 0, w, I a„ — «;„ I < e. n>-m. Hence a^-e0, m, \a„ — b„\ /S when there exists a positive rational number r and an index w, such that ci'n — ^n^f- n>m. (S We say similarly, « < ;S, if 6„-a„>r, w>w. (4 or a„ - 5„ < — r. (5 3. Numbers of 'Si which are > are called positive ; those < are negative. 40 IRRATIONAL NUMBERS 63. It can be shown that from this definition of equality and inequality the usual properties of these terms can be deduced. For example, 7/ « = /3, /3 = 7, then a = y. For, setting « = («!, a^, ...) we have «« - ^« = («« - ^«) + (^« - = 0, since by hypothesis. 64. If a = (aj, a^, ...) = (aj', a^', .-.), we say (a^, a,^, •••) and (aj', ag'' •") ^^^ different representations of the same number «. Every number a in ^ admits an infinity of representations. In fact, there are obviously an infinity of rational sequences 2j, ^2' ^31 '•* having zero as limit. Then ^ , , n represent an infinity of representations of «. 65. 1. We wish to appl}- the definition of 62 to the case that one of the members, sa}^ /3, is a rational number h. Let a = b. For yS = 5, we can take the representation ^ = 5 = (6, 6,...). (1 Then 62, 1) requires that lim (a„ — 5) = ; whence lim a„ = b. Thus the definitions of 60 and 62 are in accord for this case. CANTOR'S THEORY 41 2. Let a>h. Since J„ = ^ by 1), the relation 62, 3) becomes here a„ — b>r, n>m. (2 Let a < b. Then h — a^>r, n>m. (3 or a„-b<-r. (4 3. If cc= (aj, flfgi "0'^ ^1 ^^^^^6 exists an index m, and two positive rational numbers A, B, such that Am. (5 and conversely . For, set 6 = 0, then 2) gives, replacing r by J., a„ > J. > 0. n>m. (6 On the other hand, 59 gives \a„\=anm. (7 From 6) and 7), we have 5). The second half of the theorem is obvious, by 2. 4. Similarly, we have If a=^(a^^ ag, •••)<0, there exists an index m, and two negative rational numbers —A, — -B, such that — Am. and conversely. 5. From 3 and 4 we have If a = («!, ^2' **')t^^5 the7'e exists an index m, and two positive numbers A, B, such that A<\a„\m. 6. In any number a = (^a^, a^, '■•')^0, the constituents a„ finally have one sign. This follows at once from 3 and 4. 42 IRRATIONAL NUMBERS 66. 1. Let a = (aj, a^^ •••). // a„>a, n>m. (1 Then a^a. (2 For, suppose a0, and an m, such that a — an>r. n>m. Hence a > a„ + r > a^ ; and therefore a„ < «, which contradicts 1). Hence 2) holds. 2. Similarly, we show : Let a = (m, then a0, m, |a^— a^|m. But then I ^''i^ — ^ij < f 1 r, s < «, ix > m. Hence 2) is regular, and defines a number /3. We show now a= ^. Since 2) contains only a part of 1), t„^/i, n=\, 2, 3, ••• Since 1) is regular, \a„ — a, \m. Hence, by 62, 1, « = yS. CANTOR'S THEORY 43 2. As corollary we have : The number «= (a^ a^^ •••) is not altered, if we remove from or add to the numbers in the parenthesis, a finite number of rational numbers. 3. We have also : If in «= («!, fljg, ...), yS= (b^, ^2' •") <^n=K^ n>m; then a = /S. 68. 1. If (t= (^j, ^21 *••) =5^ 0, ^Aerg cannot be an infinite number of constituents a^=Q. For, say Then, by 67, 1, But (a,„a,^, ...) = (0, 0, ...)=o. Hence a = 0, which is a contradiction. 2. If a ^0, we can choose a representation (a^, ag, •••), ^w which all the a„^0. For, let / r r \ /^t be any representation of «. By 1, it contains but a finite number of zero. If we leave these zeros out of 1), we do not change the value of «, by 67, 2 ; and get thereby a representation of «, none of whose constituents are zero. 69. 1. Having ordered the elements of ^, we proceed to define the rational operations upon them. Addition. be two elements of 9J, different or not ; then « + /3 = («! + &!, ^2 + ^2' •••)• G 44 IRRATIONAL NUMBERS To justify this definition of addition, we show first that is a regular sequence. Since a^, a^, ••• is a, regular sequence, we have e > 0, m, I a„ — a^ I < e/2, n, v>m. (3 Since 6j, b^, ••• is regular, we have e<0, m, |5„-5,lm. (4 Evidently we can take m so large that 3), 4) hold for the same m. Now I («n + ^n) - Q^v + ^.) I = I {an - «.) + Q^n " ^v) I < |a„ - a,| + 16^ - 6,|, by 37, 3) ; <| + |=6,by3),4). Thus 2) is regular, and defines a number. 2. We show next that, if «, /S are rational numbers, say « = a, /3 = 5 ; then a + yS, as defined by 1), is a + 6. Since a is a rational number a, lim a„ = a, by 60. Similarly, lini6„ = &. lim (a„ + ^,) = lim a„ + lim 6^ = a + 5, by 49. Thus by 60, , , , , , . , , Hence by 1), , o . a 70. 1. If /3>y, then a + ^>a + y. Let 7 = (cp (?2' "■)■ Since /3> 7, there exists, by 62, 2, a positive rational number r, such that ^i >(^n + r. n> m. Hence, adding a„, dn + K > «n + Cn + ^' Hence, by 62, 2, „ CANTOR'S THEORY 45 2. From 1, we conclude, as in 26, that Tfa-\-^=a + j, then y8 = 7. 71. 1. Subtraction. This is the inverse of addition ; we define it as we did in % and M, viz. : The result of subtracting ^ from a is the number or num- bers I, in 9?, which satisfy «=/3 + ^ (1 There is at most one number ^. For, suppose a — ^ + 7]. (2 Then 1), 2) give, by 63, y8 + | = /3 + 7;. Hence, by 70, 2, 7; = |. To show that 1) admits one solution, we prove just as in 69, 1, that , J. «i — ^i, a^ — b^, ••• is a regular sequence, and thus defines a number If we put this value of | in 1), the equation is satisfied. For, ^ + I = (^1, b^j, •••) + («! - Jj, a^- b^, •••) = (&i + flj - ^1, 63 + «2 - ^2' •••)^ by 69, 1) = («!, flg, •••) = «. 2. Thus subtraction is always possible in 9?, and is unique. The result of subtracting /3 from a we represent by a — /3 ; we have then n r r 7 n « - P = («i - Oj, ^2 - 62, •••)• 3. We represent — « by — a. Evidently, , . — «=(-«!, —«2' —^3' •••)• We observe that «+(—«) = ; a+(-/3) = «-/3; -(-«) = «• 46 IRRATIONAL NUMBERS 72. 1. -Z/« is positive^ — « is negative; and if a is negative, — a is positive. For, if cc= (aj, a^, •••) >0, we have, by 65, 3, a,i>A>0. n>m. Now _ «= (- a^, - ^2, •••)' i^y "^ii 3. Hence, by 1), — a„< — yl<0. n>m. Hence, by 65, 4, - a < 0, which proves the first part of the theorem. The second part is proved similarly. 2. All the numbers of dl ^0 are of the form a or — «, where a is a positive tiumber. Let /3 be a number ^ 0. We need to consider only the case that /3 is negative. and by 1), — yS is positive. 73. 1. Multiplication. The product of a by /3 we define by a^ = (a^b^, a^b^, •••)• (1 We have to show that is a regular sequence. Let e be a positive rational number, small at pleasure. Then, by 59, there exists a positive M., such that, |a„|, \b„\m. (3 Also, since the sequences |«„(, \bn\ are regular, we can suppose m in 3) is taken so large that 1««-«J, |5„-6J<^. n, v>m. (4 CANTOR'S. THEORY 47 Now d^ = aj)^ - aj)^ = a^h^ - 5 J + hX^n - « J- .-. |(^„|^|a„||6„-6J + |5J|a„-aJ, by 37, and 2) is regular. 2. If a, y8 are rational, say «=(x, /3=J, we show that ayS as defined in 1) is ah. For, since a and ^ are rational, lim a„ = «, lim 5„ = 5, by 60. But then, by 50, lim ajbn — lira a„ lim 5„ = a5, which states that a^= ah. 74. 1. The formal laws for addition and multiplication are readily proved. We illustrate this by establishing the associative law of multiplication. We wish to show that a . ^7 = a/3 . ry. (1 We have, by 73, 1), Hence «./37 = (aj, ^2, ••')(h^c^, h^c^, .••) Similarly, «/3 • 7 = (^1*1 • (^p «2^2 ' ^2' •••)• (3 Since multiplication is associative in i2, the two numbers repre- sented by 2), 3) are identical, which proves 1). 2. As a consequence of the associative law, we have, m, n being positive integers, which expresses the addition theorem for integral positive exponents. 48 IRRATIONAL NUMBERS 75. 1. The properties of products, relating to greater than, less than, are readily established for numbers in 9?. If a> 13, and 7 >0, then ay > /3y. For, since a>yS, we have, by 62, 2, Since 7 > 0, there exists a positive rational number c, by Qb, 3, such that Cj^>c. n>m. By taking m sufficiently large, we may take the same m in both these inequalities. They give ««Cr>0. Then, by 62, 2, ay > /37. 2. From 1 follows : lfa>^>0, then a" > /3". w positive integer. 3. From 2 we conclude : If a, /3 > 0, and a"- = y8% w being a positive integer, then a = 13. 4. IfO 0, yS > ; then a/3 > 0. B}^ 65, 3, there exist two positive numbers A, B, and an index m, such that a„>^, K>^- n>m. Hence aj)^ >AB>0. Thus a/3=:(ajb.^, a^b^, •••) >0, by 66. Ze^ « > 0, /3 < ; ^Agw aj3<0. For, by 65, 3, 4, a„>A, h„< — B. n>m. .'. aJ)„<-AB<0. Thus, by 66, «/3<0. In a precisely similar manner, we can treat the other cases. 77. 1. TJie product of any two numbers in 9? vanishes when, and only when, one of the factors is zero. In the product a/3, suppose « = ; then «/8 = 0. Then « = (0, 0, 0, ...), ^ = ib„b^,-). Conversely, if «/S = 0, either « or /3 = 0. This is proved, as in 25. 2. If a^ 0, and «/3 = a<^, then /S = 7. Proof same as that for 26, 3). 78. 1. Division. The quotient of « by ^ is the number or numbers f , in 9^, which satisfy « = /3f (1 There are two cases, according as ^=0, or ^ 0. 50 IRRATIOXAL NUMBERS Case I; /3^0. Since y8 ^fe 0, we may suppose, by 68, 2, that in all 6„ ^ 0. To find a solution of 1), consider the sequence Zi^ is regular. For, ^ ^ ^ _ ^ ^ <^rf>v - ^v^n _ ««(^,. - ^n) - ^«(«^ — «n). (2 Hence By 59, \a^\m. (4 By 65, 5, we have A<\hj^\m. (5 By taking m sufficiently large we may suppose it to have the same value in 4), 5). Then 4), 5) gives in 3), Since the sequences {a„|, \h^\ are regular, we may now suppose m taken so large that also Then 6) gives Since 2) is regular, it defines a number = (a^, Og, •••) = «, by 73, 1), 5 satisfies 1). That this is the only solution of 1) follows as in 30, 1, from 77, 2. Since CANTOR'S THEORY 51 Case II; 0=0. We can reason precisely as we did in 30, 2. Hence, when the divisor /3 = 0, division is either impossible or entirely indetermi- nate. For this reason division by is excluded. 2. We have thus this result : in the system 9^?, division is always possible and unique, except when the divisor is 0, when division is not permissible. 3. The result of dividing « by yS, we represent by a/0 and have therefore a I a^ a,^ ^'""« 1= (1,1,1,-), Kh'h' This is called the reciprocal of 0. ■} 79. 1. The system 9? is now completely defined ; its elements have been ordered, and the four rational operations upon them have been defined. As a perfect analogy exists between the sys- tems H and 9?, we are justified in calling the elements of 9? num- bers. In the future, when speaking of numbers, without further predicate, we shall mean the numbers m 9?. As already stated, they are called real numbers. 2. In the e,m test, given in 57, we were obliged at that stage to take e rational. This is now quite unnecessary, and we shall therefore, in the future, suppose e is any positive number in 9?, small at pleasure. 80. 1. We have shown in 61 that 9? contains all the numbers of M ; but we have not shown that it contains other numbers. To this end, we show that there is a number a which satisfies x^ = 2. (1 This is easily done. For in 54 we determined a rational sequence «^ = 1, a^= 1.4, a3 = 1.41, ^.. -2 such that , • 9 o ^o iim a„^ = z. (o 52 IRRATIONAL NUMBERS The sequence 2) is regular by 58, 1. Hence , . is a number in 9^. But, by 73, 1), «' 2 /// 2 /y 2 {a^, «2 ' •••)• Hence 3) shows, by 60, that a2 = 2. Hence « is a solution of 1). 2. As we saw in 52 that a is not rational, we have shown there is at least one number in 9? not in R. But the reasoning we have just applied to V2 applies equally to "v^a, when this latter is not rational. There are thus an infinity of numbers in 9? not in R. Some Properties of 9? 81. If a> 0, there are an infinity of positive rational numbers < a, and also an infinite/ of rational numbers > a. If a is rational, the theorem is obviously true by 33. Let , . a — {a^, a,, •••). Then, by 65, 3, < J. < a„ < ^. w > w* (1 But from . a„>A, we have, by Q6, 1, _ a^A. Since there are an infinity of rational numbers between and the positive rational number A, the first half of the theorem is established. Using the other part of the inequality 1), we prove similarly the rest of the theorem. 82. Between «, y8, lie an infinity of rational numbers. For, let a < /8 ; then, by 81, there exist positive rational num- bers A, B, d, such that A/3, d<^—a. SOME PROPERTIES OP 9t 53 we can, by 34, 2, determine the positive integer n so great that I> 7 ~^. Since the system B. is Archimedian [34, 1], there exists an integer w, such that , But , , ry na >na, and o> p. Hence ^ na> jd. 2. For any pair of positive numbers a < /S there exists a positive n such that „ P - k = i-{-r n are irrational. For, if y were rational, i=nj is rational. This is a contradiction. Similarly, if k were rational, i=k — r is rational, which is a contradiction. 54 IRRATIONAL NUMBERS This established, suppose first that « is rational and positive. Let i be any positive irrational number. Then, by 83, 2, we can take 7i so large that -<^-oc. n But then «<« + -< /3 ; n and a -f- - n is irrational. Suppose now that a is irrational and positive. By 81, there exists a positive rational number r, such that 00 is taken; then „ «= p. The demonstration is the same as in 46. 88. If « = (aj, a^, •••), then Since the sequence a^ a^, ••• defines a number, it is regular. Hence ^ , , € > 0, m, I a„ — a^ I < e. n, v>m. 56 IRRATIONAL NUMBERS From this we conclude that the sequence is regular. For, by 87, 4), (1 I «„ I — I «„ I I < I a„ -— a„ Hence 1) defines a number. Tq shoiv that /S = I «|. First suppose a = 0» Then Hence Therefore lim a„ = 0. lim |a^|= 0. /3= 0, and y8 = |«|. Suppose « :^ 0. Then, by 65, 6, the constituents a^ of a are of one sign, for n>m. a„=\aA. n>m. If «>0, Hence If «<0, Hence "n I ^n I = a = !«!, by 67, 3. a„= — |a„|. n>m. ^^ V 1^1 I' '"'' l^wih *m+l' '^«»+2i "'J = — a, by 67, 3 89. In the following articles we give certain equalities and inequalities which are often useful. Let < «< 1 ; then 1 + a 1 >l+a. (1 (2 NUMERICAL VALUES AND INEQUALITIES 57 To prove 1), let us suppose the contrary, viz.: 1 =1 1 + a Clearing of fractions, l0awc?P,„=(l + «i)(l + «2)...(l + a,J. P,„>1 +(«! + ... + 0- m>l. Pm > 1 + («i H h a„0 + («i«2 + «1«3 H ^- «m-l«/«)- Wl > 2. In fact, Pg = (1 + ftj) (1 + «2) = 1 + («j + «2) + «i«2 >!+(«! + «2) ; -^3 = A(l + «3) = 1 + («i + «2 + "a) + «l"2 + "l^'^S + "2^3 + "l"2"3 > 1 + (rtj + «2 + "3) + "l«2 + "l"3 + "2^3 > 1 + («j + tta + ttg). In this way we can continue. 2. Similarly, we can prove : Let < «j, a2, ••• «„j< 1, and ^„, = (l-«l)(l-«2)-(l -«,„). 2. 91. The demonstration of the following identities is obvious : -1— = l + « + «2 + ...+ ««-! +-2!L. (1 1 — « 1— a « + e _ tf /3e- aS • 58 IRRATIONAL NUMBERS 92. Let \8^\, \b^\<^S, and /3 =^ ; let P = 1. Let e>0 5e small at pleasure ; tve can take S>0 so small that ' P=^ + -. \a\0. Let I«l' l/3l<^, 1^1, \^\-8>h. Then 2) gives 0" < — ^• ' ' P Hence, if we take «0. 93. Let «p «2, ••• «„ he n arbitrary/ numbers. Let ^i-/3„; 7i-7«>0. then For, from ^ < 7i«i + --- + 7A < (nr_ ~7AH ^Jn^n^ "l>i, ..., '^>x, /3i - /e„ 7i«i H h 7««« >^(^i7i H ^ /3„7n), which gives the first half of 1). The rest of 1) follows similarly. we have Adding, NUMERICAL VALUES AND INEQUALITIES 69 94. 1. Let « > ^ > ; and n > 1, « positive integer. Then n{a - /S)/3"-i < a" - /3" < n{a - /3)a"-l. (1 For, by direct multiplication, we verify «« _ ;3" = (a _ /3) (a"-l + a»-2y3 ^ a"-3y32 _, ^ y3«-i ^^ _ ^2 In the second parenthesis, replace a by /3. Then, since « > /3, a" - yS'^ >(« - /3)(yS«-i + /S"-' + ••• w terms), or an — ^""^^ n{a — ^}/3"-~'^, which is a part of 1). If in 2) we replace /S by «, we get the other half of 1). 2. In 1), set a = 1 + S, S > 0, ;8 = 1, we get (1 + 8y > 1 + nS. If we set « = 1, yS = 1 — 5, 1) gives (1 _ By >l-n8. We have thus (1 + «)" > 1 + na, « ^fc and y. — 1, n positive integer. (3 3. We observe that 1) can be written a" > y8"-i [/3 + w(« - yS)] , (4 /3" >«"-![« -M(a-yS)]. (5 95. Let «j ••• a,j be any n numbers. ^»"~ ^^ is called their arithmetic mean. Let «j • • • a„ 5e positive, and P^ = ^\ ' f'-^ " ' ^w Then P„ < ^„", unless the as are all equal, when P„ = A^. If aj = a2 = ••• = «„, vl„ = «i and P„ = ai". Hence P„ = ^/. Suppose now the a's are not all equal. 60 IRRATIONAL NUMBERS a.=(^bjt^X-(^^J^]< "i + '^a Y Let n = 2 We have Hence i^2<^2'- Let n = 2"^. Since the a's are not all equal, at least two of them, say «], ag- are unequal. Then <'.«.<(^J. and / , Hence «:W.<(H^J(H^J- O On the other hand, applying our theorem to «T + a-i «3 + «4 2 ' 2 ' we have , , / , , , „ \2 «1 + «2 . «.S+«4 ^ / «T + «^ + «,s + «4 Y. /2 2 2 ~V 4 y "^ Hence 1) and 2) give r> ^ j 4 In the same way, we may continue for any power of 2. Let 2'"-! < w< 2'". Set /i = 2'% 2"'-n=v. We have, by the preceding case. Then 3) gives Set here a ^^^...^a^ + .A^J^^^^ (4 A* Since — — Jiff Dividing in 4) by A^, we get LIMITS i61 96. From algebra we have the Binomial Theorem, (a + /3)" = a^ + na^-^^ + ^ ■^^-'^ a^^-^^ + n-n-l-n-'l ^^,3^3 1 • 2 1 • 2 • o where w is a positive integer. The binomial coefficients n-n — l-n — 2--'n — m-hl l'2-"m we denote by \mj We have obviously, 'n n W\ /■ 71 N _ /'^ + 1 m) \m —l) \ m If we set a = )S = 1 in 1), we get 2»=i+(«)+(«)+...+(^«i)+(:: If we set a = 1, /S = — 1 in 1), we get n\ , fii\ . ^ ^N«^W n 0=1- 1 + 2 -••• + <:-i)"i It is often convenient to set and \mj ' if m>n. Limits 97. We extend now the terms sequence, regular sequence, limit, etc., to numbers in 9?. This is done at once ; for the definitions given in 40, 42, and 57 may be extended to 0?, by simply replacing the term rational number by number in SR. 62 IRRATIONAL NUMBERS For example, the sequence of numbers in 9t «i, «2j «3> ••• is regular when, for each positive e (not necessarily a rational number now) there exists an index w, such that I «„ — a^ I < e, for every pair of indices ?i, v>m. Or in abbreviated form, when € > 0, m, I «^ — a^ I < e. n, v> m. (2 This definition, we see, is perfectly analogous to that given in 57, 1 for regular rational sequences. Evidently the reasoning of 57, 3, 4, can be applied to the sequence 1). Thus the e,m test given in 2 may also be stated in the form : e > 0, m, I «„ — «„i I < e, w > m. (3 Similarly, X is the limit of the sequence «!, ao, «3, ••• when e > 0, ?7^, I X. — «,J < e, n > m. (4 As before, we write X = lim a„. or X = lim We say also a„ converges to X or approaches X as limit. This may be indicated by the notation «„ = X. 98. Let lim a„ = a and lim ^^ = /3. Then lim («„ ± /3„) = « ± yS ; (1 lim a„/3„ = a/3. (2 //"yS, ySj, ySg, •••=?^0, ■^^^e have also lim-^ = ^- (3 The demonstration is precisely similar to those of 49, 50, 51 ; and thus does not need to be repeated here. LIMITS 63 99. We prove now the important theorem : Let a = (aj, a^, •••), the as ratiojial ; then lim a„ = a. We must show that e>0, wi, |a — a^]m. ("1 Since the sequence is regular, we have cr > 0, m, I a„ — a^ I < cr, w, 1/ > m. (2 Now we can write, by 60, Hence, by 71, supposing n to be fixed for the moment. By 88, Hence, by 2) and QQ, 2, I a — a„ I < cr. Thus if we take o- < e, we have 1) . 100. If a sequence . ^ ^ = «i, «2' ••• has a limit X, A is regular. For, by definition, e > 0, m, I X — «,j I < e/2, n>m. \\ — «^|m. Adding, by 87, 3, I '^re "■ f^r I < ^' W, y > 7W. Hence A is regular, by 97. 101. 1. Conversely., if ^ = a^, a^, ••• is a regular sequence, there exists one., avid only one, number «, such that lim «„ = a. (1 To show that A cannot have two limits, we need only to repeat the reasoning of 47, 64 IRRATIONAL NUMBERS We show now A has a limit. Let 8^, §2' ^3' ••• (2 be a sequence of positive numbers whose limit is 0. We choose the S's now, such that «,. = «» + ^«^ w=l, 2, ••• (3 are rational. This is evidently possible by 82. The sequence dfji ^2' ^3' *** C is regular. For, a^-a^= a„ -«,+ (§„- 8 J . (5 Since A is regular, e>0, m, |«„ — aj<-, n,v>m. (6 A Since, by 100, the sequence 2) is regular, \^n — ^v\<^l^' n,v>m. (7 In the inequalities 6), 7), we may take m the same. Then 5), Hence 4) is regular. We set a= (a^, a^^ •••). Then, by 97, But, by 3), Hence, by 98, lim a„ = a. lim a„ = lim a„ — lim B„ 2. As a result of 1 and 100, we have : In order that a sequence a^, a^^ ••• has a limits it is necessary and sufficient that it is regular. 102. Let A = «!, a^ sequence LIMITS 66 be a sequence. Let us pick out of ^ a 'r '•2' where t^ < tg < ^3, •••• We call B a partial sequence of A. EXAMPLES 1. A = l, 1 2' 1 1 1 3' 4' 5' .., B = l, 1 3' 1 1 5' 7' ■" C=l, 1 22' 1 1 23' 2*' •• D = l, _\_ 1 ' c' < 2-3 2.5 2.7 Here B, C, Z) are partial sequences of A. 2. ^ = 1, -, 1, I, 1, 7, — '2 3 4 5=1, 1, 1, ... ^-2' 3' 4' 5 and C are partial sequences of A. 103. 1. Among the symbols given in 42, to indicate the limit of a sequence .Ji. CCj, Cinj one was lim a„ A Analogously, we shall denote the limit of a partial sequence B z= a,, a^ ., ... of A, by lim a„. B 2. We have then, obviously : If A is regular, so is every partial sequence B; and lim a„ = lim «„. A B 66 IRRATIONAL NUMBERS 3. From this, we conclude at once : The sequence A cayinot be regular, if it contains two partial sequences -S, C, such that lim «„ =^ lira a„. B c 4. The sequence A cannot be regular, if it contains a partial sequence B which is not regular. 5. It is sometimes a difficult raatter to show that a sequence A is or is not regular. The theorems 3, 4 enable us often to show with ease that A is not regular. Thus, in Ex. 2, 102, lim «„ = 1, lim a„ = 0. B c Hence A is not regular. 6. Unless the contrary is stated, it is to be understood that lim «„ » has reference to the whole sequence A. 104. 1. From 98, we conclude the following theorems, which are often useful: If lim («„ ± /3n) = o"i and lim a„ = a ; then lim /3^ exists and equals ±a-T a- 2. If lim a„/8„ = tt, and lim a^=: a^O; then lim ;S„ exists and equals 7r/«. 3. If lim -—■ = p, and lim yS^^ = /3 ; then lim «„ exists and equals ^p. Pn 4. If lim -^ = p^O, and lim a^= a; then lim y8„ ercis^s and equals The demonstration of these theorems we illustrate by proving 1. We have ^« = ± («« ± /^n) T «„. Applying 98, 1), lim ;S„ = ± lim (a„ ± yS„) =F lim «„ = ± o" T «. LIMITS 67 105. 1. Let \ixn a^^ = a ; let ^, ^ he tivo numbers^ such that ^ ni. The demonstration is the same as in 48. 2. Let lim «„ = a, and «„ < a. Let j3 be any number m. 106. 1. Let lim «„ = « ; if )\.m ; then X < « < /i. For, suppose a> fx. Let /S be chosen so that /i < y8 < «. Then, by 105, 1, a,, > yS. n>m. Hence fin > /*» which is a contradiction. 2. /f «„ 0, w, 7„ — «„ < e. n>m. Then, by 2), /8„-«,'\+i^ n = l, 2, ... it is a decreasing sequence. If it is either one or the other, but we do not care to specify which, we may call it a univariant sequence. If, on the other hand, ««<««+^ -^ = 1, 2, ... A is said to be a monotone increasing sequence. it is a monotone decreasing sequence. If A is either one or the other, but we do not care to specify which, we may call A a monotone sequence. Univariant sequences are special cases of monotone sequences. 2. If there exists a fixed positive number Gr^ such that |a„| < (r, w= 1, 2, ••« A is said to be limited^ otherwise unlimited. 109. A limited monotone sequence is regular. For clearness, let A = ccj, a^, • • • be an increasing monotone sequence, and let a„0, w, 0m. (2 LIMITS 69 Since A is monotone increasing, < «„ - a^. To show the rest of 2), take tn^ at pleasure. Either there exists an infinite sequence of indices m^Km^Km^K-" (3 such that _ or there does not. Suppose such a sequence 3) exists. Then, however small e has been taken, we can take the integer p so large that Adding the first p inequalities 4), we get Hence, by 5), „ -^r "■nip ^ "^1 which contradicts 1). We thus conclude that there exist but a finite number of indices Wj, such that 4) holds. Thus we can take m so large that «« — ««i < e^ 'W > ^h which proves tlie other half of 2). 110. 1. A limited increasing sequence of great importance is «, = (!+ 2)". « = 1,2,... (1 To show that 1) is increasing, i.e. that «« > ««-i, (2 we employ the relation 94, 5), viz. : ^»>a»-i[«-w(a-/S)]. (3 Set a=l + -J_, /3=1 + -, n—\ n in 3) ; we get 2) at once, for w ^ 2. 70 IRRATIONAL NUMBERS To show that 1) is limited, we set a=l + — -, /3=1, w = m4-l, zm in 3) ; we get l>ifl + ^T; m = l, 2, 2\ 2mJ or squaring, H^-£T- Thus «2«<4. But, by 2), ^2m-l<^2»i- As all positive integers are of the form 2 m or 2 m — 1, 4) and 5) give a„<4. w = l, 2, ••• (4 (5 2. Since the sequence 1) is limited and monotone, it has a limit by 109. We set e=lim(l + -). 71=1, 2,3, ••• As the reader already knows, e = 2.71828 •••, and is the base of the Napierian system of logarithms. 111. 1. Let A = a^, Kg, ■•• be a regular sequence, whose limit is a. In A exist partial monotone sequences B ; and for each such sequence, lim a„ = a. B Then are two cases: 1° « — «„, n>m has one sign, when not zero ; 2° a — «„ may have both signs, however large m is taken. Case I. To fix the ideas, suppose a— «„^0. n>m. (1 In this relation, it may happen that for some m' >m a — a„ = 0. n^m'. LIMITS 71 In this case, = «, a, ... is a sequence required in the theorem. Let us suppose now that there are in 1) an infinite number of indices w„, such that «-«»,> 0. Let Vj be one of the indices n^ ; then Let )8j lie between these, so that a.. 0, (2 let us denote those values of n for which 2) holds, by Wj, Wj, Wg, •••. Then the partial sequence of ^, 72 IRRATIONAL NUMBERS belongs to Case I. Hence in A' lie an infinity of sequences of the type B. 2. The demonstration of 1 shows : If, in the regular sequence A = a^, a^, •••, the «„ do not finally become all equal, there exists in A an infinity of partial univariant sequences B which have all the same limit as A. The Measurement of Rectilinear Segments. Distance 112. In 39, 43, 44, we have made use of the graphical representa- tion of the numbers in J?, by points of a right line. We wish now to extend the considerations to numbers in 9?. With this end in view, we proceed to develop the theory of measurement of recti- linear segments and the associated notion of distance. 113. 1. Let AB, CD be two rectilinear segments. We say ^5 is greater than CD, when, if superimposed, AB contains CD as a part ; while CD is said to be less than AB. If, when superimposed, AB and CD coincide, we say AB and CD are equal. 2. We assume, with Archimedes, that if the segment AB is laid off a sufficient number of times on the line L, we can obtain a c D d" t. segment CD' greater than any given segment CD. And conversely, that it is possible to divide a segment CD into a sufficient number of equal parts, so that one of them, as CE, is less than any given segment AB. MEASUREMENT OF RECTILINEAR SEGMENTS. DISTANCE 73 114. 1. Let S = AB be a segment we wish to measure ; and let TJ = CD be a segment which we take as a unit of comparison. If we ca^ divide *S' into I equal segments, equal to U, i.e. if we say I is the measure of *S', or I is the length of S. 2. If it is impossible to do this, it may happen that n segments S are equal to m segments U ; i.e. n- S=m'U. We say then, that n is the measure or length of S. 3. In both cases we say S is commensurable with U. The segment AB being commensurable, we say its length I ex- presses the distance of A from B, or B from A. We write I = Dist (A, B), or more shortly l = AB. 115. We show now that the number I, just determined, is unique. This is evident when I is an integer. We suppose, therefore, that nS = m f/, (1 ^"^ 71^8 =m^U. (2 Multiplying these equations respectively by Wj, n, we get nn-^S = n^m Z7, Jin^S = nm^ U. .'. n-^mU— nm^U. .'. n^m — nmp or , m m n n' Thus, the two equations 1), 2) lead to the same value of I. 74 IRRATIONAL NUMBERS 116. 1. Let I oiS. nS s m .asure ~n ^ be the me Then = mU. (1 V Let us divide U into n equal parts, and call V one of them. Then nV=U. This in 1) gives nS = mn V. Hence S=mV. This shows that by taking a new unit V, whose length is 1/n of the old unit, the length of S can be expressed as an integer. 2. The above considerations also give us a new way for defining the length of S. In fact, suppose it possible to divide U into s equal parts V, such that S=rV. (2 Then Z=-. (3 s ^°'' sV=U; hence, multiplying 2) by s, we get sS= rsV= rU; so that the length I oi S is indeed given by 3). 117. Let S = AB, T = BC be two segments whose lengths are respectively ^ „ b a A B C a, 0, c, d being positive integers If we put them end to end, we get a segment W= A O whose length, we show, is , n=^l-\-m. By definition we have bS=aU, dT=cU. MEASUREMENT OF RECTILINEAR SEGMENTS. DISTANCE 75 Multiplying these equations respectively by d, 6, and adding, we sret ^ hd'S+hd-T=adU+hcU=(iad + hc)U. (1 hd-S + hd-T=hd'W. (2 'Hence 1), 2) give hdW=(ad + hc')U. Hence ad + hc a , c , , " = -^- = 4+5 = ' + ™- 118. 1. We turn now to the measurement of segments which are incommensurable with the segment chosen as unit. An example of such segments is the diagonal of a square, the side being taken as unit. I 1 An Bn To measure AB^ we begin by marking off points on the right line L, at a unit distance apart, starting with A. By the axiom of Archimedes, 113, 2, B will fall between two consecutive points of this set, say between A^^ B^. Let Zj = Dist (A, A{). On the segment A^B^ we mark off points at the distance 1/w apart, where n is an arbitrary positive integer. Then B will fall between two of these points which are con- secutive, say between A^, B^. l^= Dist (^, .42). We may continue in this way, subdividing each interval A^, B„i into n equal parts, without end. The point B will never fall on the end point of one of these intervals, for then AB would be commensurable. The sequence of rational numbers ^1' ^2' 3' *" V-*- so determined is monotone increasing, and limited. In fact, all its elements are < Zj -f- 1. Thus, by 109, the sequence 1) is regu- k 76 IRRATIONAL NUMBERS lar, and so defines a number X. We say X is the measure or length of AB, and we write as before X = Dist (A, B) = A, B. 2. If we had taken the numbers IJ = I)ist (A, B J, /c = l, 2, ... where 5^ denotes the right-hand end of the interval in which B falls, instead of the numbers l^, we would have got a monotone decreasing limited sequence whose limit X' = X. F«^' l'-l= 1 K whose limit is 0. n' K-V 119. We have defined the length X of AB by a process which subdivides each interval A^, B^ into 7i equal parts. The question at once arises : would this process lead to the same number X, if we had divided each interval into m instead of n equal parts ? We prove the following general result : Let us modify the above process so as to divide the first interval into n^ equal parts, the second interval into n^ equal parts, etc. This system of sub- division leads to a sequence which we denote by The limit of 1) being X', we show it exists and X = X'. For, each point A J will fall in a certain interval A , B of the old system of subdivision, where i^ is the lowest index for which this is true. ^^^^"^ Dist (^^^ J ^ Dist (A, AJ) < Dist iAB^J. But, by 118, 2, lim Dist (J., A^^ = lim Dist (^, B^J = X. Hence, by 107, ,. ,^. , . , . ,. ^ •^ lim Dist (J., J.,„') = X. MEASUREMENT OF KECTILINEAR SEGMENTS. DISTANCE 77 120. The process explained in 118, 119 is obviously applicable to the case when AB is commensurable. The only difference is that after a certain number of steps the point B may fall on one of the end points of the little segments A„,, B^. In this case the corresponding sequence 1' ^2' '" ^S1 ^5' ^si '" would have all its elements the same after a certain one. 121. We have now two methods for measuring a commensurable segment; viz. those given in 114 and 120. Let I be the length of AB as given by 114 ; and \ its length according to 120. We show l=\. Since AB is, by hypothesis, commensurable, we have, by 117, preserving the notation already employed, I = Dist CAA,J + Dist (A^B) 0, and to the left of 0, if X < 0, and such that |X| = Dist(0, P). This we shall call the Cantor-Dedekind axiom or the axiom of con- tinuity of the right line. As we proceed, it will be made evident that many apparently simple geometric ideas are extremely subtle and complex. One of the most elusive of these is the notion of continuity. To say the right line is continuous because it has no breaks or gaps^ is simply to replace one undefined word by another. 3. We have now established a one to one correspondence between the numbers of '^ and the points on L. We may consider the points as images or representations of these numbers. 124. 1. The correspondence which we have just defined is a generalization of that given in 35 for R. The considerations of 39, 43, 44 can now be extended to 9^ without any further com- ment. The graphical interpretation of sequences and their limits which we thus obtain wall illuminate greatly the section on limits., 97-111. We recommend the student to go over the demonstra- tions which we gave there, employing graphical representations as an aid to the reasoning. Indeed, some of the theorems, when interpreted geometrically, seem almost self-evident. 2. Consider, for example, the theorem of 107. We have there three sequences, A= |a„|, B= |/S»|, 0— {7»|. The relation states that the point ^^ ^i^s in the interval /„= («„, 7„). Since now both end points of i„ converge to the point \, evidently any point in /„, as /3„, must also converge to the point X. 125. As another example, consider the theorem of 109. To fix the ideas, let J.= ^«„| be a monotone increasing sequence. The points in the figure represent a^, ag' "■' ^® have drawn a curve through them, as the eye seizes more easily the law of 80 IRRATIONAL NUMBERS increase or decrease of a sequence when such a curve is drawn. The reader will observe that since the sequence is monotone, this curve can have segments parallel to the axis X. As A is limited, all the points of A lie be- q tween a certain line G, and a line JEJ drawn through the first point a^ of the sequence. To see now that A must have a limit, let us suppose the line Gr moved parallel to itself toward X. Evi- dently there is a line F below which G cannot move without getting below points of A, and which the points of A approach as an asymptote. If A, is the distance of F from X, evidently lini a„ = X. Fig. 1. 126. As a final example of the helpfulness of graphical repre- sentation, let us consider the theorem of 111. The two cases we considered there are represented in Figs. 1 and 2. The heavy curves represent the law of increase and decrease of the sequence A. The points «j, a^, •'• lie on" these curves, but are not indicated. The straight lines A represent the limit a of A. The light curve in Fig. 1 indicates an increasing sequence which one could pick out of A. By the aid of such a representation the theorem becomes almost self-evident. Fig. 2. CANTO R-DEDE KIND AXIOM 81 127. 1. Let A==\a,l B ={^J, C = \y,\. Let A he monotone increasing, and C moriotone decreasing. Let f^n<^n72>73-" (4 From 1), «n < 7n- Hence, by 4), «n<7r w=l, 2, ••• Thus A is limited. Similarly, C is limited. Thus A and C are regular, by 109. Let lim «„ = X. (5 Then 2) and 5) give, by 104, 1, lim 7„ = X. Then, by 107, lim /3„ = X. 82 IRRATIONAL NUMBERS 2. The preceding theorem may be put in geometrical language, and gives : Let Ij^ = (a^^ 7„) he a sequence of intervals w = l, 2, 3, •••. Let In lie in -Z„_i, and let the lengths of these intervals converge to 0. Let /8„ he any point iii I^ (including end points'). Then the sequence f/3„| is regular, and all such sequences have the same limit X. The point \ lies in every I. 3. The reader should avoid the following error: Let {a„|, {/3„5 be two sequences, such that lim-(«„-^J=0. (6 Then lim «„, lim /3„ exist and are equal. That this conclusion is false is shown by the following example : «„ = (-l)« + i, ^„ = (-l)^ Here neither limit exists, although 6) is satisfied. Dedekind's Partitions 128. We proceed now to establish the notion of partition,* introduced by Dedekind, to found his theory of irrational numbers. Let a be any number of 9? ; we can use it to throw all numbers of 9"? into two classes A, B. In A we put all numbers < a ; in ^ all numbers >a. The number a we may put in A or B. This division of the numbers of 9? into two classes we call a partition, and say, a generates a partition (A, B). Geometrically, the point a may be used to throw all points of a right line into two classes. In class A we put all points to the left of « ; in ^ all points to the right of a. The point a we put in either ^ or 5 at pleasure. Example. Let a = v^. In A put all numbers < \/2 ; in B put the numbers ^ y/2. This partition may also be generated as follows : in A put all numbers whose square is < 2 ; in B all numbers whose square ^ 2. * The German term is Schnitt. DEDEKIND'S PARTITIONS 83 129. 1. More generally^ we shall say that any separation of the numbers of 9? into two classes J., B, such that 1° Any number of J. is < any number in B, 2° Any number of 5 is > any number in J., constitutes a partition (^, B^. 2. The condition 2° is really redundant, as it follows at once from 1°. In fact, suppose 2° did not follow from 1° ; i.e. suppose there were a number /3 in B., ^ some number a in A. Then there is an « in tI which is not < any number in J5, for « is not < /3. This is a contradiction. 3. Two partitions (J., 5) and (C, i>) are the same or equal only when A and C contain the same set of numbers; or only when B and D contain the same numbers, excepting possibly the end numbers. 130. 1. We consider now this question : suppose a law given which throws all numbers into two classes J., B^ such that every number in A is less than any number in B, and every number in B is greater than any number in A. Is there a number X in 9? which will generate this partition (J., B^ ? We show there is. To this end we construct two sequences S=a-^., ttg, "3, ••• y = y8i, yS^' ySg, -.• S being monotone increasing, and T monotone decreasing, as follows : Let ttj be any number at pleasure in yl, and /3j a number in B. Their arithmetic mean _ 2 lies between a^, /3j. If it lies in A, we set if it lies in B, we set 84 IRRATIONAL NUMBERS We build now the arithmetic mean of «2, ^^.^ and reason with this as before. Continuing this process indefinitely, we get the sequences S and T. By 127, 2, the sequences S, T have a common limit, which we call \. Let X generate the partition (^', B'). We show that (^A, B^ = {A' , B-), by showing 1° that every number in A' lies in A ; and 2° that every number of A lies in A' . 1°. Let a' 4^\ be any number of A'. By 105, 2 there are an infinity of numbers «„ lying between «' and X. But a,^ is in A, by hypothesis. Hence a' < «„ is in A. 2°. Let « be any number of A. We have to show that cc^X. Suppose the contrary, X < «. Then ^ ^ e = « — X > 0. We can take n so great that A.-«. = ^^<- (1 On the other hand, by supposition, «„ < X < a < ^„. Hence which contradicts 1). 2. We have thus this theorem : Every partition can he generated hy a number in $}?. 131. 1. A partition (^, -S) cannot he generated hy two different numbers X and yu,. To fix the ideas, let X< /jl. In the partition ( C, i>) generated b}^ /u,, C contains all numbers < fi. It therefore contains numbers > X, and hence numbers not in A. Hence {A, J5), ((7, D) are different. 2. Since each number generates one partition, and each partition is generated by one number, we can establish a uniform or one to INFINITE LIMITS 85 one correspondence between the numbers of 9? and the aggregate of all possible partitions. In fact, to the number a shall correspond the partition {A, -6), that a generates. To the partition (i^, (7) shall correspond the number X, which will generate {F, 6r). Infinite Limits 132. Let A = ttj, ttg, ••• be an unlimited sequence [108, 2]. The following cases may occur : 1°. For each positive number G, arbitrarily large, there exists an m, such that «„ > G, for every n > m. In symbols (r>0, w, a„>(x, n>m. We say the limit of A is plus infinity/; and write lim «„ = + oo , lim «,j = + ao , a„ = + go . Such sequences are i o o -L, -^5 ^1 •" 1!, 2!, 3!, ... 2°. For each negative number (r, arbitrarily large, there exists an m, such that a„< (r, for every n^m. In symbols ^ < 0, w, Uj^kG-, ny^m. We say the limit of A is minus infinity ; and write lima„= — Qo, lima„= — QO, a„ Such a sequence is -10, -100, -1000, ... In both these cases, we say the limit is definitely or determinately infinite. 3°. The elements a„ do not finally all have one sign, but still lim |«„| = 4- Qo. We say the limit of A is indefinitely or indeterminately infinite. Such a sequence is 1. _2, +3, -4, +5, ... 86 IRRATIONAL NUMBERS 133. 1. Case 3 is of little importance. We shall therefore in the future, when using the terms the limit is infinite, or certain variables become infinite, always mean definitely infinite, unless the contrary is expressly stated. The symbol + qo is frequently written without the + sign. The symbol ± qo means that the limit is either + qo or -co, and one does not care to specify which. The limits defined in the previous sections are called, in contra- distinction, finite limits. The symbols +ao, — oo are not numbers; i.e. they do not lie in ^. They are introduced to express shortly certain modes of variation which occur constantly in our reasonings. 2. Finally, we wish to state, once for all, that the terms, the limit exists, the limit is X, etc., or an equation as lim «„ = \, always refer to finite limits, unless the supplementary phrase, '■'■finite or infinite,''^ is inserted. 134. A sequence cannot have both a finite and an infinite limit. For, if ^ = |a„| has a finite limit, the numbers a„ lie between two fixed numbers, by 105, 1. It is thus limited. It cannot therefore have an infinite limit. 135. Let A= fa„}, and let B be any partial sequence of A. If lim «„ = ± 00, A then lima„ = ±Qo. B The demonstration is obvious. 136. If the limit of a sequence A=\a^\ is indefinitely infinite, its positive and negative terms each form sequences whose limits are respectively + go and — qo . For, let B=\^n\ be the sequence formed of the positive terms of A; and C= |7„| the sequence formed of the negative terms. By hypothesis, (?>0, m, |a„|>(r, n^m. INFINITE LIMITS 87 But then, for a sufficiently large m', Hence lim ^n = + ^1 lini 7„, = — 00. 137. Let lim «„ = a, lim /3„ = ± oo ; ^A«w 1. lim(a„±/3„) = ±00, lim^=0. 3. //«:^0, ^>0, aw(^ lim/3„ = 0; then lim -^ = 4- 00. The demonstration is obvious. 138. Let «!, a^, ttg, ••• ySj, (^2' ^3' "• be two sequences. Let I3n ^cc„. n = 1, 2, — If lim «„= +CO, then lim /S„ = + oo . For, by hypothesis, G>0, m, an> Gr, n>m. Since y3„ > «„, we have, a fortiori. Hence lim ;8„ = + 00. 139. Let A = ttj, ttgi ••• ^^ <^ monotone increasing sequence. Let he a partial sequence of A. If lim «„ = + 00, then lim «„ = + oo, u A 88 IRRATIONAL NUMBERS For, by hypothesis, Gr>0, m, a > G-. n>m. 're But since A is monotone increasing, Hence «r> (r, and lim a^= -\- cc. 140. Let Wj, ^2, ••' Je a sequeyiee of integers whose limit is +go. Then lim a™" = 0, zj < « < 1. (1 For, let «>1. We set Then, by 94, 3), where We apply now 138. Since we have, from 4), Let 0l. (3 « = 1 + S. 8>0. «^ = (i + sy«<>^^, (4 lim /3^= 4-00, lim a^ = +(». 1 P>1. a™» = INFINITE LIMITS 89 141. We consider now a few examples involving infinite limits. Example 1. a^=l + l + l+..-+-. ri = l,2,-. 16 n Here lim«„=+Gfo. (1 For, let /*=2™-l. w = l, 2, ... Then Om—l ■•)m 1 Each parenthesis is >-• For, 1,1 11 1 - + ->- + - = -' 2 3-^4 4 2 l^l^l^l 1,1^1,1 1 , 4 + 5 + 6 + 7>8-^8 + 8 + r2'^*^- Thus ^ m As lim m= +00, we have, by 138, lim « = + 00 . But \c(.^\ is a partial sequence of the increasing sequence la^]. Hence, by 139, we have 1). 142. Example 2. «n = - + -T^ H —. + ••■ + a a-(-l a + 2 a + w where a=?^0, —1, —2, —3, ••• Then ,. ^ lira «„= +O0. (J ie^ a>0. Then there is a positive integer m, such that w — 1 < a^. ^7=0, 1, 2, .^. Hence ^ ^ ^ m m+1 m+n But, by 135, 141, ,. ^ Hence, by 138, ,. •^ lim«,j = +QO. Let a<0. Then there exists a positive integer m, such that 0<« + m< 1. Hence 11 1,11 a-\-m a + m+1 (t-\-m-\-n Z n Then, by 141 and 138, lim 7^=+ GO. But {7„| is a partial sequence of Ja^}. Hence we have again 1), by 139. 143. Example 3. Q. _ «(« + 1) ••• (« + »i) where /S=9S=0, -1, -2, ••• 1°. a>/3, /3>0. Leta = a-/3. Then -— ! — = 1 + — m=0, 1, •••w. p + 71(1 p + m Hence by 90, 1. Hence, by 142, lim^„=+Qo. 2°. a>^,/3<0. If a is a negative integer or 0, Q^ finally becomes and remains 0. DIFFERENT SYSTEMS FOR EXPRESSING NUMBERS 91 Hence i- /-» n n ^ n Otherwise, let the positive integer m be taken so large that yS + m>0. Then Q _ «(« + l)"-rct 4-m — 1) (« + m) •••(« + n) _ T>a ^" /3(/S+l)-..(/3 + m-l) * (^ + ^)...(;3 + ri)~ "• The first factor ^ is a constant. In iS'^, set a' = a + m, /3' = ^ + m, n— m = n'. Then ,^ , .^ . , ,^ '^ yg'(^' + l)...(/3'+ri')* As a' > 13' > 0, iS'„ falls under case 1°. Hence .. ^ hm ^„ = ± Qo, the sign being that of B. 3°. a1. Then we can give a the form a = a„m" + a!„_im"~^ H 1- «o' O where 0 «Q + a^m + •• • + a„_im'*~^ (1 This is obviously true for w= 1. We apply now the method of complete induction. Supposing 1) is true for n = s, we show it is true for n= s + 1. Let, then, m^>aQ + a-^m + ••• + a^^-^nf'^. Then, since both numbers on the left and right are integers, 771* — (aQ + a^m + • • • + «5_iWi*~0 ^ !• Hence ,+i . , , ,_ix ^ 94 IRRATIONAL NUMBERS Subtracting an integer 5=0, 1, ••• m — 1, from both sides, w'+^ — (5 + a^m + a^m"^ + h as_inf) > 0. Hence, changing the notation slightly, w'+^ > «o' + «i'^M h ajm% (2 if the a' are ^m — 1. This established, let iV= 5(j + 5jm + Sgm^ + • • • + b^m^ where a^^^O, b^^O. Then 1) shows that if r>s, then M^JST. Hence, if Mia to equal iV", it is necessary that r = s. If r = s, then 1) shows that iltf^iV", according as a^^bj.. Thus, if M= N, it is necessary that a,. — b,.. In this way we may con- tinue, and so show that when M= N, a^ = b^, a;= 0, 1, ... r. 2°. Let Q 5^. Then iV>/*. For, Since P is written in the normal form, there exists an s^r, such that , ^ bg 5^, 2 -^-^ 2- Hence . iVj + iVg > iV^j + Pg' and a fortiori^ ^ p CHAPTER III EXPONENTIALS AND LOGARITHMS Rational Exioonents 146. Having developed now the number system 9? with suffi- cient detail, we shall in this and the subsequent chapters represent numbers in 9t indifferently by Greek and Latin letters. 147. Up to the present we have defined the symbol a*" (1 only for positive integral values of the exponent fi. We proceed to define it for any value of /a, supposing a > 0. We begin with rational values. The numbers 1) are then called roots or radicals. 148. 1. Let a>0, and let n he a positive integer. There exists one and only one positive number satisfying x^ = a. (1 Let (^B, C) be a partition such that B contains all positive numbers h such that b"' < a. Let p be the number which generates (.6, (7), 130, 2. By 130, 1, we can pick out of 5 a monotone increasing sequence \b^\ and out of C a monotone decreasing sequence {c^} such that lim b^ = lim c^ = p. As l>l the radical -y/a shall always be a positive number. 2. The equation x^ = admits one and only one solution, viz., X = 0, in $R. We write i 70 = 0^=0. 150. 1. If m, n are positive integers, and a>0, then 1 1 . (a")™ = (a"'y. (1 Set 1 p = a'^K RATIONAL EXPONENTS 97 Then p« = a, and 1 r = («")"•; (2 also The equation 3) shows that p"^ is the positive solution of a;" = a*". Hence, by definition, i p'« = {dPy. (4 Comparing 2) and 4), we have 1). 2. We write i i m {ccy = (a"')" = a". We have now the definition of a*", a > for positive rational exponents. 151. Let ft be a positive rational number. We define the symbol a"** by the relation ^ a-'^ = — . We also set „ ^ a" = 1. 152. ie^ r, s 5e rational numbers, and a > 0. ?%gW' a'-a* = a'-^'. (1 This equation expresses the addition theorem for rational ex- ponents. It is a generalization of 74, 2. To fix the ideas, suppose r, s > 0. Let I m r = — , s = — , n n where I, m, n are positive integers. Let p=a'^ ; then p"" = a'. (2 r of a = a' ; tjien o-" = a'". (3 98 EXPONENTIALS AND LOGARITHMS Multiplying 2), 3), we get {pa-y = «'+'". This shows that /> • o- is the positive root of Hence i+m p(T = a '^ . (4 But 2), 3) also give pa = a^'a\ (_5 Comparing 4), 5), we get 1), since Z + m = r + s. n 153. Let fi be a rational number, and a > 0. Then „ ^ a'">0. 171 For let ytt = — > ; m, n positive integers. We have, by 148, i a">0. Hence ^ l l I (t =z a^ ' a'' ••' a" > 0. m factors. If fjL0. Then i a'" = — , a" and as a" > 0, so is a*^ > 0. If /u, = 0, we have a'^ = 1, by 151. 154. Let fjL be a positive fraction and a>0. Then a'^ = 1 according as a = l. Let /i = — ; m, n positive integers. 1 If a > 1, then a" > 1. For, suppose the contrary, i.e. let RATIONAL EXPONENTS 99 Raising to the nth. powers, by 75, 2, al, a">l. m Hence a" > 1, by 75, 2. The other cases are similarly treated. 155. Let n be a positive integer and CL>0. Then a^^ a according as a = l. Let a < 1 and suppose 2 Then, by 75, 2, a^a\ (1 But when a a. The other cases are easily treated now. 156. 1. Let «>0 and let /m be a positive fraction. If a>'>b>0, then I a>b''. (1 For, let /A = — ; w, w positive integers. n From TO a" >b follows, by 75, 2, a'">J™. (2 Suppose now 1) were not true, i.e. suppose . Then _ a'^ < J", which contradicts 2). 100 EXPONENTIALS AND LOGARITHMS 2. Let a > and let fi be a negative fractioyi. If a^>b> 0, then a<.b^. (3 We can set fi = — v^ v > Then -j a- This reduces 2 to 1. 157. Let fjL

0, Then . 7. \-, ^-1 a^^a^ according as a^l. (^1 We can set r s u = - V = -t t t where r, s, ^ are integers and ? > 0. To fix the ideas, let a > 1, and /a, i'> 0. Suppose for this case, 1) were not true, i.e. that a*^ ^ a". Then a'' ^ a^ by 75, 2, which is absurd, since r 0, w = 1, 2, •••. If lim a„ = 1, (1 then lim < = 1. (2 To fix the ideas, let r ... /i, = - ; r., s positive integers. If «„<1, «„<«,/l, 1 <«„"<«„, 1 by 154, 155. Thus in either case a,j* lies between 1 and a„. Ap- plying 107, we have, using 1), lim aj— 1. IRRATIONAL EXPONENTS 101 Since r i i . «/=«„*■«,/•••«„% r factors. we have, by 98, ?• lim a„^ = lim < = 1. The other cases are now easily treated. 159. Let n he a rational number; let lim a^ = a > 0, a„ > 0. Then hma'^= ai^. (1 For, < = «.(^J. (2 But, by hypothesis, lim^=l, by 98. a If we apply now 158 to 2), we get 1). Irrational Exponents 160. Let R = r J, r^^ •-• he a sequence of rational numbers whose limit is 0. Then^ if a > 0, lima''» = l. (1 We show : /^ i -i . i ^o e > 0, 7n, 1 1 — a'^" I < e, w > Tn, (2 which is the same as 1). Let a > 1, r„ > ; then, by 154, a'^">l. Since r„ > and as small as we please, n being sufficiently large, we can take m so large that 1 — >g, n>m, rn however large the positive integer g be chosen. But, by 91, 3), ^^ ^ ^ 102 EXPONENTIALS AND LOGARITHMS We can also take g so large that 1 +g€>a. Then ,-. x„ (l + ey>a. (3 On the other hand, by 157, (l + e)^«>(l + 6)^. (4 Hence 3) and 4) give (l-fe)'"">a. n>m. This gives, by 156, 1), 1 + e > ar\ Hence 2) holds in this case. Let a < 1, r„ > 0, We set 1 a=-; then b>l. By the preceding case 6''«-^>l-e, J'^" 1 + e by 89, 1). This gives 1 — a^« < e. Hence 2) holds in this case. Let r„ < 0. This case reduces to the case that r„ > 0, by observ- ing that We consider now the case that the r^ do not all have one sign. We divide li into three sequences, Mq, M+, R_. In the first, we throw all r„ = ; in the second all r„ > ; in the third all r„ < 0. Should any of these sequences contain only a finite number of elements, it can be neglected. For we have only to consider a partial sequence of R, obtained by omitting the terms of R up to a certain one. IRRATIONAL EXPONENTS 103 Consider ^+. We have seen there exists in it an index m' such that 2) holds for every n>m' . Similarl}^ in jB_, there exists an index m" such that 2) holds for every n>m" . Consider finally R^. As r„ = 0, 2) holds for every n of R^^. Thus if m be taken >w', m" , 2) holds for every n>m'va. R. 161. Let A — aj, a^^ ••• he a regular sequence of rational numbers, and let 5 > 0. Then b% 5% ••• (1 is regular. We have to show : e>0, m, 15""— 5'''»|m. (2 Set d^ = 6"» - b"'" = 5«"'(6««-«'» - 1). (3 Since A is regular, we have 8>0, m, |a„— a,„|<8. n>m. But if 8 is taken small enough, by 160, |5a«-«m_l|<^, (4 where ?; > is arbitrarily small. Since A is regular, there exist, by 65, 5, two rational numbers, Q, R, such that Ql; 1, we take i] = € 5^' if 5 < 1, we take '^ — To Then in either case 2) holds. The case that 5 = 1 requires no demonstration. 104 EXPONENTIALS AND LOGARITHMS 162. Let ap a^^ ■•■ and a^, a.^, •• he two sequences of rational numbers having the same limit. Then, if 5 > 0, lim b"" ~ lim 5"». (1 By 161, both limits in 1) exist. Let c?^ = 6"« - 6"« = 5«"(1 - J""-""). (2 We have only to show that lim d^ = 0. (3 But lim (a„ - «„) = 0. Hence, by 160, lim (1 - 5»»-«'') = 0. (4 Hence, 2), 4) give 3). 163. We are now in the position to define irrational exponents. Let , . /I = (7-1, 7-2, •••) be a representation of fi. We say a*" = lim a''". (1 By 161, the limit on the right of 1) exists ; by 162, it is the same whatever representation of /x is taken. 164. 1. Let r^, r^, ••• be a sequence of rational numbers having a rational limit r. Then, if 5 > 0, lim b''^ = b\ (1 In fact, the sequence r'l, r'a, r'3, •••; 'r'^ = r, w=l, 2, •• has r as limit. ^ ' hm 5% = hm h'n But lim h^n = lim b^ = b'^. (2 This in 2) gives 1). 2. The object of 1 is to show that the definition of a*^ given in 163 does not conflict with that given in 150, 151, in case /m is rational. IRRATIONAL EXPONENTS 105 165. 1. Let fi be an arbitrary number, and r, s, two rational numbers, such that r < /u, < s. Then for 5 > 0, b^'^b^^ b% according as b^l. (1 For, let fi = (mj, m^, •••), the m, s being rational. Then, by 105, 1, r < m„ v. Hence, by 157, b'-l. Then passing to the limit, by 106, 1, b''< b\ (2 Here another the equality sign must be suppressed, rational number such that For, let r' be Then, But r ; then a*^ vanishes for no value of fi. In fact, the relation 1 shows that a'^ always lies between two positive numbers, by 153. 166. 1. The properties given in the preceding articles for rational exponents hold for irrational exponents also. We illus- trate the demonstration in a few cases. Let X < yu., and 5 > 0. Then 5'^ ^6% according as b^l. (1 106 EXPONENTIALS AND LOGARITHMS To fix the ideas, suppose 6 > 1. Let r be a rational number, such that \ 1, we conclude from that whereas, if 0 0, and 6 > ; then b'^ = 1, according as 6 = 1. 167. *""=f- *>0- For, let Then — « = (— ffj, — ^2, "Oi by 71, 3. Since, by 161, we have 6-»= lim6-''™ = lim 5"" 6*' since 5* :^ 0, by 165, 2. IRRATIONAL EXPONENTS 107 168. If a>0. This is the addition theorem for any exponents, and is a generali- zation of 152. Let Then Hence \ = lim X„, ^l = lim /a„. a^ = lira a-^", a*^ = lim a*^". a^a'^ = lim a^^^ lim a*^" = lim a^"**^" = lim a^n+f*", by 152, 169. iei Xj, Xg, ••• 5g a sequence whose limit is + go, lfa>0, + CO, ?/ a>l, lim a^n= -, 0, ?/ « < 1. For, let a>l. Let ^„ be the greatest integer in X„. Since lim X„ = + oo, lim ^„ = + oo. Then, by 140, lim a'n = + 00. As lim a'^'i = + 00, by 138. Let a< 1. Set h. = -. Then 6 The demonstration follows now at once. 170. Let aj, a^, ••• he a sequence of positive numbers whose limit is 1. Then lim a„^ = 1. 108 EXPONENTIALS AND LOGARITHMS Let r, s be rational numbers, such that r<\ 1 ; O an < a/ < <^ «n < 1- (2 Let us apply now 107. Since, by 158, lim ttjf = lim a,/ = 1, we have, from 1), 2), lim a/ = 1. 171. Let aj, a^, ••• be a sequence of positive numbers tvhose limit is a > 0. Then lim a^^ = «^. (1 Since, by 170, lim^=l, a 1) follows from 2) at once. 172. Let \j, Xj' •■* ^^ ^ sequence whose limit is X. 7/ a > 0, lim a^« = a^. (1 For, let ^11 ^2' '"•> ^1' ^2' °" be two sequences of rational numbers whose limits are X, and such that '>'n 1 ; then, by 166, a^'n < a'^n < a'n. (2 By 162, lim al'n z= lim a^n. The application of 107 to 2) gives 1). The case that a ^ 1 is now readily treated. LOGARITHMS 109 Logarithms 173. Let a, 5 > 0, and b^l. The equation h' = a (1 has one, and only one, solution. To fix the ideas, let 6 > 1. We form a partition ((7, i)) in which C contains all numbers c, such that h'a. This separation of the numbers of 9? into the classes C, D is indeed a partition. For every number of C is < any number of B. In fact, from ^^ ^ ^,^ follows, by 166, 2, Let ^ be the number which generates (C, i)) ; let ^l>^2 >... be the monotone sequences of 130, whose common limit is ^. Tlien ,, o« = a. (2 For, by 171, t 7. i- 7^ 7t xo On the other hand, , ,v b'n<:a< ¥n. (4 From 3), 4) we have, by 106, 2, lim ¥n = a. (6 From 3), 5) we have 2). The equation 2) shows that | is a solution 1). Let r) be also a solution, so that 5'' = a. (6 From 2), 6) we have b^ = b\ Hence, from 166, 2, 110 EXPONENTIALS AND LOGARITHMS 174. 1. As we have just seen, the equation 1^^= a, a>0; 6>0 and =#= 1, admits one, and only one, solution. This uniquely determined number |, we call the logarithm of a, the base being b; and write I = logj a, or when we do not care to indicate the base, I = log a. 2. We shall suppose, once for all, that the base b is ^ 1 ; also that the numbers whose logarithms we are considering are > 0. 3. From log u = log w, follows u = v. The demonstration is obvious. 175. log ab = log a + log b. This is the addition theorem of logarithms. Let the base be c. If (t = log a, ^— log b, then c« = a, c^ = b. Multiplying, we have c'^c^ = c'^+P = ab. From the equation c"+^ = ab, we have log a6 = a + /3 = log a + log b. 176. By using the properties of exponentials we may deduce in a similar manner all the ordinary properties of logarithms. As this presents nothing of interest, we pass on. We note, however, the following important relation. Let a > 0, and b be the base of our logarithms. Then LOGARITHMS 111 For, by definition, jioga^^^^^ (2 But . , log a>^ = II log a. This in 2) gives 1). 177. Let aj, a^, •" he a sequence whose limit is 1. Then lim log a„ — 0. (1 To fix the ideas, let 5, the base of our logarithms, be >1. Let€>0, then, by 166, 3, b^>l. (2 Hence g=l-->0. (3 Since , . ^ hm a„ = 1, we have _ ^ _ ^ c. 6 > 0, m, — 6 < a„ — 1 < 0, n>m; which gives -, c^ -. c^ ^a ^ 1 - S < a„ < 1 + S. (4 From 3) we have ., 1-8 = 1. b' This in 4) gives ^ On the other hand, «-,>- (5 by 3), 2). This gives l + 8<6.. (6 Then 6) and 4) give ^_^^j._ ^^ From 5), 7) we have finally This may be written, by 176, 1), b-'<¥°^"''m, which is another form of 1). 112 EXPONENTIALS AND LOGARITHMS 178. Let Vun a^ = tt>0. Then lirn log a„ = log a. a„>0. (1 For, Hence As lim^=l, we need only to apply 177 to 2) to get 1). 179. Jjet a^a^ -•• be a sequence whose limit is -{- ao . Then limlog,a,= | Q .^^^^^ Let h>l. Let m^ be an integer, such that h"'n w„. (1 But lim m„ = + Qo, since lim a„ = + oo. Hence, by 138, using 1), lim log an= + 00. The case that 5 < 1 follows at once now. Some Theorems on Limits 180. Let A = a J, «2' "* ^^ any sequence^ such however that its limit is ±30 when it is 7iot limited; let B—h^^ h^,'--he an increasing seqxience tvhose limit is -{- co. If is finite or infinite, then lim p = I. (2 On SOME THEOREMS ON LIMITS 118 Proof, r. I finite. Set bn+i-b„ '" b„^p-b^ From 1) we have : S > 0, m, \l — q„\<8, n^m. Hence To these inequalities apply 93, setting the 7's equal 1. Then or qm,p-BQ being small at pleasure, and m fixed, we have, by 92, 1 K. S>0^ i>o> ^ = 1 + Si; \8^\<8, p>p,. (4 ''m+p U4 EXPONENTIALS AND LOGARITHMS r a m+p Now Also, by 3), Qm,p ^m+p\ -*- 7 From 5) we get, using 4), 6), m+p^ 3'i0, w, f^>^. n>m. Hence Applying 93, we get qm,p>9- ^ = 1,2,... This shows that lim a„ = + 00. Then 5) shows that, taking 7} such that < 77 < 1, 9'»»,P = ^(1+V); |V|<^,i'>i?o' by 92. Hence ^m.+p qm,p T> 'm+p 1 + 77' 1 + 77 (5 (6 .0 SOME THEOREMS ON LIMITS 115 If we take where Cr is arbitrarily large, we have -=^>Gr. n>m+p. o„ n This proves 2) for this case. 181. Let aj, a^^ ■■■ be a sequence whose limit, finite or infinite, is a. Then lim ^i + ^^ + -- +^- = a. n Let A = «! + ••• + ««• Then Hence, by 180, lim -^ = a. 182. Let aj, ag' •" ^^ ^ sequence whose terms are positive, and whose limit, finite or infinite, is « > 0. Then ^^ \\va.^a^a^---a^= a. (1 1°. a finite. Consider the auxiliary sequence log a,, loga„, •••; base>l. By 178, lim log a„ = log a. (2 By 181 and 2), lim - (log a.-\ h log a«) = log a. (3 n But -(log «!+•••+ log a„)=l0g^«l«2-"^«- (4 n From 3), 4) we have 1). 2°. a infinite. To fix the ideas, let a = + oo. By 179, hm log a„ = + 00. 116 EXPONENTIALS AND LOGARITHMS By 181, Thus lim log Vai«2 •••(«« = «= + CO. lim Vaj •••«„= + 00. Hence 1) is true in tliis case. 183. Let aj, a^, ••• he a sequence of positive numbers. Let Then For, lim — ^ = « ^ ; finite or infinite. Inn V«,, = «. " » /^ /7 /» '■ Apply now 182. (^n-X ^n-'l "-1 184. Let a J, a^., •■• he a sequence whose limit is 0. Let 5j, 621 ■•• he a decreasing sequence whose limit is 0. Let linir^^: 7^^^ = I ; finite or infinite. Then hn - ^«-l 1°. I finite. As in 180, we have e>0, m'. Since by 92, ^m ^m+p "m "rii+p m lim '"■ "m+p ^TO p=" 6, we have ^m+p ^1)1 €' ^0' a^, — a m "^nH-j> ^«i ^m ^m+p ^m € <2' P>?o- Adding 2), 3), we get (1 (2 (3 -I < e ; w > ?n'. SOME THEOREMS ON LIMITS 117 2°. I iyifinite. Let Z = + oo Then «.„,. — a a>0, m', 5„,^=-^!^L_^>^. ^ = 1,2,... m>m'. TO ^m+p But for sufficiently large jd, Hence Hence ^.«,p = |^.+ Si; |Si|g\ m>m'. and ^ is large at pleasure, since G is. 185. EXAMPLES J lim logw _ Q For, log w- log (n -1) ^ j^g n ^ jj^ 71 — (?i — 1) ° n — 1 2. lime»^^^_ For, fLzurL= en (l -'-]=+ ^ I— (n — 1) \ ey 3. li™\/n=l. For, = 1. n-1 4. li™ v/n ! = + 00. For, •TS C. X ^ < -i w-o o2 W o DO 3^ CHAPTER IV THE ELEMENTARY FUNCTIONS. NOTION OF A FUNCTION IN GENERAL FUNCTIONS OF ONE VARIABLE Definitions 186. The functions of elementary mathematics are the following : Integral rational functions. Exponential functions. Rational functions. Inverse circular functions. Algebraic functions. Logarithmic functions. Circular functions. The reader is already familiar with the simpler properties of these functions, which we may call the elementary functions. We wish, however, to restate some of them for the sake of clearness. 187. In applied mathematics we deal with a great variety of quantities, as length, area, mass, time, energy, electromotive force, entropy, etc. In a given problem some of these quantities vary, others are fixed or constant. The measures of these quantities are numbers. In certain parts of pure mathematics we study the relations between certain sets of numbers without reference to any physical or geometrical quantities they may measure. In either case we find it convenient to employ certain letters or symbols to which we assign one or more numbers, or as we say, numerical values. A symbol which has only one value in a given problem is a constant. A symbol which takes on more than one value, in general an infinity of values, is a variable. 118 DEFINITIONS 119 188. The set of values a variable takes on is called the domain of the variable. It is often convenient to represent the values of a variable by points on a right line called the axis of the variable, as explained in 123. The domain of a variable may embrace all the numbers in 9?, or, as is more often the case, only a part of these numbers. Very frequently the domain is, speaking geometrically, an interval ; i.e. the variable x takes on all values satisfying the relation a one or more values to ?/. We say i/ is a function of x, and write ^=/(2;)» or ^ = (a^)' etc. If 1/ has only one value assigned to it for each value of x in D, we say ?/ is a one-valued function, otherwise y is many-valued. The variable x is called the independent variable or argument; y is called the dependent variable. 120 ELEMENTARY FUNCTIONS. NOTION OF A FUNCTION We must note, however, that y may be a constant. The domain of the independent variable x which enters in tlie law defining a function f{x) is also called the domain of definition of the function. The above very general definition of a function is due to Dij'ichlet. 190. The reader is already familiar with the graphical repre- sentation of a function, by the aid of two rectangular axes. Let ^, . be a given function whose domain of defini- tion call D. The graphical representation of i) is a set of points on the x-axis. Let a be a value of x to which corresponds the value b of y. The point P in the figure whose coordinates are a, h represents the value of the function for x = a. As x runs over the values of its domain 2), the point P runs over a set of points, which we call the graph of fix). 191. 1. Another representation of a function is the following: We take two axes as in the figure ; one for x, one for y. x 1 In this representation, the graph of /(a;) is a set of points i© J^ on the 3/-axis. 2. The reader will observe this important difference between the two modes of representation just given. In the first we know for each point P of the graph the corresponding values of both x and y. In the second mode of representation, we do not know in general the value of x corresponding to a point P of the graph, and conversely. In spite of this deficiency, we shall find that this second representation is extremely useful. This is especially the case when we come to consider functions of n variables. INTEGRAL RATIONAL FUNCTIONS 121 192. Ex. 1. Let D be given by 04 cannot, in general, be solved by the extraction of roots, or, as we say, do not admit of an algebraic solution. 200. All fuiictions which are not algebraic functions are called transcendental functions. The terms algebraic and transcendental may also be applied to the numbers of 9?. Any number a which satisfies an equation of the type a;" + a^x""-^ + a^x"'"^ -\ \- a^_-^x + a„ = 0, (1 where w is a positive integer, and the a's are rational numbers, is called an algebraic number. All other numbers of 9^ are transcen- dental numbers. When w = 1, the equation 1) defines a rational number ; the rational numbers are special cases of algebraic numbers. Circular Functions 201. As the reader already knows, the circular functions may be defined as the lengths of certain lines connected with a circle of unit radius. Thus, in the figure sin x = CE, cos x — OE, tan x = AB, etc. We have shown in Chapter II how the rectilinear segments AB, CE, etc., are meas- ured. It has not yet been shown, however, how to measure arcs of a circle, i.e. how to each arc as AC, a number x may be attached, as its measure. 126 ELEMENTARY FUNCTIONS. NOTION OF A FUNCTION This will be given later. No inconvenience can arise if we assume here a knowledge of this theory inasmuch as the reader is perfectly conversant with its results, which are all we need for the present. Arcs measured in the direction of the arrow are positive ; those measured in the opposite direction are negative. If we suppose the point to move around the circle in a positive direction starting from a fixed point A as point of reference, it has described an arc whose measure is 2 tt, 7r= 3.14159265... when it reaches A again. If it still continues moving around the circle, it has described an arc = 4 tt when it reaches A for the second time. On reaching A for the third time the arc described is 6 TT, etc. Thus to each positive number in 9? corresponds an arc; also, conversely, to each arc measured in the direction of the arrow corresponds a positive number in 9?. With arcs measured in the negative direction are associated the negative numbers of 9?, and conversely. 202. From this mode of defining the circular functions we con- clude at once the following properties : The domain of definition of sin a;, cos x embraces all the numbers of 9ff. The domain of definition of tan x embraces all numbers of 9? except 5 + ^^' (1 where m = 0, ±1, ±2, •••. In fact, for these arcs, the secant OB is parallel to the tangent line AB, and therefore cuts off no segment on it. Thus for these values of the argument a;, tan x is not defined. Similarly, sec x is not defined for these same values 1) ; while cosec X is not defined for X =17117. w= 0, ± 1, ± 2, ••• (2 From similar triangles we have for all x, except these singular values in 1) oy 2), sin X 1 1 tana:= -, seca;= , cosec 2; = cos z cos X sin x CIRCULAR FUNCTIONS 127 We observe that these relations involve division by 0, for the singular values 1) or 2). From the above definition of the circular functions we see that they are one-valued functions of x. 203. The graphs of the three principal functions sin a:, cos a;, tana;, are given below. \^ A y J A J A X .Z IT Air -IT -Y/ 2 \ f tanx Sir 2 204. The next most important property of the circular function is their periodicity . In general we define thus : Let &) be a constant t^O. Let f(x) be a one-valued function whose domain of definition D is such that, if x is any point of 2), so is -, r. x+mo), m=±l, ±2, ••• ^^ /(^ + «)=/(^) (1 for every x in D, we say /(a;) is periodic, and admits the period to. If 6) is a period of /(a;), so is mco. m = ±l, ±2, ... For, fCx + 2 to) =/[(a; + a,) + «] =f{x + a,) =/(a:), ^ hence 2 co is a, period. Similarly, 3©, 4 to, ... are periods. Qn the other hand, /(a; = 0)) = /((a; - «) + a>) = /(a:). 128 ELEMENTARY FUNCTIONS. NOTION OF A FUNCTION Hence f(x) admits the period — to, and so —2 ft), —3 ft), etc. If all the periods that f{x) admits are multiples of a certain period S, this is the primitive period of /(a;), or the period oi f(x). From trigonometry we have : The period of sin x^ cos a;, is 2 tt ; the period of tan x is ir. 205. 1. If f{x)^ g(x) admit the period &), then f(ix-)±g(ix\ (1 fix-)g and let r be any period of g(x). Then g{x + T)=g(x^, '''' /«a: + T)) =/(«:.). This gives, setting _ /0 + aT)=/(0. Thus ar is a period of f(x) ; and therefore ar = mat. m an integer. Hence T = m— . (3 a As - is obviously a period of ^(a;), it is *Ae period of g(x), by 3), CIRCULAR FUNCTIONS 129 3. Suppose in 1, that &>, instead of being any period of / and g^ is the period of these functions. It is important to note that we cannot infer that therefore a> is the period of the functions in 1), 2). Ex. 1. Let fix) = sin X, g(x) = 4 sin x cos'^ x. The period of these functions is 2 ir. Yet the period of h(x) = g{x) —f(x) = sin 3 a; is 2^ 3 Ex. 2. Let f{x) — sin x, g(x) = cos x. Then h{x) ~f{x)g{x) = i sin 2 x. The period of / and gr is 2 tt ; the period of h is ir. Ex. 3. Let /(x), ry)= z. — =^» 1 — tan X tan y 208. 1. Let fipc) be a one-valued function whose domain of definition, i>, is such that if x is any point of D, so is — x. Let /(-a:)=/(a;), for every x in D. We say, then, that /(a;) is an even function. If we sa,jf(x') is an odd function. Obviously, The functions sin x, tan x are odd, while cos x is even. 2. Letting 0, 0^, 0^ represent odd functions, and E, E^, E^ even functions, we have: 0±0^ = 0,^, E±E, = E^, 0'0^ = E^^, EE^ = E^, 0'E=0^, For example : ONE-VALUED INVERSE FUNCTIONS 131 , The Exponential Functions 209. Let a > be a constant ; the exponential functions are defined by y = w'. The domain of definition of y is 9^?, and ^ is a one -valued function. When a = 1, the corresponding exponential function reduces to a constant, viz. : y = l. The graphs of y fall into two classes, according as a^l. y An important exponential function is that corresponding to ^^^' e = 2.71818 - 210. 1. The only properties of the exponential functions which we care to note now are the following : The exponential function is noivhere equal to 0, or any negative number. See 165, 2. 2. The addition theorem is expressed by One-valued Inverse Functions 211. 1. The two remaining classes of functions, viz. the logarith- mic and inverse circular functions, are inverse functions. Before considering them, we wish to develop the notion of inverse func- tions in general. 132 ELEMENTARY PUXCTIONS. NOTION OF A FUNCTION 2. Let /(a;) be a one-valued function defined over a domain D.* If fix")>f(x') for every pair of values x" > x' in i), we say f(j>c) is an increasipg function in D. If on the contrary f(x")a/, we say/(a:) is a decreasing function. If/ is either an increasing or a decreasing function in 7), but we do not care to specif}^ which, we say it is univariant. These definitions are extensions of those given in 108. Tlie corresponding extension of the terms monotone, monotone increas- ing, monotone decreasing to function is obvious. 3. Ex. 1. For the domain D = ( — - , ~ ] , sin a^ is an increasing function. For the domain Z) = [ -, — ^ ), si" a; is a decreasing function. Ex. 2. For the domain 3 = ^^,0^ is an increasing function if a > 1 ; it is a de- creasing function if a < 1. Thus whether a^\, a^ is a univariant function in 1R. 212. Let „ , . ^. y=f(.x) (1 be a one-valued univariant function, defined over a domain D. Let E be the domain over which the variable y ranges. We put the points of D and E in correspondence with each other as follows : two points x, y shall correspond to each other, or be associated, when they satisfy 1). Then to a given x corresponds only one y, since /(a:) is one- valued. On the other hand, to a given y corresponds only one x, since /(a;) is univariant. Thus to any x oi D corresponds one, and only one, y oi E -, conversely, to any y oi E corresponds one, and only one, x of D. 213. The considerations of the last article have led us to one of the most important notions of modern mathematics, that of correspondence. * Such an expression as this will be constantly employed in the future. It does not mean that D includes all the values for which /(a;) may be defined, but only such values as one chooses to consider for the moment. ONE-VALUED INVERSE FUNCTIONS 133 Let A and B be two sets of objects. Let us suppose that A and B stand in such a relation to each other, that to any object a oi A correspond certain objects 6, b', b", ••• of B; and to any object b oi B correspond certain objects a, a', «",••• of A. Then A and B are said to be in correspondence. If to each a corresponds only one b, and conversely, the corre- spondence is one to one (1 to 1), or uniform. If to each a correspond m objects of B, and to each b correspond n objects of A^ the correspondence is m to n. In many cases, to each element of A correspond an infinity of objects of B, or conversely. 214. Let us return to 212. The correspondence we established between the points of J) and ^ is uniform. This fact may be used to define a one-valued function ^(^), over the domain U. In fact, let X correspond to i/. Then ^(y) shall have the value x, at the point y. Then ^ . The function (/, just defined, is called the inverse function of f. Evidently - it x' < x'\ y' x" . We cannot have x' = x" ; for then y'=y'\ which contradicts 1). We cannot have x' > x" ; for then y' > y" , which again con- tradicts 1). We haA^e thus the theorem : Let y =^ f{x) be a one-valued univariant function., defined over a domain B. Bet E be the domain of the variable y. Then the inverse function., x = g(^y), is one-valued and univariant in B. 134 ELEMENTARY FUNCTIONS. NOTION OF A FUNCTION 215. The notion of inverse functions developed in 212 and 214 is quite general. It will perhaps assist the reader if we take a very simple case. For the domain D let us ^ take an interval /= (a, J). For /(a;), let us take an increasing function, with graph as in the figure. The domain of y is then the in- terval J=(a^ yS). That the correspondence between the points of I and J, as defined in 212, is uniform, is seen here at once. For, to find the points y corresponding to a given a?, we erect the ordinate at X. This cuts the graph but once, viz. at P. There is thus but one point y hi J corresponding to the point x in /. Similarly, to find the points x^ corresponding to a given y, we draw the abscissa through y. This cuts the graph but once, viz. at P. There is thus but one point x in /corresponding to a given y in J. That the inverse function is one-valued, and is an increasing function in J, is at once evident from the figure. The Logarithmic Functions 216. 1. We saw in 209 that the exponential functions y= a^, a>0, T^l are one-valued univariant functions for the domain '?R.. The domain of the variable y is the interval 1= (0*, + oo). See 188. Then, by 214, the inverse of the exponential functions are one- valued univariant functions, defined over /. By 174, these inverse functions are a: = log„^, (1 and are called logarithmic functions with base a. In higher mathematics it is customary to take a = e= 2.71818 ••• When MANY-VALUED INVERSE FUNCTIONS 135 no ambiguity can arise, we may drop the subscript a in 1). Unless otherAvise stated, we shall suppose the base is e. 2. The graph of the logarithmic function y=\ogx is given in the figure. 3. The only other property of log a; which we wish now to mention is their addition theorem, log xy = log X + log y. Many-valued Inverse Functions 217. The circular functions give rise to many-valued inverse functions. It is easy to extend the considerations of 212 and 214 so as to arrive at the notion of many-valued inverse functions in all its generality. Let 2/=/(^) (1 be a one or many valued function, defined over a domain D. Let the domain of the variable y be E. We put the points of D and E in correspondence as follows : two points a;, y shall correspond to each other or be associated when they satisfy 1). Then, to each y oi E correspond one or more values of a;, say (2 We define now a function y(^y} over E by assigning to y the values 2) of x associated with each point y of E. Then ^ = 9W (3 is the inverse function, defined by 1). The equation 3) may be considered as the solution of 1) with respect to x. 136 ELEMENTARY FUNCTIONS. NOTION OF A FUNCTION /s y j^ — - — n y -K"fe a "TT" i" i i : « ax X' X" X"'b 218. To illustrate the rather abstract considerations of the last article, let us consider the following simple case, from a geometric standpoint. Let the graph of y =/(^) be that in the figure. Then D = (a, 5), and U= («, yS). The greatest number of values of y for a given x in i> is 3. Hence ^ is a three-valued function. Let y be a point of -27. To find the points of D associated with it, we draw the abscissa through ?/. Let it cut the curve in the points P, P', P", ••• The projections x, x\ x", ■■• of these points P on the a;-axis are the points sought. The greatest number of values x corresponding to any y ot E is 4. Hence the inverse function is a four-valued function. ^=9iy) 219. Let us consider the function y=f(x) defined by or y Its graph, given in the figure, is a hyperbola. To a value of x>\^ or x< — \^ correspond two values of y, marked y and y' in the figure. The domain 2> of a: is marked heavy in the figure, and embraces all the points of the a;-axis, except (— 1*, 1*). The domain E of y is the whole ^-axis. To any point y oi E correspond two values of x, falling in D. The inverse function x = g{if)^ thus defined, is a solution of 1) or 2) with respect to a;, viz. : x = ± Vl + y"^. ■ The correspondence which the equations 1) or 2) establish between the points of D and ^ is a 2 to 2 correspondence. THE INVERSE CIRCULAR FUNCTIONS 137 220. The preceding example illustrates the fact that The inverse of an algebraic function which is not a constant is an algebraic function. To prove this theorem, let y=f(x^ be defined by P,(x)y- +P,(x)r~' + - +^n(^)= 0, (1 where the P's are polynomials in x, with constant coefficients. The inverse function ^=9i:y) (2 also satisfies 1). Let us arrange 1) with respect to x. If m is the highest degree of x in this equation, we get ^o(i/)^™ + ^i(y)^™-' + - + Qmiy^ = 0. (3 As 2) satisfies 3), the inverse function 2) is an algebraic func- tion also. In this example, ?/=/"(.r) is, in general^ an w- valued function, while x = g{jy) is an w-valued function. The correspondence that the equation 1) or 3) establishes be- tween the points of D and ^, the domains of the variables a;, y, is thus an n to m correspondence. The Inverse Circular Functions 221. These are the functions sin~^a;, cos"^a;, tan^^a;, etc. We prefer to follow continental usage, and denote them respec- tively by Arc sin a:, Arc cos a;, Arctga;, etc. We shall not take the space needed to treat all these functions ; we take one of them. Arc sin, as an illustration. The others may be treated in the same way. We start with the equation y — sin a;, (1 whose graph is given in 203. 138 ELEMENTARY FUNCTIONS. NOTION OF A FUNCTION The domain i), over which sin x is defined, is 9? ; the domain of Let ?/ be a point of M. If x^ is one of the associated points of Z>, all the points of D associated with y are given by Xq+ 2 mir, (2 m=0, ±1, ±2 ... TT— Xq-\-2 WITT, (3 as is shown in trigonometry. Thus, to a given value of y there are a double infinity of values of X. The inverse function defined by 1), viz, : X = Arc sin y, has the interval ^ = (— 1, 1) for its domain of definition. It is an infinite-valued function whose values for a given y are given in 2), 3). 222. The graph of y = Arc sin a; is given in the adjoining figure. The reader will observe that this graph can be got at once from the graph of sin x (see 203) by turning it around and changing the axes. This property is obviously true of the graph of any inverse function. Thus, if the graphs of e*, cos a;, tana;, etc., are given, we may get at once the graphs of log X, Arc cos X, Arc tg x, etc. 223. The treatment of many- valued functions is much simplified by employing the notion of a branch of the function. This will be explained when we have considered the notion of continuity. For the present, however, we wish to define what are called the principal branches of the inverse circular functions. I THE RATIONAL AND ALGEBRAIC FUNCTIONS 139 Looking at the graph of Arc sin x given in 222, we see we can define a one-valued function over the interval (—1, 1) by taking those values of Arc sin x which fall in the interval /_7r 7r\ l~2' 2J' The function so defined is called the principal branch of the Arcsin futiction. We shall denote it by arc sin x. Its graph is given in the adjoining figure. 224. 1. The principal branch of Arc cos x is formed of those values of this function which fall in the interval (0, tt). The one-valued function so defined over the interval (—1, 1) is denoted by arc cos x. Its graph is given in Fig. 1. 2. The princii^al branch of Arc tg x is formed of those values of this function which fall in the interval ( — 77, -r ) • The one- valued function so defined over ( — 00, oo) is denoted by arc tg X. +1 arc cos x Fig. 2. Its graph is given in Fig. 2. IT 2" 00 /-"^"^ 00 _7r T arc tg z FUNCTIONS OF SEVERAL VARIABLES The Rational and Algebraic Functions 225. In the list of the elementary functions given in 186, the first three, viz. the integral rational, the rational, and the alge- braic functions, are, in general, functions of several variables. 140 ELEMENTARY FUNCTIONS. NOTION OF A FUNCTION For simplicity, we treated them first as functions of a single varia- ble. We wish now to define them in all their generality. At the same time we shall consider the general notion of functions of several variables and certain related geometric ideas. 226. 1. An integral rational function of n variables x^, x^ ••• Xn is an expression of the type y = Ax^^x^-i • • ■ a;„'"" + Bx^^x^^ • ■ ■ xjn + • • • + Lx^-^-x^-^ ■ ■ • a;„*». (1 Here AyB^-L are constants, and the exponents m, Z, • • • e are positive integers or 0. Such functions are ax^x^x^ + bx^x^ + cx^ + dx^'x^Xy (3 We may write 1) in the form where the summation extends over all the terms of y. A still shorter notation is y = ^Ax{"^x^''^- ■ ■ ■ a:„,'"« (5 which may be employed when no ambiguity can arise. The greatest of all the sums of the exponents m^ + 7n^-\ h nin, li + l^'^ 1- ^« ••' is the degree of y. Thus the degree of 2) is 3 ; the degree of 3) is 13. 2. When the degree of each term of 1) is the same, it is said to be homogeneous. F=^Ax{''x^'"'---x,,'^n he homogeneous and of degree m. If in F we replace x^ by \x^, ••• x„ by Xx„, and denote the result by F, we have F= X^'F. For, _ F= '2A\"''^-^"'"x^"'' ••• x/^n. THE RATIONAL AND ALGEBRAIC FUNCTIONS 141 But, for all the terms of F, *^i + • • • + wz„ = Wi. Hence _ F= V^Ax{^t ••• x^"n = \'"F. 3. When 3/ is of degree 1, we have y = aj^x^ 4- ^2X2 H 1- a„x„ + a^. It is said to be a linear integral function of the rr's. If aQ= 0, it becomes ^ = a^X^ + • • • + CLnPCiii which is the general type of a linear homogeneous integral function of the x's,. In algebra, integral rational functions are called polynomials. 227. 1. To get a value of y = l.Ax{''---x,,"'n (1 we give to each of the variables x a certain numerical value, as, X-^ = a^i ^2 ^^ '^2^ " "^w ^^ ^re* \ These values put in 1} give the corresponding value of ?/, say y = h. When w = 1, 2, 3, we can represent geometrically the values 2) by a point on a right line, a point in a plane, or a point in space, respectively, viz. the point a whose coordinates are a^, or a^, a^, or a^, a2, a^ If we give the a;'s different sets of values, we get different points in 1, 2, or 3 dimensional space. As in the case of one variable, we can say y has the value h at the point a. 2. It is convenient to extend these and other geometric terms, employed when the number of variables n = 1, 2, 3, to the case when w>3. Thus any complex of n numbers, a^, a^., ••• a„, is called a point; a^, a^., ••• are called its coordinates. We denote the point by _ ^ . . The aggregate of all possible points, the a;'s running over all the numbers in 9?, we call an n-dimensional space or an n-way space; and denote it by 9?^. Later we shall extend the terms distance, sphere, cube, etc., to 9?„. Cf. 244. The reader is not to 142 ELEMENTARY FUNCTIONS. NOTION OF A FUNCTION suppose for a moment that there really is an 7i-dimensional space, or an w-dimensional cube, in the ordinary empirical sense of the word ; but to bear in mind that these terms are merely names for certain numerical aggregates. 3. Employing this geometrical language, we may say that, The integral rational function of several variables, say n variables, is a one-valued function whose domain of definition embraces all the points of 9t„. 228. As in the case of one variable, the rational function of several variables is the quotient of two integral rational functions in these variables. Its general expression is, therefore, Its domain of definition embraces all the points of 9^?^, except those points at ivhich Gr vanishes, which we call poles of R. For all points of this domain, R is a one -valued function. If m' is the degree of F, and m" is that of Gr, the degree of R is the greater of the two integers m' , m" . When the degree of R is 1, it is called a linear rational function. Its general expression is flfq -j- a^x^ -|- . . . -|- a^Xj^ _ ,^2 We say R is homogeneous when F and Gr are homogeneous. We have evidently, as in 226, 2, R(\X^, \X^, ••• \x^ = \^R(x^ '" Xn)j (3 where t is an integer, positive, negative, or zero. 229. The definition of an algebraic function of n variables is an obvious extension of that given for one variable, in 197. Thus g is an algebraic function of x^, x^, ••• x^, when it satisfies an equa- tion of the type g- + i^ir "' + • • • + ^n-iy + i^. = 0, (1 where the coefficients R are rational functions of x^-^-x^, and n is a positive integer. FUNCTIONS OF SEVERAL VARIABLES IN GENERAL 143 For any point x = a in Q^t^, for which none of the denominators of the It's vanish, 3/ has at most n values. 27tus y is at most an n-valued function. Its domain of definition embraces all points of 9fJ„ except the poles of the coefficients H, and those points for which i) has no real root. Functions of Several Variables in General 230. We can give now the definition of a function in n vari- ables. Let x = (^x^, x^, ••• Xn) range over the points of a certain domain i), viz. over 9?„ or a part of it. Let a law be given which assigns to y one or more values for each point of D. We say 1/ is a function of x^, x^, •■• x^, and write ^=/(^i"-^n> or y = , ?/ will ordinarily trace out a surface in D^g. 233. The other mode of representation is by means of a plane and an axis. The domain of the independent variables we represent by points in the x-j^x^ plane, wdiile ^ is represented by points laid off on a separate axis, as in the figure. 234. When n = 3, we may employ the following representation. Let i/ = /(x^x^x^) be defined over a domain D. To represent D, we take three rec- tangular axes. To the set of values ',a Xn Ctn corresponds the point a, whose coordinates are a^, a^, values of y we lay off on a separate axis, as in the figure. The 235. From the elementary functions of one variable we can build an infinity of functions of several variables. We give some examples which illustrate the various domains of definition that a function of several variables may have. We shall take w = 2. COMPOSITE FUNCTIONS 145 Ex. 1. For points within the ellipse E^ whose equation is 22 ^'-1 = 0, x" y the argument of 2 is negative. For points on E the argument is 0. As the logarithmic function is defined only for positive values of the argument, the domain of definition i), of 3, is the region shaded in the figure. Its edge, or E^ does not belong to B. Cs— 1 ) = log- uv. 236. Ex. 2. Since log uv is not defined, unless mu > 0, m and v must be both positive, or both negative. The domain of definition i>, of 2, is thus the region shaded in the figure. Since Mz; = on the edge of i>, these points do not belong to D. 237. Ex. 3. z = tan \ TTxy. Since tan u is not defined when u = — h niTT, 2 m = 0, ±1, ±2, ... we see the domain of definition of z includes all the points of the xy plane, except a family of hyperbolas ^ ^ "^ ^ xy = lm + 1. Composite Functions 238. 1. An extremely useful notion in many investigations is that of -A function of functions^ or composite functions. Let 146 ELEMENTARY FUNCTIONS. NOTION OF A FUNCTION be defined over a domain X in w-dimensional space 9?„. Let be a point in an w-dimensional space ^^. While X runs over JT, let u run over a domain U. Let y = 4>(u^'--Um^ (1 be defined over JJ. Then y is defined for every point x in X. We may, therefore, consider y as a function of the x's through the w's. We say ?/ is a function of functions, or a composite function. 2. When speaking of composite functions, we shall always sup- pose, even without further mention, that the domain of definition of 1) is at least as great as U. 3. When x ranges over X, u, as we said, runs over the domain U. It is convenient for brevity to call U the image of X. Example. ^ Ml = xiX2-i M2 = sec Xi, Us = e^i. y = log Ml + tan — • Here mi, U2, ms are defined for all the points of 9?2, for which a;i T^ or ^ + rtnr, m = 0, ±1, ±2, — while y is defined for all the points of 9?3, for which Ml ^ 0, and —=^- + mr. n = ± 1, ± 2, .- 239. The notion of a composite function is sometimes useful in transforming a function as follows. Let i/ = F(x^-"xJ. The variable x may enter F in certain combinations, so that if we set y goes over into ^ ^ . Example. Let then M = ^, X2 y = aM2 - l^t-^ + log M = G(u). LIMITED FUNCTIONS 147 Limited Functions 240. Let/(a;j •••rr^) be detined over a domain i). If there exists a positive number M, such that \f\a>0 in D, then ^ 9 is limited in D. Since /, g are limited in D, let \n\9\<^- ^^"'^ i/±^i J5 or ^ < J.. 3. If A does not exist, i.e. if it contains no points, we write ^ = 0. 148 PRELtMlNARY DEFINITIONS 149 As the symbol also stands for the origin, we shall write, in case of ambiguity, A = (0), when we wish to indicate that A consists of the origin alone. The fact that A contains at least one point, we indicate by A>0. 4. Let A, B be two point aggregates having no point in com- mon. The aggregate formed by their reunion is called their swm, and is denoted by , ^ -^ A+ B. 5. If B is a partial aggregate of A, the aggregate formed by removing all the points of B from A is called the difference of A, B. and is denoted by , ^ ■^ A — B. It is also called the complement of B. 6. If a or X, for example, are general symbols for the points oi an aggregate, we can represent the aggregate by \a\ or ]x\. Thus, if ■^ — 1' 2' 31 we can write Or if A = Oi, 02, as,'" we can write A — {«„}. 244. Definitions of configurations in n-way space. Cf. 227, 2. 1. Let a = (aj •••«„), h = (h^---h^') be two points of Qf^n. We say is the distance between a, h ; we denote it by Dist (a, 5) or a, h. 2. The points x satisfying 2^1 - «i = >-(*i - «i) ••• a:,/, - a^ = X(5„ - a„) (2 lie on a rigJit line i, viz. the line determined by the two points a, h. Here X runs over all the numbers of 9?. 150 FIRST NOTIONS CONCERNING POINT AGGREGATES When X = 0, x= a\ when X = 1, x — h. Points x^ for which 0<\<1, form a segment or interval (a, 6) of i. Such points are said to lie between a, h. An aggregate lying on a right line is called rectilinear. 3. If three points a, 5, c lie on a right line, we have from 2) that _i ' = f ^; t, « = 1, 2, ••• n. C^ 0, - «, o« - «« and conversely, if 3) holds, a, 5, c lie on a right line. 4. Let a, 5 be two points on the line i, and r= Dist (a, 5). Then «i - *i . «„ — ^. Aj = • • • A„ — r r are the direction cosines of the line L. Obviously, 5. The points x defined by (a;^_a,)2+...+(a;„-a„)2=r2, r>0. lie ow a sphere 8 whose center is a and whose radius is r. The equation of S may also be written Dist (a, x^ = r. The points a;, such that Dist (a, a;) r, a: lies without S. If a; lies on or within tS, it lies «w /S^. 6. The points x, such that form a CM6e, with center a and edge e. PRELIMINARY DEFINITIONS 151 7. The points a;, such that form a Tectsnigular parallelopiped or cell whose edges are of length 8. The cube kl-«li<~ ••• kn-««|<-^ is called the inscribed cube of the sphere S, of radius r, and center a. Every point a; of C is in S. For, 9. Let the cube (7 be given by The points f 1 = «! ± I o- • • • Vn = a„ ± I fl- are called the vertices. is one vertex, v' = C-v^ + 2a^, v« + 2a„) is called the opposite vertex. The line joining a pair of opposite vertices evidently passes through the center of C. It is called a diagonal. The length of a diagonal is Vo-2 + 1- 0-2 = cr Vw. 10. The distance between two points a, 5 in C is greatest when they are opposite vertices. For, each terra (a^ — b^ in 1) has then its greatest value, viz. a^. 11. If ^j, ^2' "• ^m ^'I'e the lengths of the edges of the parallelo- piped in 6, we say the product gj • ^2 • • • ^rt is its volume. In case the parallelopiped is a cube of edge o-, its volume is o-". 152 FIRST NOTIONS CONCERNING POINT AGGREGATES 12. The points x defined by «i.ri + 1- a,,x„ + (f = (4 lie in a plane. The two planes 1) and a^x-^ + • • • + (in^n + e = are parallel. 245. Let a, 5, c be three points in $R„. Let A = Dist (b, c), B = Dist (a, c), 0= Dist (5, a). When w = 1, 2, 3, we have A^B + C. (1 Here the inequality sign holds unless A, B, C lie on a right line L. We show now that 1) holds for every n.* To this end, set «i = ^i — Cli /3t = ftt — ^o 7i = ^i — «t, where «^, 6, (?^ ; t=l, 2, ••• w, are the coordinates of a, 5, c. J'hen Now, A^ = a^^+--- + «„2 ^ ;s«^2 . C2= 7^2+... +7,2^^7.2. (4 From 2), we have also ^2 = 2(^^ + 7^)2 = v^^2 + ^7,2 + 2 E/3,7,. (5 Thus to prove 1), we liave to show that A^0. (7 By 244, 3, the inequality sign holds for at least one pair of indices t, a:, unless a, 6, c lie on a right line. From 7), we have Let us form all the relations of this type, letting i, k run over the indices 1, 2, -••, w, and keeping L=f^ k. If we add these, we get 2^,V>2 2/3A7c7.- ^^>C' (9 On the other hand, 2yS,2 . S7,2 = (^^2 + . . . + ^^^2) (^^2 + . . . + ^^2) ^ ^^2^^ + 2^,2^,2. Hence, by 9), 2A' • 27.^ ^ 2^^2^^2 + 2 ^/3j3^ry^y^. (10 But '^-^ (SyS.x)' = C/3i7i + • • • + /3„7 J' = 2/9,27;^ + 2 2^^^,7,7,. This in 10) gives 6). 246. A point x for which Dist (a, a;) is small, is said to be near a. What is to be considered as small, depends on the problem in hand. The points x, such that Dist (a, x') < p, p >0. form an aggregate called the domain of the point a, of norm p. It is denoted by t^ ^ -. -n, -. t, •^ Dp(a), D{a), Dp. For example, in 9?j, Dp(^a') is the interval (a — p, a + p')- In 9?2' Dp(^a') embraces all points in a circle of radius p, and center a j in 9^3, it embraces all points in a sphere of radius p. 154 FIRST NOTIONS CONCERNING POINT AGGREGATES We sometimes wish to exclude the point a from its domain. When this is done, the domain is said to be deleted; we denote it by I)*{a) or i)*(a). 247. Let ^ be a point aggregate in 9?^. Let 'p be any point in 9?„. We say 'p is an inner point of A if every point in some domain of p lies in A^ i.e. if there exists a p > such that every point of i^p( jo) lies in A. The point p is an outer point of A if no point of DpCp} li^s in A., however small /> > is taken. Finally, p is a, frontier point of A if in every Dp(^p^)., however small p > is taken, there is at least one point of A and one point not in A. Every point of 9^„ is either an inner, an outer, or a frontier point of A. The frontier points of a cube or parallelopiped form its surface. V-i Pi Pi 248. Ex. 1. A=(a,^). ^ ^ ^ -^ Here any point pi, such that « /3 or j?3 < a, is an outer point. The frontier points are a and /3. Ex. 2. A embraces the rational points in (a, j3). Here all points p, such that j9 < a or p > ^, are outer points. The points of A are all frontier points. For, if a be any point of A, there are irrational points in every Dp{a), however small p> is taken, by 84. In this example A contains no inner points. Ex, 3. A embraces all the points in 9t2, both of whose coordinates are rational. Here every point p of Siz is a frontier point. In fact, consider a little circle G of radius p > and center p. Evidently p contains points in A and points not in A, however small p is taken. In this example there are no outer and no inner points of A. 249. Let h he an inner point of Then A = A(5) lies within S if /tJ + S < 0-, where /3= Dist (a, 5). PRELIMINARY DEFINITIONS 155 The theorem is proved if we show that the points y of A satisfy the relation t>.- ^ x Dist (a, y) < a. (2 But, by 245, Dist (a, y) < Dist (a, V) + Dist (6, y)=p-\-h. Thus, by 1), the relation 2) is valid. 250. 1. Let vl be a point aggregate, and je> any point in 9?„. The points of A^ lying in -Z>p(j9), form the vicinity of p^ of norm p. It is denoted by rr ^ ^ tt-^ x ^ V^p} or V(p). Thus D(p^ embraces all points near ^, while F(j9) includes only points of A, near p. Example. Let A = 1, I, ^, ••• Here Z>p(0) is the interval (— p, p), while Fp(0) is the set of points J_ 1_ 1_ ?w ' m + 1 ' OT + 2 where m is the least integer such that — < />• m The point p may or may not lie in F^jo). We sometimes wish expressly to exclude it. When this is done, the resulting aggre- gate is the deleted vicinity of p; it is denoted by V*{p) or r*(^). 251. When treating functions of a single variable a:, we have often to consider the behavior of the function on one side of a point a. This leads us to split the domain and vicinity of a into two parts, forming a right and left hand domain ; a right and left hand vicinity of a. The right hand domain and vicinity we denote respectively by BI)(a~), RVCa}. The left hand domain and vicinity are denoted by XZ)(a), LV(a). The point a lies in both the right and left hand domain. It lies in both the right and left hand vicinity if a lies in F(a). It should be remembered that these terms refer only to rectilinear aggregates. 156 FIRST NOTIONS CONCERNING POINT AGGREGATES 252. 1. A point aggregate is said to be finite when it contains only a finite number of points. Otherwise it is irifinite. A point aggregate A is said to be limited when all its points lie within a certain sphere or cube, having tlie origin as center. This definition is equivalent to saying that the coordinates a^ a^-, ••• dfii of every point of Jl, are numerically less than some i)Osi- tive number M. It A is not limited, it is said to be unlimited. Obviously : Every finite aggregate is limited. Ex. 1. A = l, 2, 3, ... is an infinite unlioiited aggregate. Ex. 2. A = 1, -h, I, ••• is an infinite limited aggregate. Ex. 3. A = points of the interval («, /3) is an infinite limited aggregate. 2. In the case of a rectilinear aggregate A, it may happen that the coordinates of all its points x are less than some number M. We say A is limited to the right. If the coordinates of all the points x are greater than some number AT, we say A is limited to the left. Ex. 1. ^ = 10, 9, ..., 2, 1, 0, -1, -2, -3, ... is limited to the right. Ex.2. A^-5, -4, -3, -2, -1, 0, 1, 2, 3, ... is limited to the left. 253. 1. It is sometimes convenient to divide an interval into equal subintervals or a square into equal subsquares, and, in general, an wi-dimensional cube F into equal subcubes. For m = 1, 2, 3, this needs no explanation. When m is > 3, the matter is still very simple. The cube a:^ ^^ F is graphically represented by m ^^ ^^ equal segments on the x-^ ••• x^ axes. ^2 ^ — ^ " ^ ' ' We divide F into cubes whose sides -^^ ' ' are 1/wth those of F by dividing ^ '^~ each of these segments into n equal ^•^' ' parts. One of these subcubes is then represented by the points which fall in a set of m segments as a^^^ • ■ ■ aj„^„^. LIMITING POINTS 157 2. Instead of a cube F in 9=?^, we may wish to divide the whole of dt„i into cubes. The meaning of this is now evident. 3. Let A be any point aggregate in di^. Let us divide dim into cubes of side S. This also, in general, divides A into partial aggre- gates. This division of A into partial aggregates we shall call a cubical division of A, of 7iorm h. 4. If instead of dividing 9^^ into cubes, we had divided it into rectangular parallelopipeds whose edges are ^ S, we shall say that we have effected a rectangular division of '^^, of norm h. 5. The partial aggregates, into which A falls after a cubical or rectangular division, may also be called cells. Limiting Points 254. 1. One of the most impoitant notions connected with point aggregates is that of a limiting point. Let ^ be a point aggregate in 9?„j. Any point p of $)i,„ is a limiting point of J., if however small /3>0 is taken, Dp{p^ contains an infinity of points olA. If every domain of p contains at least one other point, p is a limit- ing point of A. For, let a be a point of A different from p. Let 0p(jt?). ^l") ^2' '^3' 2. The following may also be taken as definitions of a limiting point : If Vp{p^ is infinite, hoivever small p is taken, p is a limiting point of A; or, If I^*(j3)>0, however small P *'« taken, p is a limiting point Of A. 158 FIRST NOTIONS CONCERNING POINT AGGREGATES 3. If JO is a limiting point of A and p itself lies in A, it is called a proper limiting point. If p is not in J., it is called an improper limiting point. An 3^ point of an aggregate A which is not a limiting point is an isolated point. 4. Let ^ be a rectilinear aggregate, and a one of its limiting points. If no point of A falls in (a*, a + 8) or in (a — S, a*), B>0 sufficiently small, a is called a unilateral limiting point. Other- wise a is a bilateral limiting point. 255. Ex. 1. A — ■{ 1 1 1 .^ — -"^i 25 T' ¥' Here the origin is a unilateral limiting point of A. As does not lie in A, it is an improper limiting point. Ex.2. ^ = 0, 1, 1, i, ^,... The origin is a proper unilateral limiting point of A. Ex. 3. A = totality of rational numbers. Every point p in 9t is a bilateral limiting point. If ;9 is a rational point, it is a proper limiting point of A. If p is an irrational point, it is an improper limiting point. Lirniting Points connected ivith Certain Functions 256. We give now a few examples of point aggregates which come up in the study of certain functions. Let y = sin -• The domain of definition of this function embraces all points on the a;-axis except x = 0. It oscillates between — 1 and + 1. The points for which y takes on a particular value, as 2/ = 0, form a point aggregate whose limiting point is a;= 0. In any domain of this point, y oscillates from +1 to— 1 aii infinite number of times. LIMITING POINTS WITH CERTAIN FUNCTIONS 159 257. Let j^ = sin -. (1 sin- When a; = 0, or when * sin-=0, (2 y is not defined, since for these points, 1) involves division by 0. The points x for which 2) holds are ±i, ±^, ±^,... (3 This is a point aggregate whose limiting point is a; = 0. As X approaches one of the points — , y oscillates with increas- nir -J ing rapidity. At the same time these points, — , become infinitely dense as x nears the origin. The domain of definition of y is the a;-axis except the origin and the points 3). •> 258. Let 1 y = sin - 1 sin 1 sin- X This expression does not define y, because of division by 0, when x = 0, (1 or when x satisfies sini=0, (2 X sin_ij = 0. (3 sin - X The points x defined by 1) and 2) are ^ = 0, ±1, ±-L, ... considered in 257. IGO FIRST NOTIONS CONCERNING POINT AGGREGATES It is easy to see that the points x defined by 3) form an aggregate B such that each of the points of vl is a limiting point of B. In fact, let X approach the point ± As it does so, sin - becomes ^ nir X smaller and smaller ; hence — - becomes larger and larger. sin - X 1 Thus in the domain of the point ± — ■-, sin. sm- oscillates infinitely often between — 1, 1, and in particular 3) is satisfied infinitely often. Thus, the domain of definition of y includes all points of the a;-axis except the points A and B. About each point of B^ y oscillates infinitely often. These points of infinitely frequent oscillation, themselves cluster infi- nitely thick about each point of A ; while the points A cluster infinitely dense about the origin. Let the reader try to picture to himself how the graph of y looks about the points ± — , and 0. 259. 1. The functions of 257 and 258 are formed from that of 256 by a process of iteration. In fact, let then Similarly, y= sin-= 6(^x'): . r 1 sm . 1 sm- sm. . 1 sm- . 1 sm- Xj = oiecx^l = e\0ie >0, while J.c^+i) = 0, A is of order m. 264. Every limited infinite point aggregate has at least one limit- ing point. 1. For simplicity let us consider first the case that the aggre- gate A lies in the interval /= (a, 5). We divide I into halves. One of these halves, call it J^ contains an infinity of points of A. Divide I^ in halves. One of these halves must contain an intinity of points of A. In this way we may continue bisecting each suc- cessive interval, without end. We get thus an infinite sequence of intervals ^ each lying in the preceding, whose lengths converge to 0. By 127, 2, the sequence 1) determines a point a. This point a lies in every interval of 1). Since each D(ol) contains some /„, it contains an infinity of points of A. Hence « is a limiting point of ^. 2. The extension of this demonstration to 9?„ is now readily made. Since A is limited, it lies in a certain parallelopiped P, by 252. We divide now P into two parts «! < 2^1 < K^i - a^), ag < 2^2 < ^2 "•• «„ < a:„ < J„, (2 l(5i-ai)<2;i< 5j, a^ > 0, which contradicts the hypothesis. 266. Let A he any point aggregate. Then A"J^'p) contains an infinity of points of A, however small e is taken. To this end, let /3 < e. Let 5- be a point of A! in I)p{p). Then D^iq^) contains an infinity of points of J., however small a is. But if /3 + o- < e, Da(q^ lies in i>^(jt?), by 249. Hence B^Qp) contains an infinity of points of A. 3. We have just shown that A" lies in A' . It is, however, not necessary that A' lies in A. Thus, if J. = 1, -|, i, ••., A' = (0), and this does not lie in A. 267. Extreme values of a domain. 1. Let the variable x range over a rectilinear domain D which is limited to the right. We form a partition (^, B^ as follows : in A we put all num- bers of 9? which are ^ any number in D \ \\\ B we put all numbers of 9^ which are > any number of D. Let this partition be generated by fi [130]. We call fi the maximum of x or of i), and write yu, = Max X = Max D. The fact that a domain E is not limited to the right may be denoted by ,, ht tt ^ Max X = Max ^ = + 00, where x ranges over E. 2. Let D be limited to the left. We form a partition (tI, B^ by putting in A all numbers of 9? which are < any number of -Z>, 166 FIRST NOTIONS C0NCI:RNING POINT AGGREGATES and in B all numbers which are ^ any number of L. If the num- ber X generates this partition, we call \ the minimum of x ov of i), and write ^ at- tit- -rv \ = Mm X = Mm JD. The fact that a domain E is not limited to the left may be denoted by ,,. t.,. ^ •^ Mm X = Mm i> = - oo, where a; ranges over ^. Ex. 1, D=(a, b), a = 0. MaxZ) = l. In Exs. 3, 4, X takes on neither its minimum nor its maximum values. 268. 1. The maximum and minimum values of x are called its extreme values or extremes. Let e be an extreme of D. If the point e is an isolated point of D, e is called an isolated extreme, otherwise e is a non-isolated extreme. Evidently an isolated extreme of D lies in D. 2. When, however, e is a non-isolated extreme, it may or may not lie in D. In this case we have the theorem : If e he a finite non-isolated extreme of i), it is an extreme of D', the first derivative of D. To fix the ideas, let n.*- -r. e = Max B. Since e is not isolated, it is a limiting point of D, and hence lies in D' . Since no a; of D is > e, no x of D' is > e. Hence g=Maxi)'. VARIOUS CLASSES OF POINT AGGREGATES 167 3. We have obviously the following : Let every x of D he ^fi, while for each e > there exists in D an x>a — €. Then , , ^ fjL = Max 2>. A similar theorem holds for a minimum. 4. Let 9)? be such that Mina;<9W CHAPTER VI LIMITS OF FUNCTIONS FUNCTIONS OF ONE VARIABLE Definitions and Elementary Theorems 273. 1. We extend now the notion of limit, by defining limits of functions. We begin by considering functions of a single variable x. Let fQc) be a one-valued function defined over a domain D. Let be any sequence of points of Z), such that lim a„ = a ; a finite or infinite, a„ =f= a. If the sequence /(«l). /(«2)' /(«3) - (2 has a limit 77, finite or infinite, always the same, however the sequence A be chosen, we say ?; is the limit of f(x) for x = a and write 7; = lim /(a;), or, more shortly, ?7 = lim/(a;). We also say /(a:) approaches or converges to rj as a limit, when x approaches a as a limit. This may be expressed by the symbol fix) = V ' 2. If for some sequence 1) the limit of 2) does not exist, we say the limit oif(x) for x= a does not exist. 171 172 LIMITS OF FUNCTIONS 3. Since the limit of 2) must be r} however the sequences 1) are chosen (provided, of course, they have a as limit anda^^a), we have the theorem : Let A = \an\t B = \hn\ he two sequences lying in D; let a„ = a, hn^a. If lim/(a„):#.lim/(5„), then limf(x).,for x = a, does not exist. 274. 1. It is sometimes convenient to restrict the sequences A = a-^, a^, ••• so that all the jjoints a„ lie to the right of a. In this case we call r] a right hand limit and write 7] = lim /(x) or t] =f(_a + 0) or r] = R lim f(x} or tj = R lim fQx}. x=a+0 x=a If we restrict the sequences A to lie to the left of a, we call rj a left hand limit and write T] = limf(x^ or t] =f(a — 0) or tj = L lim /(a;) or rj = L lini/(.t;). x=a-0 x—a Obviously if lim /"(a:;) = i], finite or infinite. (1 then L lim f(x) = R lim fQc) = 77. (2 Conversely., if 2) holds., 1) does also. 2. Right and left hand limits are called unilateral limits. If we do not care to specify on which side of a the limit is taken, we can denote it by U\imf(x). 275. 1. When considering infinite limits or limits for a;= ±qo, it is often convenient to suppose the axes terminated to the right and left by two ideal points + 00, or — 00, respectively. We call these the points at infinity. We call the interval ((7, + 00) the domain of + 00, and denote it by i>«(+^)- (1 We call G- the norm of -Z)(+ 00). DEFINITIONS AND ELEMENTARY THEOREMS 173 Let J. be a point aggregate lying on our axis. Those of its points which fall in 1) we call the vicinity of + oo for the aggre- gate A. We denote it by VcX+oo). (2 Similar definitions hold for Dg(- oo) and Fe(-oo). (3 2. When G- increases, the intervals ((r, + oo) or ((r, — oo) are, in a way, diminishing. It is convenient, for uniformity, to say that ^ -Z>(;(±oo), Va(±cc} are arbitrarily small when (r is taken arbitrarily large, positively or negatively, according to the sign of oo. 276. Corresponding to the two ideal points ± oo, Ave shall intro- duce two ideal numbers, which we also denote by ± oo. These numbers are respectively greater, positively or negatively, than any number in 9?. We say they are infinite. The system formed by joining ± oo to the system 9fJ we denote by ^. We shall perform no arithmetical operations with these ideal numbers. 277. 1. Most of the theorems established in Chapters I, II for sequences may be extended easily to theorems on limits of functions. For convenience of reference we collect the following. The reader should remember that a theorem relating to limits for a point x = a may be changed at once into one relating to a left or a right hand limit at a. 2. Let lim /(a;) = a, lim g(x) = /3. a finite or inf. x=a x=a Then lim(/±^)=a±yS, lim/^=«^. See 49, 50, 51, 98. 174 LIMITS OF FUNCTIONS 3. In F'*(a), a finite or infiyiite^ let /(^)<5'(^)<^(^). lim/(a:) = lim h(x) = \. x=a x=K^)- lim g(x) = + oo, lim f(x') = -f oo. See 138. '^ SECOND DEFINITION OF A LIMIT 175 8. In F*((i) let f(x) he limited and monotone. Then /(« + 0), /(a-0) exist and are finite. See 109. Second Definition of a Limit 278- 1- ^ r ^^ ^ ^ v • ^ iim j{x) = ?;, a finite or inf. x=a there exists for each e > a vicinity V*(a) such that \V-Kx')\, if any such exist, for which 1) is not satisfied. Let us suppose at first that a is finite. Let Min || — a[ = /x. If /A > .0, let < S < /i, then 1) holds in F'5*(a). If ^ = 0, let t t t 61" 62' 63' be a sequence in A, whose limit is a. Then n=co and this contradicts the hypothesis. Thus, when a is finite, there exists always a vicinity V^*(a) for which 1) holds. Suppose a = + oo . Let Max I = /x. If fi is finite, let G- > fi. Then 1) holds in F'(;(4- oo). If /x = + Qo? let t t t 6l'> 62' 63' be a sequence in A whose limit is + x . Then and this contradicts the hypothesis. A similar reasoning applies when a= — ao. 2. We wish expressly to note that in passing to the limit x= a. the variable x never takes on the value x= a. 176 , LIMITS OF FUNCTIONS 279. The converse of the theorem 278 is obviously true, viz. : If for each e > there exists a vicinity V^*(a)^ 8 > 0, a finite or infinite^ such that k-/(a^)| there exists a 8 > 0, such that \f(x)-v\0, 8>0, \f(x)-v\ there exists a G->0, such that 1) holds in F^(-( + ao). This condition we shall express thus : e>0, (7>0, |/(a;)-7;| 0, there exists a G<0, such that 1) holds in Vg(^— oo'). This condition we shall express thus : e>0, a<0, \f(x)-v\0 a vicinity F'*(a), such that f(x^>a (1 in F*(a). For, let A = m be the points of i>, if any such exist, for which 1) does not hold. SECOND DEFINITION OF A LIMIT 177 1°. Let a he finite. Let Min [| — a| = /u.. If /A > 0, let < S < ^. Then 1) holds in F£*(a). If /A = 0, let be a sequence in A whose limit is a. Then and this contradicts the hypothesis. 2°. Let a=+cc. Let Max f = /i. If fx is finite, let Gr> /x; then 1) holds in VqC+cc). If /u. is infinite, let ?v ?2' ••• be a sequence in A whose limit is + oo. Then lii"/(l«) ^ + Qo ; and this contradicts the hypothesis. A similar reasoning holds when a = — ao. The reader will observe that this demonstration is analogous to that of 278. 2. The converse of 1) is obviously true, viz.: If for each (r >0, there exists a vicinity V*(^a)., a finite or infinite., such that ^^ ^ ^ •fi^}>(^ in V*(^a')., then lim/(2;) = + GO. 282. 1. From 281 we see that the following definitions of limits may be taken : r ^^ ^ , -f lim/(2;) = + CO, II x = a M>0, S>0, f(x)>M, Vs*(ia), which in full means : if for each M> 0, large at pleasure, there exists a 8>0, such that f(^x}> Min V^*(a'). 178 LIMITS OF FUNCTIONS 2. lim/(.r) = — oo, if X = a i.e. if for each M<0 there exists a S>0, such that /(a;) < iHf in 3. lim/(2;) = + oo, if ar=+oo M>o, a>o, f(x-)>M, r^(+oo). 4. lim/(a;) = — GO, if x=+cc 5. ]im/(a;) = -f oo, if a;=— 00 6. lira /(a;) = — go ; a- = — CO i»f<0, (7<0, /(a;)0 there exists a vicinity ]^*(«), such that i/(^i)-/(^2)i there exists a F*(a), such that h-/(^)l<| for any x in F'*(a). Let a;^, x^ be two points in F*(a). Then k-/(^l)l<|' i'?-/(^2)l<|- Adding these two inequalities, we get 1). It is sufficient. For, let a^, a^, ■-• be a sequence of points in V*(^a}, having a as limit. Then the sequence /(«i). /(«2)' ••• is regular by 1). It therefore has a limit t}. Then e>0, m', 1 77 -/(«„) I < |- w>w'. (2 Let B = b^, b^, ••• be any sequence of points in V*(^a^ whose limit is a. Then, by 1), |/K)-/(^„)lm". (3 Adding 2), 3), we have 1 7; — /(5„) I < e. n>m, m>m',m". (4 But since B was an arbitrary sequence, the relation 4) states that 7) = lim/(a:). 180 LIMITS OF FUNCTIONS Grajjliical Representation of Limits 285. The graphical representation of limits of sequences ex- plained in 43, 44, and 124 may be readily extended to limits of functions. Let the graph oif(x) be referred to rectangular coordinates. Let D be the domain of /(a:), and let lim fQx) — I. Then the condition e>0, S>0, \f(x')-l\ falling in the S-interval. In general, as e is made smaller and smaller, S becomes smaller and smaller. But for each e-band, however small, there corre- sponds a S-interval of length > 0. y I ^^B^^^ e ^^^^^^^^ ~~o 5 a 5 286. Let lim /(a;) = -f oo. Draw the line y = M, where ilf> is large at pleasure. Then there exists, corresponding to this Jf, a S-interval, marked heavy in the figure, such that f(x) falls in the Jf-band (shaded in the figure) for each x=^a of D, falling in the S-interval. As iHf is taken greater and greater, the corresponding S becomes, in general, smaller and smaller. But for each ever large, there corresponds a S-interval of length > 0. -interval M, how- 287. Let lim f(x) = I. J GRAPHICAL REPRESENTATION OF LIMITS 181 Draw the line y = 1^ and construct an e-band, as in figure. For each e there exists a (r>0, such that f(_x) falls in the e-baiid for each x of i>, falling in the interval ((r, + oo). These examples will suffice to illus- trate the graphical interpretations of limits, when f(x) is plotted in rectangu- , lar coordinates. 288. 1. When the graph of y=f(x) is given by means of two axes, as explained in 191, the geometric interpretation of limits of /(a;) will be made clear by the tollowing : « — \ 1 — I — ' Let lim fQc) = I. (1 p e l e x=a y ' "■■ I I About y = 1 we mark off the e-interval ; about x = a we mark off the 3-interval. Then 1) requires that /(a;) falls in the e-interval for each value ot x=^a in I), falling in the S-interval. 2. Let .. .. . limf(x) = 4- GO. x=a On the 2/-axis we mark off at pleasure the point M> 0. Then for each M there exists a S-interval, such that /(x) falls in the interval (iHf, + oo), for each x=^ a oi D falling in the S-interval. lim f(x) = 1. I finite or infinite. X = a + hu, S =/= 0, (1 ^^'"^ lim fix) = I (2 For, while x ranges over the domain D on the a;-axis, w ranges over a domain A on the w-axis. The two axes x and u stand in 1 to 1 correspondence by virtue of 1). To the point a;=a on the a;-axis corresponds the point w = on the w-axis. 182 LIMITS OF FUNCTIONS T pf f(x)=Aa^-hu) = ,^(u). (3 Then if x and u are corresponding points, / has the same value at 2; as has at u. To fix the ideas let I be finite. From e>0, S>0, \l-f{x~)\{u)\ transform ^ lim into lim, x=a u=b R lim into R lim, L lim, or lim. i GRAPHICAL REPRESENTATION OF LIMITS 183 Similarly, lim into Rlim or Llim. 2. In particular, any limit x=a or ±00 may be transformed into one with respect to a; = 0. 292. Let u = ^(x^, and lim u = h. a^h finite or infinite. (1 Let y = /(w), and lim y = 7). T) finite or infinite. (2 Then if <\>(x)4-h in F*(a), lim y = 'r). (3 To fix the ideas, suppose a, 5, t] are finite. S a 3 Then since 2) holds, I I I ' ' ■ e>0, cr>0, |y-77|0, S>0, 0<|m-5|0, S>0, |y-7;| be arbitrarily small, and e x' a a — e < x' < a. Let y' correspond to a;'. Let S > be such that b-8>y'. Then, while y remains in LV&*{b'), x remains in LD^(a). 2. Let y =/(a;) he univariant in F'*(a), where a is a bilateral limiting point of V*. If lim y = b^ then lim x = a. y = b The demonstration is analogous to that of 1. Examples of Limits of Functions 295. 1. lim sin a: =0. For, however small e > is taken, there exists an arc S > such that lsinaj|0 is taken, there exists an arc S>0 such that 1 — cos2'0. sin x As tan X = •, cos a; and lira sin x = l, lim cos a: = 0, we have 1), by 277, 6. 2. Similarly, i2 lim tan x = — oo. 298. 1. lime^=l. x=0 This follows at once from 172. 2. lim e^ = e". For, let Then, by 289, x= a + u. lim e^ = lim e'^+« = e'^ lim «« x=a «=0 «=0 = g«, by 1. 186 LIMITS OF FUNCTIONS 3. lim e^ = + QO . This follows at once from 169. 4. lim e-^ = 0. For, let _ H u Then lim e^ = i2 lim -, by 290, 2 ; x=—a3 M = - = 0, by 277, 5. 5. Obviously, as in 1, 2, lim a^ = a^". I =10 1 e^-1 6. /(^) = -T e^ + 1 i? lim/(a;) = + 1, L lim/(2;) = - 1. x=0 1=0 299. 1. Let lim/(a;) = ?;. ■J? > 0. lim(/(2:)y = 7;^ z=a This follows directly from 171. 2. In F*(a), Zef /(a;) > 0. Let \\mf(x) = 0. x = a Then lim(/(a;)y=0, /i>0. = 1, ^=0. = + 00. /t<0. 300. 1. lim log a; = log a. a>0. x = a This follows at once from 178. EXAMPLES OF LIMITS OF FUNCTIONS 2. lim log a; = + 00 . This follows at once from 179. 3. For, set Then Ji lim log x= — oo, x = 1 X= -' u R lim log X = lim log - = — linj log m = — oo . 4. Let Then lim/(a;) = t; > 0. lim log/(a;) = log r) = log lim/(a;). x=a x=o This follows at once from 178. 187 301. 1. T Sin a: ^ lim = 1. z=0 X From geometry, we have Area 0^C< Area 05a< Area OBD. Hence, for < x < 7r/2, ^ sin X cos x<^ x<^ tan x ; or As Set in 1), Then cos2;< Sill X 1 cos a; i^lim x=0 cosa; = R lim ^ = x=0 COS X 1, 277,3 1 Blim x = X 1. sin X x = — u. i21im x = X i lim u = u sin a; sin w = 1. (1 (2 188 LIMITS OF FUNCTIONS From 1), 2) we have lim — = 1. x=o sin X Whence, by 277, 2, T sin a; -, lim = 1. x = X 2. From 1 we have readily For, 1 . sin ax a in lim—- = -• 5 9^0. x=o bx sm ax « sin ax hx h ax a sin u setting h u u= ax. 302. lini*?:IL^=l. (1 x=0 X For, But tan X _ sin x \ (2 X x cos X 1 . sin X ^ -,. 1 -, iim = 1, lim = 1. x=o X 1=0 cos X Thus, passing to the limit in 2), we get 1), by 277, 2. «fto T • sin (x + K) — sin x ,., 303. lim ^^ — -—^ = cos X. (1 ft=o h For, sin (x-\-K) — sin x _ '2 cos (ar + ^ A) sin \ h l h ^^^^ lim cos (a; + 1 A) = cos a: ; ft=0 lim 2li5_y:= lim ^1^ = 1. Passing to the limit in 2), we get 1). (2 304. 1. For, 2. For, 305 1. Here while EXAMPLES OF LIMITS OF FUNCTIONS 1- 1 — cos a;_ 1^ 05=0 x^ 2 1 — cos a: _ 2 siii^ \ x _ 1 /sin \ x ^ x^ ~ x^ ~^ \x )' 189 (1 lim a:=0 tan X — sin x _ 1 ^3 -^ tan x — sin x tan x \ — cos 3? lim e""' cos re = 0. (1 lim e-^ = 0, by 298, 4, lim cos x does not exist. We cannot, therefore, apply the theorem 277, 2, that the limit of the product is the product of the limits. We therefore proceed thus : — e~' < e'" cos X < e~^. Apply now 277, 3. This gives 1). 2. We may see the truth of 1) geometrically. Let «/l=e"^ ?/2 = cosa:. and y = e " cos rr = y^y^. Let us draw the graphs (7, C of i^/j. To get ^, we multiply y^ by the factor y^, which takes on all values between — 1 and + 1. Thus y oscil- lates between the curves (7, C. As (7, C approach nearer and nearer the a;-axis, the amplitude of the oscillations converges to 0. 190 LIMITS OF FUNCTIONS Tlie Limit e and Related Limits 306. 1. We saw in 110 that lim imfl + -)=e; 7i=l, 2, 3, •.• (1 Let us consider now the more general limit 1^ lim 1 + Each X will lie between two integers w, w + 1 ; or n 1 + ^, n X n-\-\ and But and Thus 3), 4) give in 2), ^n + 1 Now, by 1), , ^v , . . , lim(l + -) =l™(l + -l3) . liml + - =lim = — =1. w+ 1 Hence 5) gives THE LIMIT E AND RELATED LIMITS 191 307. lira liin('l+iy = e. (1 For, let Then x = — u. (^-r=(-r=(^-.4iT =(^-D(^-^)> <^ where But, when a; = — oo, m = + oo, and hence t) = + go. ^^ Umfl + ^') = 1, Hmfl+iY=e, we get 1), on passing to the limit in 2). 308. From lira(l+- =e, x=+oo V xJ we get, setting 1 x = -, u 1 R\\m(l + uy=e. «=0 From lim(l + -) =e, we get, setting 1 x = --, u 1 i lim (1 + uy = e. «=0 From 1), 2) we have 1 lim (1 + xy = e. x=0 (1 (2 (3 192 LIMITS OF FUNCTIONS 309. For, where But 1 lim (1 + xuy = e^. (1 1 J_ V = ux. 1 lim (1 + vf = e. ti=0 310. l™ 1=0 For, Hence, by 299, ,. _ •^ liiu y = e^. log: (1 + ^') _ -I X lim log(^ + ^ _ |i„^ Yog (1 + rc)^ x=0 X 1=0 1 = log lim (1 4- xy, by 300, 4) 1=0 = log e, by 308, 3) = 1. a* — 1 311. lim = log a. a>0. (1 x=o a; ° Set M = a-^ — 1. Then lim M = 0, a:=0 by 298, 5. Then, by 292, j.^log(l + »)^j.^log«' ,4=0 M a^ a^— 1 = 1, by 310. But log a'' = x log a. This in 2) gives 1). DEFINITIONS AND ELEMENTARY THEOREMS 193 312. ,.^(l + .)>-l^ From 310 we have lini^Ml+^=l. Let u = (i + xy — i. n^o. Then, by 299, lim w = 0. x=0 Hence, by 292, we get from 2) ^.^log(l+^^ =^ a + a^y-i (2 or since But, by 310, =^ (1 + ^y (l + a:y-l „ log (1 + ^) log(l +xy=fM log(l + x). lim - — ,-, ^ . = 1. C4 log(l + a;) Now (1 + a;)*^ - 1 _ (1 + xy - 1 X log(l+a;) ~ X log(l + a;) Passing to the limit and using 4), we get 1) for the case that fjL^O. The case that ft = is self-evident. FUNCTIONS OF SEVERAL VARIABLES Definitions and Elementary Theorems 313. For the sake of clearness, we have treated first the limits of functions of a single variable. We consider now the limits of functions in m variables. The extension of the definitions and results of the preceding sections is, for the most part, so obvious that we shall not need to enter into much detail. Should the reader have trouble with the case of general m^ let him first sup- pose m = 2 or 3, when he can use his geometric intuition as a guide. 194 LIMITS OF FUNCTIONS 314. In the case of a single variable, we have seen how useful the ideal points ± oo proved. In the treatment of limits of func- tions of several variables, we shall find it extremely advantageous to adjoin an infinity of ideal points to dim as follows : Let A = aj, a^^ a^, ••• be an infinite sequence of points in QfJ^. Let . , ,, lim a's = «!, ••• lim a/"'' = a^; ^v ••• ^mi finite or infinite. We say the limit of the sequence A is and write a = lim a„. If any of the coordinates of « are infinite, we say a is an infinite point. This fact may be briefly denoted by the symbol oo being without sign. There is no point in dim corresponding to an infinite «. We therefore introduce an infinite system of ideal points, one for each complex, ^_ a^,a^,'"a^, (2 in which one at least of the symbols, a^, is ±oo. Such ideal points we represent also by 1), and q call the m symbols, 2), their _oo< t^ ^oo coordinates. If we employ the graphical representation ~°°^ ^ ^'^'^ of 231, we suppose, according to 275, that each axis is terminated by the ideal points +qo and — oo. Thus, any complex of m points, one on each axis, such that at least one of these m points is an ideal point, is the representation of an ideal point in 9?^. The system of points, formed of 9?^ and the ideal points, we denote by 9?^. These ideal points are also called points at infinite/. DEFINITIONS AND ELEMENTARY THEOREMS 195 315. 1. The domain of an ideal point a = (a^ '"<^m) is the aggre- gate of points X =(,x^^"X„^'), whose coordinates lie respectively in the domains It may be represented by Dp^...p^(a). Example. Let m=4, and a=( — oo, 1,2, 4-00). The domains in which the _oo.«— —i i2_ >.'ioo coordinates xi, Xo, xg, Xi range, are ''i ^ marked heavy in the ligure. -oo<: i — -^ • ^ > ^-oo Here px is an arbitrarily large nega- -cx3< f^ \ ' ii , i m tive number; p2 and pz are arbitrarily _^^ o » a small positive numbers; pi is an arbi- Pt trarily large positive number. 2. The points of an aggregate A^ which lie in Dp^...p^{a'), a being a point at infinity, form the vicinity of a, for that aggregate. We represent it by 316. 1. Let y =f(x-^'--Xj^) be defined over a domain D. Let A = aj, a2, ••• be a sequence of points in i), and let lim a„ = a. a finite or infinite. If lim/(a„) = 77, 7) finite or infinite. is always the same however A be chosen, a remaining fixed and a„ rjfc a, we say r] is the limit of y for x= a; and write r] = \imf(x^...x^'), or, more briefly, 7] = lim/(a:j ...a;^), or 77 = lim/(a;); or, 2. Just as in the case of a single variable, we can show that this definition is equivalent to the following : 7? = li m / (a^j • • • x^') , 7] finite or inf. x=a when, taking D{r)^ arbitrarily small, there exists a vicinity V*(^a), such that y remains in D(j]^ when x is in V*(^a). See 278-283. 196 LIMITS OF FUNCTIONS 317. 1. The theorems of 277 and 284 hold for functions of several variables as well as for a single variable. 2. The generalized theorem of 292 may be stated thus ; Let w^ = (/)j(a:j...a:„), • • • it„, = <^,„ (a^i • • • a:„) ; and lim Mj = 5j, ••• lira m„j = h^. x=a x=a Let and lim y = Tf). Letu^b in F*(a). Then lim y = 7], x—a Here a, J, 77 may he finite or infinite. The demonstration is perfectly analogous to that in 292. A Method for Determining the Non-Existence of a Limit 318. To determine whether 7] = \\vQ.f(x-^'-'X^, a, 77 finite or inf. even exists, is often a difficult matter. The following simple con- sideration analogous to 273, 2, 3 will sometimes show very easily that 77 does not exist. Let Whe some partial vicinity of a exclud- ing a. We may denote the limit, when it exists, of f(x-^^--'X^ for x = a when x is restricted to W by ^=\imf(x^-"X^). w Then K, must exist, finite or infinite; and however Tf is taken, we must have Thus, in case ^ does not exist, or is different for different TF's, we know that t] does not exist. DETERMINING THE NON-EXISTENCE OF A LIMIT 197 We ask, does lim/ i,y=0 exist ? As partial vicinity of the origin, take points on a line L; y = ax. X ^ 0. Then lim/(a;, y) = lim /^^ = -^^ i x=o x-(l + a^) 1 + a^ which varies with a ; J.e. with L. Hence the limit in question does not exist. 320. Ex.2. 2 Ax,y) = -^. x^ + 2/* Does lim/ a x,y=0 exist ? If we take as partial vicinity of the origin points on the line L ; y = ax, we get 2 lim f(x, y) = \imx- " ^ .^ = Q. (2 L x=o 1 + a*x^ Thus, however L is chosen, the limit 2) is always the same. We cam ot, how- ever, infer that the limit 1 ) exists, since our method only shows the non-existence of the limit. Instead of the family of right lines L, let us take a family of parabolas P ; y- = ax. lim/(a;, ?/)=lim ^^ ^ x=ox^(l + a^) 1 + a^' which varies with the particular parabola chosen. Hence the limit 1) does not exist. 321. Ex.3. Does f(x,y) = log- — -. x,y0. Then Hence which varies with X. /(x, y)= logX. lim/(x, y)= logX, 198 LIMITS OF FUNCTIONS Iterated Limits 322. 1. Let /(zj • • • a;^) be defined over some domain D\ and let a = (a^"-aj^. Then lira f(x^ "•x^^= /^ will be in general a function of all the variables except x^^. Also lim /^ = /^., = lim • lim f(x^ • • • x^) '2 '2 '1 '1 will be in general a function of all the variables except x^^, x^^. Continuing, we arrive at lim • • • lim • lim /(a^j • • • a;^), s < w, (1 «t»=<»t, ="l2=''l2 ^'l="h which is in general a function of all the m variables except Xi^t a^ij^ ' ' * x^^. Limits of the type 1) are called iterated limits. In 1), we pass to the limit first with respect to a^^^, then with respect to x,^., then with respect to x,^., etc. A change in the order of passing may produce a change in the final result. 2. Iterated limits occur constantly in the calculus ; for example, in partial differentiation, differentiation under the integral sign, double integrals, improper integrals, and double series. The treatment of these subjects by the older writers on the calculus is faulty, as we shall see, because they change the order of passing to the limit, without a careful consideration of the correctness of such a step. 323. Ex. 1. limlim^-:^ = limf^:ii^^ = -l. i,=o x=o a; + 2/ y=o\ y J lim lim 5^^ = lim f -"^ = + 1. x=0 y=0 X + y 1=0 \x/ The two limits are thus different. Ex. 2. lim lim hz^ = _ i. y=s) 1=00 1 + xy lim lim 5^^^^ = + 1. I=«, y=0 1 + SCy The two limits are thus different. UNIFORM CONVERGENCE 199 324. The following is a case where a change in the order of passing to the limit does not change the result. Let T /. lim f{x, y^=r], 7} finite or inf. (1 x=a, y=b \\VLif(x,y) = g(x), forO<\x-a\a, in F,*(a, 6). Passing to the limit, for 2: = a, we have g(x)^a, forQ<\x-a\ and each x in D there exists a S' > 0, such |/-^| 0, we say / converges uniformly to g in D ; or is uniformly convergent. Hence, if / is uniformly convergent in Z>, there exists for each e > a 2 > 0, such that for any t in F^*(t). Moreover, one and the same norm B suffices for all the points of D, e being the same. The central idea of this case of uniform convergence may be clearl}^ if somewhat roughly, brought out by saying that if the convergence is uniform the norms B for which 1) hold, e being small at pleasure, but then fixed, do not sink below some definite positive number, when x ranges over D. 2. These considerations may be extended to the case that t is infinite ; we therefore define as follows : The function /(a^j -" x^; ^^ ••• ^„) converges uniformly to g(^x■^^ ••' a;,„) in D as t = t, t infinite; when for each e>0, there exists a set of norms Pi '•- Pn-, such that for any x in i), |/(a^l ••• ^ml h •'• tn) -ffC^l ••' ^m)|, for any given value of e. 3. When f{x^ ••• x„^\ t^ ••• ^„) converges uniformly in D to g(^Xj^ ••• a;^), we denote this fact by YimfQx^ ■■■ x„^; t^ ■■■ t^^ = g(x^ ... a;^), uniformly. 4. If /= uniformly in i), we may say it is uniformly evanescent in D. 326. Ex. 1. , Z>=(0*1). A=(-h,h). h>0. Evidently for any x in D lim/(x, t) = ^=g(x). (=0 * But f(x, t) does not converge uniformly to rj{x) in D. For if it did, for each e >0 there must exist a S > 0, such that R = \f(x,t)-(J{X)\= 1^1 , . Now obviously, t being fixed, ^ can be made as large as we choose by taking x near enough 0. Hence B does not satisfy 1) as x ranges over D. In fact, as is seen at once, in order to have i? < e, it is necessary to take 5 smaller and smaller as x approaches 0. In this case then Min 5 = 0, in D. 327. Ex. 2. x -\- t D=(n,b), 00. This example is the same as Ex. 1, except D is different. As before ^ lim/(x, t) = -=g(x). t=o X But now/(x, t) converges uniformly to g{x) in D. In fact, in V^*(P') - i?< , 5 ; i.e. f converges uniformly in D. 202 LIMITS OF FUNCTIONS 328. Ex. 3. , Here Hence if we set we have (1 + X2)' D = {—a,a). A=(a, +Qo). lim/(x, t)=l + x^, x^O. t=CO = 0. X = 0. g(x)—0, for x = = 1 + x2, for X ^fc 0, lim/(x, t)=g{x). However, / does not converge uniformly to g in D. For, when x ^ 0, 1 B = \f(x,t)-g(x)\ = (1 + x2)« This shows that as x approaches 0, it is necessary to take t larger and larger in order that i? < e. There is thus no norm p, such that for each t> p, and any x in Z>. In this case, then, Maxp =00. Remarks on Dirichlefs Definition of a Function 329. The definition of a function given in 189 and 230 does not depend at all upon an analytic expression for the function. At first, the reader who has been used only to functions defined by analytic expressions, may be inclined to regard functions not thus defined as only pseudo-functions, or at least of little impor- tance. This attitude of mind must be overcome. In the first place, in certain parts of mathematical physics, e.g. the potential theory, it is of great importance to be able to assign values to a function at pleasure, totally disregarding the question of an analytic expression for it. Secondly, as the reader advances, he will find that many func- tions which he might well believe have no analytic expression, do indeed have very simple ones. We give now a few examples of such functions. DimCHLETS DEFINITION OF A FUNCTION 203 330. 1. For x>0 let 7/ = l. For x = let t/ = Q. For a; < let ?/ = — !. The graph of this function is given -to in the figure. ^^_^ An analytic expression of ^ is 2 y = — lim arctg (jix). TT 71=00 This function is much used in the Theory of Numbers. We shall call it signum x and denote it by y = sgn X. When w, V Tib an equation sgn u = sgn V simply means that the sign of u is the same as that of v ; while sgn w = + 1 is only another way of saying that u is positive etc. 331. For x=^0 let y = 1. For a: = let ?/ = 0. Its graph is indicated in the -con- figure. An analytic expression of i/ is nx y = lim \-\-nx 332. Fora;=0, ±1, ±2, ••• let y=0. For ?i<2:\, y=f{x), for 0 1, = cos 2 TTX, for < X < 1. 334. 1. For rational x, let y—a\ for irrational a;, let y = h, where a, 5 are constants. This function was introduced by JDirichlet. In any little interval, y jumps infinitely often from a to b and back. It seems highly improbable that such a function should admit a simple analj^tic expression ; yet it does. We have already seen that sgn x admits a simple analytic expression. Consider now y = (J + (5 — a) lim sgn (sin% ! irx) . (1 For any ratiojial x.,n\x finally becomes and remains an integer. Hence sinwiTra; = for sufficiently large n. Hence y= a for any rational x. For an irrational a;, n\x never becomes an integer. Hence ^\v?n\'Kx lies between and 1, excluding end values. Therefore sgn (sin^ n ! irx^ = 1 ; and for any irrational x^ y=h. UPPER AND LOWER LIMITS 205 Thus 1) is an analytic expression of Dirichlet's function. The reader should note that it is utterly impossible to intuition- ally realize the graph of this function. 2. Similarly, we see that y =/(^) + (^(3^) — /(^) ) lim sgn (sin^ n ! irx) 71=00 equals /(^) when x is rational, and equals g{x) when x is irrational. 335. A remarkable function is the following. We shall call it Cauchy^ s function, and denote it by 0(x'), viz.: _j_ C(jc) = e ^\ for x^O, = 0, for x= 0. As a limit, we can write it or (7(2;) = lime ■^'+"' u=0 C(x)= lim e Its graph is given in the figure. Its peculiarity is its remarkable flatness near the origin. Upper and Loioer Limits 336. 1. Let/(a;j ••• x„^=f(x) be defined over D. Let a be a limiting point oi D; a and D may be finite or infinite. Let A = «j, a^, •• • be a sequence of points in D whose limit is a, such, however, that L = lim/(a„) n=ao J exists, finite or infinite. There are an infinity of such sequences. 206 LIMITS OF FUNCTIONS For all such sequences, let X = Min L, 11= Max L. These are called respectively the lower and upper limits of f(x^ • ■ • x^') at a; we write X = lim inf /(a^i • • • a;^) = lim/= lim/, x=a x=a fi = lim supf(x^ '■• x^ = lim/= lim/. The lower and upper limits X, /t may be infinite. 2. When dealing with functions of a single variable, we can have right and left hand upper and lower limits^ by considering only values of a:; > a, or < a, respectively. Then R lim sup/(a;) = lim sup/(a:) = \\m.f(x) x=a 1=0+0 1=0+0 = i21im/=/(a + 0) all denote the right hand upper limit of f(x) at a. A similar notation is employed for the left hand limits. 337. EXAMPLES* 1. . 1 y — sin - • X limy = — 1, lim?/ = + l. x=0 a;=0 2. ?/= (l-x2)sini. ^ X lim2/ = — 1, Iim?/=+l. x=0 1=0 3. ?/=(! + a;2) sin - • lim y = — 1. Tim y = + 1. a=0 z=0 «"( a + sin ^— ^ + 6 + sin — i— 4- . .j, = lim-^ ^^lli ^Ili. n=o 1 +«" See 333. * The reader will do well to roughly sketch the graph of these functions. We find : UPPER AND LOWER LIMITS 207 y z= a + sin , for x > 1. X — 1 Hence y = b + sm—^, /or 0 a S > 0, such that \-€/x. 2. Obviously we have the following : Let A,, fi he the loiver and upper limits of f{x-^ • • • x^ at a. There exist tivo sequences A = a^, a^, •••, B = h^., b^, ••• whose limits are a, such that lim /(a„) = \, hm /(6„) = ^. 3. Since the maximum and minimum of a variable exist, finite or infinite, we have : The upper and lower limits of a function always exist finite or infinite. If Hrn/=lim/ = i, then lim f = l. CHAPTER VII CONTINUITY AND DISCONTINUITY OF FUNCTIONS Definitions and Elementary TJieoreyns 339. 1. Let f(x-^'-- x„^^ be defined over a domain D. Let a = {a^ •'• a^ be a proper limiting point of I). If lim /(a^i • • • a;,„) = /(«! • • . a^) , (1 the function/ is continuous at a. In words: if the limit of f at a is the same as the value of f at a, it is continuous at a. The reader should observe that a is not only a limiting point of D, but that it lies in D. 2. The condition 1) may be expressed in the e, 8 notation, giv- ing the following definition of continuity : f{x-j^ ••• a;„j) is continuous at a, if for each e > there exists a S > 0, such that Ifi^i •••^m)- /(«! •••«,„) I < e, in Fa(a) . 3. A function which is continuous at all the proper limiting points of D is said to be conthiuous in D. We suppose that D has at least one proper limiting point. 4. Consider the function n^ X _ xy at points different from J\^i I/)— ~T~: — 9' ^7 • • x^ + y^ the origin. = 0, at the origin. We saw, 319, that lm\f(x, y') 1=0, y=n does not exist. Thus / is not continuous at the origin. 208 DEFINITIONS AND ELEMENTARY THEOREMS 209 At the same time / considered as a function of x alone, or con- sidered as a function of y alone, is continuous. This example illustrates, therefore, the fact *that because /(.T J • ■ • a:,„) is a continuous function of each variable separately, we cannot, therefore, assert that / considered as a function of rTj • • • x„^ is continuous. 340. The following theorems will be found useful in determin- ing whether_/(a;j •••a;,„) is continuous at a or not. From 277, 1 and 317, 1 we have at once : Letf(x-^---Xj,^^ g(x-^--'X„^ he continuous at a. Then f±g, f-g^ f 9 are continuous at a. 341. From 292 and 317, 2 we have at once : Let ^^^^^^(^^^...^j ... u„, = (fi,X^-^---x„,') be continuous at x= a= (a^ •••«„). At x = a, let u^ = h^ ... u,n = h„,. Let /•/- N be continuous at u—b = (h^--- b,,^. Then y considered as a function of the x's is continuous at x = a. In a less explicit form, we may state this theorem : A continuous function of a continuous function is a continuous function. 342. In order that f(x^"-x,n) be continuous at «, it is necessary and sufficient that for each e>0 exists an undeleted vicinity V(a), such that for any two points x\ x" in it., \f(x')-f(:x")\0, by 300. The demonstration also may be given by 294. Discontinuity 347. If ,. .. . lim/(a;i •'• x^) does not exist ; or if it exists, and is different from /(a), should f be defined at a, we say / is discontinuous at a, and « is a point of discontinuity of f. 212 CONTINUITY AND DISCONTINUITY OF FUNCTIONS Discontinuities are of two kinds : Finite discontinuities, when f is limited in F*(a). Infinite discontinuities, when f is unlimited in every F'*(a). 348. We consider now in detail some of the ways in which a function of a single variable /(a;) may be discontinuous at a point a. Finite Discontinuities 1. /(a + 0)=/(a-0)^/(a). Such a discontinuity is called a removable discontinuity. Such a function is y = lim — considered in 331. n=co 1 + 71X 2. /(a + 0),/(a — 0) exist, but are different. Such a function is y — sgn X, considered in 330. 3. If fi^x) is defined at a, and /(«) =f{a + 0), we say / is con- tinuous on the right, at a. If /(a) =f(a — 0),/ is continuous on the left, at a. 4. Either /(a + 0), or/(rt — 0), or both do not exist. Such a function is ^ V = sin -• ^ X We considered this function in 256. Here neither /(O + 0) nor /(O — 0) exist Also / is not defined for x = 0. Infinite Discontinuities 349. 1. As X approaches a from either side, f(x^, either mono- tone increases or mono- ?/ tone decreases. Ex. 1. atO. Ex. 2. y = Y atO. 1- INFINITE DISCONTINUITIES 213 2. As X approaches a, /(a;) increases monotone on one side, and decreases monotone on the other. Ex. 3. atO. Ex. 4. at'!:. y = y = tan z, 3. As X approaches a, y oscillates infinitely often about a base curve, belonging to the types defined in 1 or 2. The amplitude of the oscillations is limited. Ex. 5. Here y oscillates about the base curve y = — ha; sin -, at x = 0. Xr X and the amplitude of the oscillations converges to as x approaches 0. Ex. 6. at X = 0. Here y oscillates about 1 . 1 y = - + sin-, ^ X x' y=-; (1 X ^ and the amplitude of the oscillations remains the same, viz. ±1 above the curve 1). 4, The discontinuities considered in the preceding three cases are such that either ,. lim y x=a is infinite, or at least the right and left hand limits at a are infinite and of opposite signs. Such points of discontinuities of fQc) are called infinities ; we also say /(a;) is infinite at such points. 5. In either or both the right and left hand vicinities of a, y is unlimited, while the corresponding (infinite) limits do not exist. Ex. 7. 1 1 w = - sin — ^ X X :0. Here y oscillates between the two hyperbolas 1 ^ = ± X The amplitude of the oscillations increases indefi- Ditely as x approaches 0. 214 CONTINUITY AND DISCONTINUITY OF FUNCTIONS Ex. 8. y = --}■ - sin — a; = 0. Here y oscillates about the base curve 1 y= — X The amplitude of the oscillations increases indefinitely as x approaches 0. 1 Ex. 9. 2/ = e^. Here L lim y = 0; BMmy =+ , I 1 Ex. 10. y — 6' sin — Here L lim y = ; i? lim y does not exist. 1=0 x=a Some Properties of Continuous Functions 350. 1. If f(x^"-x^ is continuous in a limited perfect domain jD, it is limited in D. For if / were not limited, Max I/I = + 00. (1 Then, by 269, there is a point a of i) in whose vicinity 1) holds. This is impossible. For, since /is continuous, /(«i • • • O - e 0, 8>0, |/(a;) -/(«)!<€, TsCa). Hence /(«)-€0, such that ^ |/|>p, inF-(a). 352. Let /(a;^ • • • a;„j) be defined over a domain 2>. By defini- tion, it is continuous in D when, for each proper limiting point x lim f(x^ + Aj • • • rc,„ + h„,') = f{x^ • • • a;„), ft=0 the points x + h lying in D. If f{x-^^ + Aj • • • x,,^ + 7i,„) not only converges to f(^x^ • • • a;^) in D, but converges uniformly^ we say /is uniformly continuous in D, We have now the very important theorem : If f(x-^ •"X,n) is continuous in a limited perfect domain D, it is uniformly continuous in D. Making use of the notation of 325, we have only to show that \ = Min 8, for I) is >0. Suppose it were not, i.e. let 3^ = 0. We show that this assump- tion leads to a contradiction. For, by 269, there is a point a in i), such that in V{a) MinS=So=0. (1 This is impossible. In fact, by 342, there exists for each e>0, a 8', such that for any pair of points x\ x" in V^-icC) \f(x'^-f(x"-)\ there exists a cubical division of i>, of 7iorm S > 0, such that \f(ix'^-fix")\0 be such that 1) holds for any point x" of V^(x'^. Let now the norm of the cubical division be Suppose x\ x" were a pair of points in some cell A, such that 1) does not hold. Since Dist(a;', x"~) = (0, 2), y = lim »i=x) X"- + 1 for for This function is a particular case of that in 333. For 0 small at pleasure, there is at least one point x' in any Fi(a), such that \f(x')-e\< 2 (1 Since / is continuous at a, there is a 8 > 0, such that for any x in VsQa') In 2) set x = x', and add to 1); we get ■ |/(«)-e| in the perfect limited domain D. Then ^^. ^ ^ . -r^ Min/>0, inD. 218 CONTINUITY AND DISCONTINUITY OF FUNCTIONS 356. In the interval % = (a, h') let f(x) he continuous. Let it have opposite signs at a and h. Then f vanishes for, some point c within 51. Let us form a partition (^, jB) with the points of 21. The class A is formed thus. Not only shall sgn/(a;)=sgn/(a) (1 at every point of J., but between a and any point of A shall 1) hold. In B we throw the other points of 21. Let c generate this partition. Then in any V(c^^f has opposite signs. But if f{c) were =^ 0, by 351, we could take S so small that f(x) has only one sign in Fg. This leads to a contradiction. Hence /(c) = 0. The point c cannot be an end point of 21, for at these points / T^ by hypothesis. 357. Let f(x) he continuous in 21= (a, 6). Let Minf(^x}=a^ Maxf(x) = ^ in 21. Then f(x) takes on every value in (a, /3) at least once., while x passes from a to h. By 354, f(x) takes on its extreme values in 21. Let, therefore, /(:r')=«, f(x")=^. To fix the ideas, let < a < /3. Let a<70. Hence, by 356, g vanishes at some point in (a;', a;"). At this point /(a;) = 7. 358. Let y = f(x) he a continuous univariant function in the inter- val (a, 5). Let a=f(a), ^=f(Jiy. Then the inverse function x = g(^y^ is a one-valued univariant continuous function in (a, /3). By 214, g(^y') is a one-valued univariant function in its domain of definition E. By 357, U = (a, /3). By 294, g^y) is continuous in (a, /3). THE BRANCHES OF MANY-VALUED FUNCTIONS 219 The Branches of Many-valued Functions 359. Let F(x^ ••• x,^ be a many-valued function in D. We can form a one- valued f unction /(a^j ••• a;^) over a domain A with the a;-axis, ^ _, .^ Fig. 1 Ay Ax~PW = tan <^. That is : the difference quotient is the tangent of the angle secant makes with the x-axis. that the 224 DIFFERENTIATION Suppose now y is continuous in a little interval about x = a\ if the secant PR approaches a limiting position P U^ as R approaches the fixed point P from either side, we say PU is the tangent to the curve at P. Evidently, if /'(«) is finite. Ay f'(^a} = lim -r^ = lim tan cj) = tan a, where a is the angle that the tangent line makes with the a;-axis. If f'(a')= ±00, the tangent line is parallel to the ?/-axis. /(«;=+ 00 Y Fig. 2. .TT P /'(CO =-00 Fig. 3. Such cases are shown in Figs. 2 and 3. The point ^ is a point of inflection with vertical tangent. For an example of such a function, see 388, 5. 366. 1. When the differential coefficient at a does not exist, finite or infinite, the right and left differential coefficients may. They are then different. If both are finite, we have a case illustrated by Fig. 1. Sucli a function is ^/ N e^ - 1 for x 9^ ; Here e^+ 1 = 0, for X = 0. i2/(0) = +l, i/'(0) = -l. If one is finite and the other infinite, we have a case illustrated by Fig. 2. The points P in Figs. 1, 2 are called angular points. NON-EXISTENCE OF DIFFERENTIAL COEFFICIEN'J' 225 2. When both differential coefficients are infinite, but of op- posite signs, we have a case illustrated by Figs. 3, 4. Lf{u) = - CO R/'(a) = -t-m Fig. 3. Lf (,a) = -i- CO B/'(a) = -co Fig. 4. Here I* is a cusp with vertical tangent. See 388, 3 for an example of sucli a function. 3. In Case 1 the curve has not one but two tangents at P ; viz. a right and a left hand tangent. Case 2 may be considered as a special or limiting case of 1. The curve has a tangent at P. In both cases the direction of motion along the curve changes abruptly. When we say "a curve has at every point a tangent," we exclude Case 1. Non-existence of the Differential Coefficient 367. 1. We consider now some examples of continuous func- tions for which the differential coefficient on either side of certain points does not exist. Let = 0, TT for x^Q X for a; = 0. The graph F of ?/ is given in the adjoining figure. Q Evidently F oscillates between the two lines y=±x, (1 with increasing rapidity as x approaches 0.. 226 DIFFERENTIATION For a:=5tO, ^ is evidently continuous. For a; = 0, y is also continuous, since lim X sin — = 0. 1=0 X At the origin the secant line OP oscillates between the two lines 1), and obviously does not approach any fixed position as P approaches from either side. Thus F has no tangent at all at 0. This result is verified at once analytically. For, ^y . IT . n —^ = sm -i— 1 at a; = U ; Ax Ax and as Ax = 0, sin -r— oscillates infinitely often between ± 1. dy 1 2. For use later, let us find ~ for a; = — ax n We have, setting Ax = A, ¥^l('-+k\ + h ] sm Ax h\n J 1 , - + h n 1 + wA . nir sm But sin - = sm TiTT — =— (— l)"sm- 1 + nh \ 1 + nhJ 1 + nh Hence, setting u — 1 + nh and thus ^y ^ ix« sinw Ax u T ^V ^ -«N„ T sinM lim — ^ = — ( — lynir lim = -(-l)"W7r, by 301. NON-EXISTENCE OF DIFFERENTIAL COEFFICIENT 227 368. Let y =f(x) = x^ sin—, a; ^t ; X = 0, x = Q. Evidently y is everywhere continuous even at 0. Tlie graph T oi y oscillates between the two parabolas y = ±x^ with increasing rapidity as x approaches 0. As P approaches 0, the secant OP oscillates between narrower and narrower limits, which limits converge on both sides toward the 2;-axis. Evidently, /(0) = lim^=0; and r has a tangent at 0, viz. the axis of x. This result is verified analytically at once. For, and -r^ = A3;sin— - at 0, Aa; Aa: lira Aa; sin-^— = 0. Ai=o Aa; 369. Let A = 0, ±1, ± |, ±\, ... For X not in A, let y =f(x~) = x sin— sin- sin — X For X in A, let y = 0. Here y is everywhere continuous, even at the points of A. Let C be the graph of y^ and V the graph of y^ = x^\n-, considered in 367. In Fig. 1, the full curve represents an arc of r for an interval /„ = (a„, 5„), «„ = -, 1 ^ 6„= -• The dotted curve, call it P, is n — \ symmetrical to V. Fig. 1 228 DIFFERENTIATION We observe now that y is obtained by multiplying the ordinate y-^ of r by the factor 2/2 = sin - Sin— X As X approaches an end point of i^. sin — = 0. Hence y^^ oscillates infinitely often between ± 1. The effect of the factor y^ in y = y-^y^ is thus to bend F in I„ an infinite number of times, so that the resulting curve, a portion of C, lies between F and F'. This is represented in Fig. 2, where the light and dotted curves are F and F', and the heavy curve is C. At one of the points of A, as a„, the secant a^P oscillates with increasing rapidity as P approaches a„ from either side. Since lyrnr, by 367, 2, dx Fig. 2. the tangents to F and F' are not the a:;-axis. Hence the limits of oscillation of the secant do not converge to 0, and hence the secant a„P does not converge to some fixed position as x approaches a„. Thus y has no differential coefficient at any point of A, and its graph O has at these points no tangent. Since is the limiting point of A, there are an infinity of these singular points in the vicinity of the origin. 370. Let ^ = 0, ±1, ±2, TT TT For X not in A, let y = f(x) = x^ sin — sin For X in A, let ^ = 0. ^"^ ^ The reasoning of 369 may be applied here. The graph of y oscillates between the two curves discussed in 368. y-. = ± x^ sm — , ^ X FUNDAMENTAL FORMULA OF DIFFERENTIATION 229 There is no tangent at the points ±1, ±2, • • • while at the origin there is a tangent, viz. the a;-axis. The graph O oi y presents therefore this peculiarity : in the vicinity of the origin there are an infinity of points at which has no tangents ; yet at the origin itself C has a tangent. 371. In 369 and 370, the aggregate A is of the first order, by 263, 2. It is easy by the process of iteration to form continuous func- tions which have no differential coefficient over an aggregate A, of order m. Let 6(x) = sin — and y = xd(x)d'-\x) ••• e^^+^\x^. This expression does not define y at points involving division by zero. At these points, call their aggregate A, we set y = 0. It is easy to show that y is everywhere continuous and that it has neither right nor left hand differential coefficients at any point of A, The aggregate is of order m. See 259, 260. • Fundamental Formulce of Differentiation 372. As many American and English works on the calculus derive these formula? in an incorrect or incomplete manner, we shall deduce some of them here. We shall, at the same time, prove them under conditions slightly more general than usual. As domain of definition D of our functions y, u^ v, -•• we take any aggregate having proper limiting points. The domain of definition A of their derivatives will embrace, at most, the proper limiting points of D. It is convenient to represent y(x-\-]i)^ u(x+K)^ ••' by y, u, ••• etCo, -1 dy du -I , f . and. -^, — , ••• by y, w, ••• etc. dx dx 230 DIFFERENTIATION 373. We begin by proving : If the differential coefficient f (^a) is finite^ f{x) is continuous at a. For, since A=o h we have, for each e > 0, a S > 0, such that, if | A| < S, f(a + A) —f(a') j?i r \ , I I M ^ n Hence f(a + A) =/(a) + A(/'(a) + e'). Therefore, lim/(a + A) =fCa}, which states that / is continuous at a. 374. //^ ?/ ^s constant in D, i/' = 0. For for any point of D. Ax 375. ie^ y = u±v. Let u' ^ v' be finite in A. Then y' = u' ± v' in A. For A^ _ Aw Ay ^-. Ax Ax Ax Since w', v' exist and are finite, we can apply 277, 2, to 1). 376. Let y = uv. Let w', y' he finite in A. Then, in A, y' = wv' + vu' . (1 For _ Ay _uv — uv _ (u + Am) (^ + A'^) — uv Ax Ax Ax uAv -\- vAu -Av , Au ^o = ■ = u \-v — -• {z Ax Ax Ax FUNDAMENTAL FORMULA OF DIFFERENTIATION 231 By 373, .. _• lim u = u. By hypothesis, lim — ^ = %', lim— = t/. Aa; Ax Hence, passing to the limit in 2), we get 1). 377. 1. Let y=-' Let u', v' be finite and v=^0, in A. Then f vu' — uv' . A / -1 (2 For Ay _ vAu — uAv _ 1 Aw _ w 1 Av ^ Ax vvAx V Ax V V Ax By 373, lim V = v. By hypothesis, T Aw , T Av f nm — =u', nm — = v'. Ax Ax Passing now to the limit in 2), we get 1). We observe, by 351, that v=^0 for Ax sufficiently small, since v is continuous and r^t at a;. It is therefore permissible to divide by V, as in 2). 2. Criticism. Some writers derive 1) as follows. From u they get yv = u. They now apply 376, which gives u' = yv' + vy'y (3 which, solved, gives 1). This method is incorrect. For to get 3), by using 376, we must impose the condition that y' exists and is finite. But noth- ing in this form of demonstration shows the existence of y' . The method then shows only this : on the assumption that y' exists, its value is given by 1). But this assumption of existence makes the demonstration worthless. 232 DIFFERENTIATION 3. Many writers of elementar}^ mathematical text-books are not alive to the fact that a demonstration, which involves an assump- tion of the existence of certain quantities or forms, renders the demonstration invalid. This error of reasoning is extremely com- mon in the calculus. Because determinate results are obtained by such reasoning, it is allowed to pass as conclusive. To show how fallacious this style of reasoning is, let us assume that we can write * 1 c2-4 = a sin X -\-h cos x. Granting this, it is easy to determine a and h. In fact, setting a; = 0, we get Setting a; = — , we get 4 a = 7r2 - 16 Hence ; sm X cos X, a;2 _ 4 ^2 _ 16 4 a perfectly determinate result; but also a perfectly false result. In fact, the right side of 4) is a periodic function, while the left side is not. The reader should therefore not begrudge the pains it is some- times necessary to take, to prove an existence theorem. He should also notice that by modifying the form of proof it is sometimes possible to avoid assuming the existence of certain things which enter the demonstration. Witness the demonstrations just given of 1) in 1, 2. 378. Let y=f(x'), and x = g(t). Let g'(t^ = — he finite in T. n dt Let X he the image of T. If -^=f' (a;) is finite in X, dy^ _dy^ dx ^1 dt dx dt * In treating the decomposition of a rational function into partial fractions, it is often assumed, ivithout any jitstification, that the decomposition in tlie form desired is possible. FUNDAMENTAL FORMULA OF DIFFERENTIATION 233 Before proving this theorem, we wish to illustrate two cases which may occur. Ex. 1. Let x = t sin 2 mirt. The period of sin 2 mtrt considered as a function of t is — By taking m very large but fixed, x will oscillate a great many times near m the origin. Where the graph cuts the «-axis, i.e. when A. , 1 ,1,3 2 m m 2 m we have Ax = 0. But however large m is taken, we can determine a 5 > 0, such that Ax ^t 0, in F5*(0). In fact, we have only to take 5 < 2 m What we have shown for i = is true for any other point t. That is, we can always choose 5 sufficiently small so that in V^*(J,), Ax shall not =0. Ex. 2. x-fi sin -,fovt^O; = 0, for t = 0. The graph of this function we considered in 368. For any point « ^t we can determine a 5 such that in Vs*(,t), Ax does not vanish. Not so at < = 0. Here, however small 5 > is taken, x oscillates infinitely often in Fg*(0) ; and thus for an infinity of points in F5*(0), Ax = 0. We can, however, throw the points of ^^^(O) in two sets. In one, call it Vo, we put the points for which Ax = 0. Then Vo = ±-, ± m m + 1 In the other set, call'it Fj, we put all the other points of V*. We can now show for the function y =/(x) in the above theorem, that 1) is true for each one of these sets of points, and therefore true for both together. 379. We give now the proof of 378. Let ^ be any point in T; let x be the corresponding point in X. Let Aa;, Ai/ be the increments of a;, y, corresponding to the incre- ment At of t. Case 1. There exists a V*(^t^, in which Ax=f=0. The identity At Ax' At ^ does not involve a division by 0, as Ax ^ 0. 234 DIFFERENTIATION Siiice — = g'(t') is finite at t^ Aa; = when Ai = 0. Hence, by 292, ^* lim^ = lim^ = f. A«=o Aa; Ai=o Aa; aa; Thus, lim -~ = lim -r-^ • lim -r— , A«=o Ac Ax=o Aa; Afc=o Ac and c?^ dx dt which proves the theorem for this case. (2 Case 2. Aa; = for some point in every V*(f). Let Vq be the points of V*(t)^ for which Aa: = 0. Let V^ be the remaining points of V*(f). If we show T Ay , dy ^. Aa; ^o iim-r^, and ^lim-r— , (o At cZa; Ar ^ have one and the same value for every sequence of points whose limit is t, we have proved 2) for this case. Let A be any sequence in V^. Then lim — = 0, (4 since Aa: = for every point in A. As ~ is finite at a;, dx On the other hand, dy ^. Aa; ^ -/ hm -i— = 0. c^a; A Ai lim^ = 0. ^ A^ For, Aa; being for every point of vl, y=fQc) receives no increment, and hence Ay =0 in ^. Thus, for every sequence A^ the two limits in 3) have the same value, viz. 0. FUNDAMENTAL FORMULA OF DIFFERENTIATION 235 Let now B be any sequence in F^ which =t. Let the image of the points B be the points (7, on the rc-axis. Then, by 292, hm — ^ = hm -r^ • hm -i— B iid C I!^X B ^t = ^lim^. (5 dx B ^t Thus the two limits of 3) are the same for each sequence B. It remains to show that one is 0. Now, by 4), since, by hypothesis, lim — = lim — = ; B M A M lim — = a'(0 for any sequence whose limit is the point t. Hence the right side of 5) is 0. Thus the two limits 3) have the value for every sequence A or B. These limits therefore have tlie value for any sequence, whether its points all lie in F^, or in J\^ or partly in Vq and partly in Fj. 380. The demonstration, as ordinarily given, rests on the identity •^ Ay _Ai/ Ax 'At ~ Ax ' At' The theorem is, therefore, only established for functions x=g(t)^ which fall under Case 1. If one wishes to give a correct but elementary demonstration, it would suffice to restrict g(t) to have only a finite number of oscillations in an interval T^ and have at each point of ^ a finite differential coefficient. In an elementary text-book on the cal- culus it is not advisable to consider functions with an infinite number of oscillations. 381. Let y =f(x) he univariant and continuous. Let x = g(y^ be its inverse function. Let f (x) he finite or infinite in A. Let E he the image, of A. Let x and y he corresponding points in A and E. 236 DIFFERENTIATION If f (x) is finite and ^ 0, then g' Qy^= — f (P) jf f(x)={) then a'(v')= { + °^ '^^ '' increasing. ^ ' ^ ^ ' ^ ^^^ [ -co if f is decreasing. If f (x) is infinite, g' (jy^= 0. Since / is iinivariant, Ay and therefore also -^ are 4^ 0. Hence the relation . ^ A.r _ 1 A^~A^ Aa; (1 does not involve for any point a division by 0. Since y is continuous, Ay = when Aa; = 0. We have therefore only to apply 292 in passing to the limit inl). 382. The geometric interpretation of 381 is very simple in the following case : a x■^x Xg 5 Let y z=i f(x) be a continuous increasing function in (a, 5). The inverse function x = g(jf) is increasing and continuous in (a, /3). See Fig. The graph of f{x) and g(jf) is the saqie curve G. At Pj, P^ we have points of inflection. If Pr is the tangent at P, tan = TT = 2' tan = 1 tan 4 dx _ dy 1 ~ dy dx or The consideration of the tangents at Pj, P^ illustrates the theorem for the other cases. 383. We apply the preceding general theorems to find the derivatives of some of the elementary functions, choosing those whose demonstration is often given incorrectly. i>x= ; w^ log a. a > 0, a; arbitrary. (1 For, let y = a\ Then Aa: Aa: (2 But, by 311, n^x _ 1 lim — = log a. Ax=fl Aa; ^ Passing to the limit in 2), we get 1). When a = e, 1) becomes D^e"" = e^ (3 384. 1. i>^loga; = -. x>0. (1 Let Then But From 2) we get, by 381, which is 1). >x log X = 1 x' y = ■■ log X. x = ■.e\ dx dy --e« = x. dx 1 X (2 238 DIFFERENTIATION 2. We can get 1) directly as follows : Ay _ ]og(x + Ax') — log X _ ^\ x Ax 1 X Ax log(l+ J Ax But, by X 310 and 292, log(l + lim ^ ^ Ax=o Ax Ax Hence, passing to the limit in 3), we get 1) again. From 1) we can prove again D^e"^ = ef. For, from y==e\ we have X = log y. Hence, by 1), dx _1 dy y Using 381, we which is 5). have dy (3 (4 (5 385. 1. Criticism. In either of the preceding ways of getting B^e^ and D^ log a;, we need the limit j lim (1 + w)« = e. (1 u=0 Some writers only prove 1) when u runs over the sequence 2^ ^' 4' "• Others prove 1) only for a right hand limit. As, however. Ax may have any positive or negative values as it converges to 0, the limit 1) must be established without any restriction. FUNDAMENTAL FORMULA OF DIFFERENTIATION 239 2. If the method of 384, 2 is used to get D^ log a;, we must not only prove 1), but we must show that 1 1 lim log (1 + uy = log lira (1 + m)«. u=0 „=0 This is rarely done. 3. A third method is to employ the Binomial Theorem, which is taken from algebra. The rigorous demonstration of this theorem for any case, besides that of integral positive exponents, is far beyond the limits of the ordinary high school or college algebra. Moreover, the demon- strations usually given are incorrect. The employment of the Binomial Theorem to find the above derivatives is therefore open to the most serious criticism. 386. 1. The differentiation of the direct circular functions pre- sents nothing of note ; let us therefore turn at once to the inverse circular functions. We take . ^^ y = arc sin x (1 as an example. The notation indicates that we have taken the principal branch of arc sin a;, [223]. Then -2<3'<2' ^^ From 1) we have a: = sin y. Hence , ax , = cos y = Vl — x^, CS dy ^ The radical has the positive sign, as cos^ is not negative for the values 2). Hence, by 381, -^ = i)^ arc sin a; = — •> \x\^l. dx Vl - a;2 = + 00 for a: = ± 1. 2. Criticism. In many books the branch of arc sin x which is taken is not specified. Consequently, the sign of the radical in 3) is not specified. For some branches the negative sign should be taken. 240 DIFFERENTIATION 387. 1. D^- = fxx'^- -^ a; > 0, fx arbitrary. (1 Let y = x>'. Then ^^gA^loga;^ (3 Let fl log X = M. Then y = e\ But dy _dy du ^ dx du dx and • du dx X Hence dx 2. Criticism. Some writers rest the demonstration on log(l + w) _ lim «=o U and are thus open to the criticism of 385, 2. Others proceed thus. From 2) we have \ogy = fi\ogx. Differentiating both sides, we get 1 dy fM y dx X from which we get 1) at once. This method rests on the assump- tion that -^ exists, and so is open to the criticism of 377, 2. dx EXAMPLES 388. 1. y = a + b0. For a; > 0, we can apply 387, getting, since here At = f , dx S-^^ "^ FUNDAMENTAL FORMULA OF DIFFERENTIATION 241 2. For x<0, the formula of 387 is inapplicable, since it resta on the essential hypothesis that a: > 0. We can, however, adopt a method applicable to any x=^0. Set x^ = u. Then y = a + hu^. For all X in 9? which are =/=0, u is >0. Applying 387, we have du 3 On the other hand, by 378, hu ^ dy _dy dM dx du dx since is finite. Hence du dx = 2; dy ^2 b dx 3 -^x 3. When rr = 0, even this method fails, as u must be > 0, in order to apply 387. In order to calculate the differential coefficient at this point, we must start from its definition. We have, setting h = Aa;, ^y /(A)-/(0) ,VP Aa; Here, when Aa; = 0, = h R lim — ^ = + 30, L lim -^ = — oo. Aa; Aa; The graph of f(x) has thus a vertical cusp at the origin, as in Fig. 1.: , 242 DIFFERENTIATION 4. In order to get 72/(0), i/CO), some readers may be tempted to take the right and left hand limits of the expression 1) for x=0. In the present case we would get the correct result. In general, if the expression for /'(a;) assumes an indeterminate form for a particular value of x, say x = a, the reader must avoid the temptation to conclude that /'(a)=lim/'(a:). oo=a This is only true when /' (2;) is continuous at a. Ex.1. f(x) = xs\n-, x^O; X = 0, x = 0. Here/'(0) does not exist by 367, while, for x^^O, /'(x)=sinl--cosi. XXX Thus also does not exist. lim/'(ic) x=0 Ex.2. /(cc) = a;2 sin -, aj^O; X Here = 0, « = 0. while, for a;:^0, /'(0) = 0, by 368, f'(x) = 2 a; sin i - cos 1. X X Thus does not exist. lim/'(x) x=0 r 5. Let I f{x) = x^. We find readily that i2/(0) = i/(0)=+oo. " J The graph is given in Fig. 2. yiq. 2. FUNDAMENTAL FORMULA OF DIFFERJ:NTIATI0N 243 389. 1. Let log x = l^^ log log x = l^x, log log log x = IgX, etc. Since log u is defined only for w > 0, we shall suppose that x is taken sufficiently large so that l^x has a meaning. We prove now DJr„x = - . m>l. (1 J, til J -f J V y=l^x = log log x. u = log X. y = log U. dx du dx u X X log x y=l^x = log . l^x. For, first, let Set Then Hence Next, let Set Then Hence By 2), Hence u = l^x. y = log V,. dy dy du dx du dx du T\ 1 11 dx X log X dx u X log X xl^xl^x By induction, we now establish 1) readily. 2. In a similar manner we establish 244 DIFFERENTIATION From 1), 3) we have two formulae to be used later : i>A^-^ = - -—^ J-; m>l. (4 and J>A-^ = 1 ^-^ 1 -■ ^*l- (5 ^ '^^x — a x—a^x — a In 4), 5) we suppose a;>a, such that the quantities entering them are defined. Differentials and Infinitesimals 390. 1. Since /(:.) = lim^, we have for each e>0, a S>0, such that in V^^Qc)^ ^-/'(^) or I ^y —f (a^)^a: I < e I Aa; |, or Ay=/'(2;)Aa: + e'Aa;, (1 where I e' I < e. We call fXx)^x the differential off(x)^ and denote it by dy or dfQc). The relation 1) shows that A«/ is made up of two parts, viz, dy and e'Aa;. The ratio of these two parts is , — ^. f(x^^Q. Q DIFFERENTIALS AND INFINITESIMALS 245 As /'(a;) is fixed, for fixed x, and e' can be made numerically as small as we please, by taking h sufficiently small, we see that the part e'Aa; is very small, compared with dy for all points x-\-/\x in Fg*. Thus, in the immediate vicinity of x, the principal part of Ay is di/. Differentials owe their importance to this fact. 2. To make the notation homogeneous, it is customary to replace Ax by another symbol, dx, in the expression for dy. We have then dy =/' (^x^dx, 391. The notion of a differential may be illustrated as follows : Let the graph of /(a;) be that in the figure. Let PB, be the tangent at P; and PS=Ax, QS=Ay, RS=dy. QB = e'Ax. The reader will see, if dy^Q^ that as Q approaches P, QR becomes smaller and smaller as compared with RS=dy. This is illustrated by comparing this ratio at Q and at Q' . We see dy = RS approximates more and more closely to Ay as Q approaches P. 392. A variable whose limit is is called an infinitesimal. When employing differentials, we suppose that the increment given to the independent variable Ax = dx can be taken as small, numerically, as we choose. It is thus an infinitesimal. Then both Ay and dy are also infinitesimals. In the limits considered in 301-304, 310-312, the numerators and denominators furnish examples of infinitesimals. Also the lengths of the intervals considered in 127, 2, are infini- tesimals. Many other examples of infinitesimals are to be found in the preceding pages, and many more will occur in the following. 246 DIFFERENTIATION The Law of the Mean 393. One of the pillars which support the modern rigorous development of the calculus is the Law of the Mean. It rests on Holies Theorem. Let f(x) he continuous in 31 = (*? ^)» and fi_a)=f(J>). Let f (x) he finite or infinite within 51. Then there exists a point c within 31, for which /(c) =0. a 0, /(e+A)-/(c)<0, /(c-A)-/(c)<0. Hence Kc + h^-fjc-) ^ h - ' /(^-A)-/(O ^Q^ — h /(c)^0; Those together require that /'(.)= 0. In case f(x) is a constant in 31, the theorem is obviously true 394. 1. The geometric interpre- tation of Rolle's theorem is the following : Let the graph of f(x) be a con- tinuous curve having everywhere a tangent, except possibly at the (1 (2 According to 1), according to 2), 1 THE LAW OF THE MEAN 247 end points J., -B, which are at the same height above or below the a;-axis. Then at some point O the tangent is parallel to the rc-axis. Since /'(a;) may be infinite, the graph may have points of inflec- tion with vertical tangents, as at P. 2. Let A^ B be two points at the same height above the a;-axis. The reader will feel the truth of Rolle's theorem for simple cases if he tries to draw a continuous curve V through A, B, whose tangent is not parallel to the cc-axis. F should, of course, have no vertical cusp or angular point. We say for simple cases, because we cannot draw a curve with an infinite number of oscillations or a curve which does not have a tangent at A or B. Yet neither of these cases need to be excluded in Rolle's theorem. 395. lif'(x) does not exist for some point within % the theorem 394 is not necessarily true, as Fig. 1 shows. (See 366.) 1 k a b Fig. 1. Fig. 2. lif'(x) is not continuous in 21, the theorem does not need to be true, as Fig. 2 shows. 396. 1. Criticism. Many demonstrations are rendered invalid because they rest on the assumption : 1°. In passing from a to J, the function must first increase and then decrease, or first decrease and then increase ; or on the assumption : 2°. There must be at least one point between a, h where the function ceases to increase and begins to decrease, or conversely. Either of these assumptions is true if we use functions having only a finite number of oscillations in 31. In case the function has an infinite number of oscillations in 51, neither of the above assumptions need be true. 248 DIFFERENTIATION The function of Ex. 2, 378, where 51 = (0, 1), illustrates the untruth of 1°. We shall later exhibit functions which oscillate infinitely often in any little interval of 31 and yet have a derivative in 51- Such functions show that 2° is not always correct. 2. The demonstration given in 393 is extremely simple. It rests, however, on the property that a continuous function takes on its extreme values in an interval (a, 6). In an elementary treatise this fact might be admitted without proof, since it seems so obvious. 397. 1. Laiv of the Mean. Let f(x) he continuous in 51 = («, ^), and letf'(x) he finite or infinite., witlmi 21. Then, for some point a (h - :.). Evidently ^(a)=(/(5)=0. Also at those points, at which /'(a;) is finite, /(.)=-/(.) + ^W9^; (2 while at the other points of 51, g' (x) is infinite. Thus gQx") is continuous in 2t, and g'Qjc) is finite or infinite within %. Hence, for some point a a be a point of 21. The function /(a;) satisfies the con- ditions of the Law of the Mean in :53 = (a, x}. Hence, by 397, /(a;) =/(«) + (a;- a)/ (c). a0, and /'(c) has the sign cr, when not 0, fix"} ^/(a;'), if o- is positive ; /(a;") 0, and /' (c?) ^ 0, we have f{x")>f{x'y, i.e.^f(jc) is monotone increasing in 31- To show it is constantly increasing in 21, suppose Then /(a;) must =f(a) for all points in ^ = («, /3), since it is a monotone increasing function. Since /(a;) is a constant in SQ,f'(x) = in ^, which contradicts the hypothesis. 404. Let f'^x^ be eofitinuous in the intei'val 2(. Then the differ- ence quotient — ^ converges uniformly tof'Qx} in %. Ax For, by the Law of the Mean, f{x + h~)-f(x)=hf(x+eh'). 0<6'<1. Hence Ay Ax ^f'(x + eh^. (1 But /'(a;) being continuous in 2t, is uniformly continuous by 352. Hence lim/' (x -{-dh')= f (x) . uniformly . ft=0 ' -^ ^' lim -r^=/'(x). uniformly in %. Derivatives of Higher Order 405. The first derivative of/' (a;) is called the second derivative of f(x^, and is denoted by Evidently, /'(a;), J)Jfix}, 1^. dx^ ft=o A ^^=0 ^a; this limit being finite or infinite. DERIVATIVES OF HIGHER ORDER 253 In this way we may continue to form third, fourth, ... and derivatives of any order. Derivatives of order n are denoted by 406. We add the following formulae, which will be used later. They are easily verified : i)X=/i./x-l. .../A-w + l.a;'*-". x>Q. (1 I)l(l + xy = ^i'ix-l....fi-n + l-(l + xy-''. l+x>^. (2 i>>^ = e^. (3 D^ sin X = sin [ — + x\. (4 2)!^cosa;= cosf ^ + a;]. (5 407. Let y = mw, where w, v have derivatives of any desired order. The following relation is known as Leibnitz' s formula. where fn\ n • n — \ • •••n~m-\-\ mj 1.2 — m We prove it by complete induction; i.e. we assume it true for n and prove it is true for n + 1. For n = 1, 2 it is obviously true. Differentiating 1), we get 254 DIFFERENTIATION Now, by 96, \mj \m — lj \ m J This in 2), gives which is 1), when we replace in it nhy n-\- 1. 408. 1. Let us apply Leibnitz's formula to find the derivatives of y =f(x) = tan X. We have y = sec'^a;, 1/" = 2 secure tan x = 2 t/y'. Now This gives y"=2(y'3/+y2). 3/- = 2(t/*^2/ + 4 y'"y + 3 y"2), etc. 2. Another way is the following, which will lead us to a formula that we shall need later. We have 1/ cos X = sin X ; or setting, u = sin X, z = cos a;, u = yz. Now by Leibnitz's formula, TAYLOR'S DEVELOPMENT IN FINITE FORM 255 Also, by 406, 4), 5), sin X. Hence 1) gives sin (~ + ^)={ ^'"^ - (2)^^"-^' + (^)^<"-^' - • This gives the recursion formula, sin f -— + a; ] ^n + tana; I (!^')y«-i'-('^')y-«> + ... I . Setting a; = in 2), we get /»)(0)-(^)/(-2>(0) + (^^)/-«(0)--.= sin^. (3 Taylor s Developinent in Finite Form 409. 1. Using derivatives of higher order, we can generalize the Law of the Mean as follows : In the interval % = (a, 5), let f{x) and its first n — 1 derivatives be continuous. Let f^"\x) he finite or infinite within %. Then for any x in 51, /(2;)=/(a) + ^^^/(a) + ... n~ll nl where w a(c)-^l = 0. (5 M— 1 1 As c^^a;, the first factor in 5) is not 0. Hence the parenthesis is 0, which gives r^\c)=A. Putting this value of A in 4), we have 1). TAYLOR'S DEVELOPMENT IN FINITE FORM 257 2. The formula 1) is called Taylor s development of f(cc) in finite form. It may also be written as follows : Set x = a+h, c = a-\-6h. 0<0<1. Then 1) becomes /(a + A)=/(a) + l/'(a) + |^/"(a) + ... + ~^/'"-"(<^) + ^fKa + 6h). (6 n — 11 ni a-\-h, in 21. 410. 1. Letf(x) and its first n — 1 derivatives be continuous in the interval SQ = (a — H^ a + ZT), while f^'^^x) is finite in ^. Then for any x in iS, + (^-^)""> -i)(a) + ^^=^>«)(c), (1 w — 1 ! n\ =/(a + A) =/(a) + A/' (a) + 1!/'' («) + ... where + -^/"-'^(a) + A/<")(a + dh-), (2 n — 1\ n : x = a + h, c = a + 6h, 0<^<1, \h\ 5, if we change x and c accordingly. 2. When a = 0, 1) gives fix) =/(0) + f-/'(0) + f'/'co) + -. + "^^r-Kdx). (3 1 I 2 ! nl Qx- 0? 3! 0. a"^ = 1 + — log « + ^ log^ a ' a^^. Hence, as in 311, 3. Let a^ — 1 lim = log a. x=0 X f(ix~) = (\^xY. We get from 410, 3) y^xy=l-\-fJLX-\- Hence, as in 312, (i + xy = i + fix + a?' ^'^ ^ (1 + exy-\ 0 0. We can therefore apply 378 and 387, getting df X df _ dx Va;2 + 2/2 dy Va;2 + j/2 Thus the partial derivatives with respect to x and y exist at all points different from the origin. PARTIAL DIFFERENTIATION 261 When the point x, y is at the origin, we cannot apply this method. (Compare 388.) We therefore proceed directly. We have This shows that A% Ax i?/',(0, 0)=r+l, Z/,(0, 0)=-l. Thus the partial differential coefficient with respect to x does not exist at the origin. Similarly, Bfy{Q, 0) = + 1, Z/^(0, 0) = - 1 ; and the partial derivative with respect to y does not exist at the origin. 416. In the case of two independent variables, the (first) partial differential coefficients admit a simple geometric inter- pretation. Let the graph of z=f(x, y') be a surface S. The plane y = constant intersects ^ in a curve C. Let PT be the tangent to C at P = (a:, y, 2), making the angle 6 with the a;-axis. Then bx = tan 6. Compare 365. The partial differential coefficient dy has a similar meaning with respect to the y-a.xis. 417. Let f^ (a:^ •"^m) be finite for a domain A. We may now reason on /j. as we did on /. Let a^ be a proper limiting point of A, and x' = {x^ • • • Xj_i, x^ + h, x^+i •■■x,^ any point of A different from X. 262 DIFFERENTIATION ft=o h is finite or infinite, 77 is called the second partial differential coeffi- cient of / with respect to 2;^, x^ at the point a;, and is denoted by 7^2 -C/' ~\ •/ V 1 * * * 771 J -Pf /" \ ■^ XjaTj/C^l • • ■ ^my » « a i •^ ^i^j V^l * * ' •^'n-' ' The aggregate of these t/'s will define a new function over a cer- tain domain Aj ^ A, which is called the second partial derivative^ first with respect to x^, then with respect to Xj. Proceeding in this way, we may form third, fourth, ••• partial differential coefficients and derivatives. Change in the Order of Differentiating 418. 1. In almost all cases which occur in practice, the partial differential coefficient has the same value, however the order of differentiation is chosen. For example : f" = /'" = f" = /'" = f" = /'" That this is not always true is shown by the following example : 2. f(^iy)—'^y~^ — ^' for points different from the origin. X -\- y = 0, for the origin. Then if x, y is not the origin, bx ^\x^ + y'^ (x^ + y'^yy ^ ^ = x[ ^^ "~ ^^ - "^^V I (2 dy [x^ + y^ (2-2 + 2/2)2 At the origin, ^=0, ^=0. (3 dx ' By ^ CHANGE IN THE ORDER OF DIFFERENTIATING 263 From 1), 2) we have, in particular, Consider now the second partial derivatives. From 1), 2) we have, for all points different from the origin, ^ _ :i;2 _ ^2 r 8a:y 1 _ dj dxdy x^ + y^ \ {pfi + y'^)^ j dydx At the origin, we have from 3), 4), 5), ^=-^; .•./-(0,0)=-l. (6 |S = ^; .../-(0,0)= + l. (7 Hence, at the origin, 52/ ^dj dxdy dydx 3. In connection with this example, we may warn the inexperi- enced reader to avoid certain errors he is likely to fall into. To get the equations 3), i.e. /X0,0)=0, /X0,0) = 0, it is not permissible to set a: = 0, y = in the relations 1), 2). In fact, these formulae were obtained under the express stipulation that this point x = y = Q be ruled out. To get the equation 6), i.e. /^(0,0) = -l, (6 it is not permissible to differentiate 4), i.e. fXO,y) = -^, (4 with respect to y, thus getting and in this set y=0, getting the required value of /^(O, 0). 264 DIFFERENTIATION In fact, the relation 4) was obtained under the express condition that y =5^ 0. Similar remarks apply to/^j^(0, 0). Junior students are so accustomed to differentiate with their eyes shut that they often overlook the fact that formulse and theorems are usually not universally true, but are subject to more or less stringent conditions. Compare also the example of 388, 4. 419. It is easy to see a priori why /^^(a, 5) mai/ be different from fly'^i^ah). By definition, /i.(«. 2/) = ii™ — S ' A=o n 4=0 K .. If,. fCa + h,b + k')~f(a,b + k) = lim - \ lim ^^ ^ — ^^^ fc=0 rC [ 71=0 n h^ h } Let us set ^., ,,^ /(g + Kb + k) -fja. b + h) -fja + A, 5) +/(a, 5) Then 1) gives /i;(a, 6) = lim lim FQi, k~). In a similar manner we find that fy^cQa, b} = lim lim F{h, Tc). 7i=0 k=a These formulae show that /^y(a, 6), fyxia, 5) are double iterated limits, taken in different order. It is therefore not astonishing that a change in the order of passing to the limit may produce a change in the result. Cf. 322, 323. 1 CHANGE m THE ORDER OF DIFFERENTIATING 265 420. 1. We consider now certain cases when it is possible to change the order of differentiation in a partial differential coefficient. Let f(xy') he defined in DQa, 5). Let L* be the deleted domain of JD. We suppose : a) that fl exists in 7), )8) that f'Jy exists in 2>*, 7) that lim/^^ = A,. finite or infinite. x=a,y=b fg{a,b)=\. a Then If, moreover, S) fl exists for all points of D on the line y = h ; then f;,'Xa,b) = \. (2 We suppose first, that all four conditions a-8 are satisfied, and show that then „,,. . ^., ■, f[',Ca,b:,=f';Aa,b}. (2' Let ^ /(a + h,b + k) -f(a. h + k) -fia + A, b} +fiab) ^= hk ' as in 419. We introduce the auxiliary functions aCx)=f(x,b+k}-f(x,b% (3 Hiy^^fia + Ky^-fia,y-). (4 ^^^^ hkF = aCa + h) - a^a} (5 = H{b + k')-H(b^. (6 Setting, as usual, h = Aa:, k = A2/, we have from S), A/ ^^ -/;(-, ^) = e(a;); where e = e(a;) is a function of k and x, such that lim e = 0. (7 fc=0 266 DIFFERENTIATION Similarly, by a), ' Hiy^^MMa.y^ + rtiy'yU (9 where r} = »;(y) is a function of h and ^, such that lim 77 = 0. (10 ft=0 Then 5), 8) give F=\\fl,{a + hh^-fl,(a,b:^ + eia + K)-e(a)l (11 Similarly, 6), 9) give F=\\f':,(ia,h + k~)-f'Xa,h-) + 7j(h + k-)-7^(h')\. (12 On the other hand, we can apply the Law of the Mean to 5), by virtue of a), getting kF=Gr'{c~). a (18 Similarly, 17) gives, letting first h pass to the limit, and then k, \=fjyia,h'). (19 The equations 18), 19) prove 2'). 2. If we wish to prove 1), without imposing the condition S), we have only to observe that 17) has been established without reference to 3). But, as has just been shown, we can conclude 1) from 17). 3. It is well to note that this demonstration does not postulate the existence of fyj.\ or the continuity of either of the second partial derivatives ; or the continuity of f'y in D or D*. We observe also that x and y can obviously be interchanged in the statement of the above theorem. 421. The case which ordinarily arises is embodied in the fol- lowing corollary : Let fl^ f'y^ fly he continuous in the domain of the point a, h. Then fyxici'-) i) exists, and is equal to f'Jy(a-, b}. 422. It is easy to generalize 421 as follows : Let the partial derivatives of f(x^ • • • a;^) of order ^n be continu- ous in the domain of the point x. Then we can permute the indices L in without changing its value. 268 DIFFERENTIATION Since any permutation of the n indices l'l' ••' l' 12 ?* can be obtained from any other permutation by repeated interchanges of successive indices, we have only to show that we can interchange any two successive indices as t^, t^+i in 1) without changing its value. Let us introduce the function of x, , x, '■r 'r+1 where we consider all the variables on the right as fixed, except the two noted in g. Then, by 421, ^^ ^^ dx, dx, dx, dx^ 'r '■r+1 'r+1 V Hence ^(r+l) _ f(r+l) J il'"tr— I'r'r+l "^ 4"V-l'T+l'r' Differentiating now with respect to x^^ ^•••a;^, in the order given, we get fi.n) _ fin) Totally Differentiable Functions 423. 1. If the function fQc) has a finite differential coefficient at a; = a, we saw that where h is an increment of x^ and a is a function of A, such that lim cc = 0. Under certain conditions, to be given later, an analogous theorem holds for functions of several variables. Let A/ be the increment that f(x-^ ••• Xj^ receives when we pass from the point a= (a^ — a,„) to the point a + A = (a^ + /i^ ••• a„^ + A„j). TOTALLY DIFFERENTIABLE FUNCTIONS 269 Here any of the A's may = 0. Let A/" =/4(«)^i + - +fL(j^^^m + «i^i + •" + «m^m. where the a^ are functions of Aj ••• A^, such that lim ttj = 0, ••• lim a^ = 0. ft=0 ft=0 The function / is, in this case, said to be a totally differentiahle function at a. We call df=f^^(a)h, + - +/L(«)A. (1 the total differential of f at a. Thus, when / is totally differentiahle at a, A/ consists of two parts, viz. : ,/. i t t aj and a/ii + -•• + «m"m' Here the as in the second part have the limit when h = 0. If we replace a by x and set h^ = dx^, •••^m = ^^mi 1) becomes <^f = fr^ (^) dx^^"-+ /L(^) ^^m = •- dx^-\ h T^ dx^. 424. 1. It is easy to give examples of functions which are not totally differentiahle at every point. Ex. 1. Consider at the origin. Here Hence f{x, y)=V\xy\= Vxhf /i(0,0) = 0, /;(0,0) = 0. #=0 (1 at the origin. Suppose now / were totally differentiahle at the origin. Then the increment A/ would, on account of 1), have the form A/=aA + ^^, (2 where the limits of « and /3 are 0. This is not possible. 270 DIFFERENTIATION For, we have directly A/=/(A, k} -/(O, 0) = V|M]. (3 From 2), 3) we have V^ =ah + /3k. (4 To show now that the limits of a, jS are not 0, let h, A; = 0, running over the line L, in the figure. Then h = p cos 0, k = p sin 0. 6 constant. This in 4) gives p Vsin d cos 6 = p(a cos 6 -{- /3 sin ^), or . Vi sin 2 ^ = a cos ^ + /S sin 6. (5 If now a = 0, y8 = 0, the limit of the right side of 5) is ; while the limit of the left side depends on 0. We are thus led to a contradiction. 2. Ex. 2. fixy) = - , for a;, y not the origin. ^7? + ?/2 0, for the origin, lii h-- Hence at the origin. If now / were totally differentiable at the origin, we would have A/= ah + I3k, or r cos sin = /•(« cos ^ + /S sin ^). Hence cos ^ sin = a cos ^ + /3 sin 0. Letting now h, k = 0, this gives, in the limit, cos sin ^ = 0, which is absurd. TOTALLY DIFFERENTIABLE FUNCTIONS 271 425. Let f(xy) be defined in the domain D of the point P = (a, i). We suppose that: a) f'j. exists in D, yS) fy exists at P, 7) f'x 0'^ f'y i^ continuous at P. Then f is totally differentiahle at P. For, = \f{a + h,h + k) -f(a, b + k)l + If (a, b + Jc} -f^ab) | = ^i + \- (1 By a) we can apply the Law of the Mean to Aj, getting Aj^ = hf^(e, b + k}. a(a), in order to use 430. It is interesting to note that the development may not hold if these derivatives are not continuous. Consider the function f(xy^ = ^\xy\, employed in 424, 1. We have •^•^ '/» • /I Tq' Jy~9 /j To' x.y^yj. /i(:r, 0) = 0, fl(0.y)=0. The derivatives of the first order are thus continuous, except at the origin. Let P = (x, a;), Q = (x -^ h, x + h) be two points on the line y = x, which we call L. If now Taylor's development were true in a domain about a, in which the nth. partial derivatives were finite, we could write, tak- ing here 7i = 1, /(^ + 7^, :, + A) =/(:r, x) + h\fX^. 0+fyil OK (1 where (|, |) is a point on L between P, Q. This formula should be valid for all x, h. But in the present case Thus 1) gives \x+'h\ = \x\ + h'&gn ^. (2 That this result is false is easily seen. For example, let a;= — 1, A = 5. Then 2) gives -^ 4 = 1±5, ifl^O ^^ Y ^ = 1, if 1=0. Co <^ OJ? •V^ o ^ CHAPTER IX IMPLICIT FUNCTIONS 436. 1. Let be a relation between the m -f 1 variables x^^ •" Xj^, u. Let a^j = ^j, • • • Xjj^ = a^ be a set of values such that the equation F(a^-a,,,u) = (2 is satisfied for at least one value of u ; i.e. the equation 2) in u admits at least one root. Let D be the aggregate of the points x = (x^ -•• Xjn) for which 1) has at least one root u. We may con- sider M as a function of the x's, u= (f>(x-^^ ••• x^} defined over D, where (a:j ••• x^) has assigned to it at the point x, the roots u of 1) at this point. We say u is the implicit function defined by 1). It is in general a many valued function. EXAMPLES 1. Let y=f(x) (3 be defined over a domain D. Let E be the image of D. Then 3) defines an inverse function , . x = g(y), defined over E, by 217. This same function may be considered as an implicit function, defined by ^/ \ t^^ \ a 2. Let /(«) = 1 for every sc in Z) = (01). If we set the image E of Dis the single point y = 1. The inverse function x = g(y), is defined only for ?/ = 1 ; at this point g takes on all values between and 1. 282 IMPLICIT FUNCTIONS 283 3. Let i^ = be the relation x2 + y2 + a2-r2 = 0, r=^0. (4 At each point of the domain D, a:2 + 2/2^r2, the equation 4) admits one, and in general, two values of z. The equation 4) therefore defines z as a two-valued implicit function u of x, y, over the domain D. *■ x2 + 2/2 + ^2 = 0. (5 In this case there is only one set of values, viz. x = y = z = satisfying 5). Thus z is defined only for a single point, viz. x = y = 0. At this point, 2 = 0. ^' x2 + 2/2 + 22 + r2 = 0. r^ 0. (6 This equation is satisfied for no set of values of x, y, z. The equation 6), there- fore, does not define any function z of x, y. 6. sin2 u + cos2 it — = 0. (7 y This equation admits no solution except for points on the line y = x. For all points on this line, the origin excepted, the equation 7) is satisfied for any value of m in 'St. 2. More generally, let ('S^ be a system of p relations between the m +p variables a;, u. Let D be the aggregate of points a: = (a:j ••• a;^), for which the system )S is satisfied for at least one set of values of u^--'Up. We may consider the w's as functions of the x's. where the ^'s have assigned to them at the point x, the values of the roots u■^^^■^ u^ at this point. We say u^-- Up is a system of im- plicit functions defined by the system S. These functions are, in general, many valued. 284 IMPLICIT FUNCTIONS 3. Suppose we know that a set of values X-^ = ftp ••• X„^ = d^, Wj = Oj, ••• Up = Op satisfies the system S. Let us call the set of values u^ = b^---Up = hp initial values. We wish to show now that under certain conditions, the system S defines over a region M a set of p one-valued continuous func- tions u^--- Up in the variables x^--- a:^, satisfying 6' for every point of M, and taking on the above initial values at the point x= a. Furthermore there is only one such system of functions. The method employed is due to Goursat, Bull. Sac. Math, de France., vol. 31 (1903), p. 184. It rests on a principle, having many applications in analysis, known as the Method of Successive Approximation. 437. 1. Let us first consider only two variables. Tlie method employed for this simple case is readily extended to the most gen- eral case. We begin by establishing the fundamental Lemma. LetfQx., w) he continuous., and -~ exist in the domain J), defined hy 21; |a;— a|(x), in%. which is continuous at a, and takes on the initial value u — b at x — a. IMPLICIT FUNCTIONS 285 The function (j) is continuous in 21, and remains in ^ while x runs over 31. We set Wj - 5 =/(«, ^), u^-h = f(x, u{), u^ - 5 = f(x, ^2), ••• Then all these us fall in ^. For, by 2), Wj falls in ^. Let us admit that u^-i falls in :33» and shoAv that % also falls in ^. In fact, by the Law of the Mean, Uj.— u^ = (Uy — 5) — (mj — 5) (4 Hence, by 1), I u,. — u-^\<6\ u^-i — b\ (5 <0T, (6 since, by hypothesis, Uj._i falls in ^. Thus, from u,.— b = (u^ — %j) + (?*j — h} and 6), we have \Uj. — b\ 0, m, I w„ — w^ I < e, n>m. (8 For, in the same way that we established 5), we can show that \Uj.— Uf.^l\. Since / is continuous, we get in the limit U-b=f(_x, U). We show now that Z7= ^(a;) is continuous in 5t. For, since w„ converges uniformly to (x} in 21, we have cf>(ix + h')=uXx-{-h')-\-€\ W\<1- if n is taken large enough. But Wj, u^, ■■• are continuous functions of x, since / is continuous. Thus, for sufficiently small S, |Wn(2; + A)-w„(a;)|<| for I Al < S and x + h in 21- IMPLICIT FUNCTIONS 287 Hgiicg \(x+h)-(f>(^x)\(x) = 4>^(x), in {A, B), = <^2(^)' ill (^1' A)- Then the equation 1) is satisfied by this function in {A, B{)^ and it is uniquely determined by the fact that it is continuous in (^, J?j) and has the value u = b, for x= a. In this way we can continue extend- ing on the right, and on the left, the original interval, until we are blocked by certain points beyond which we cannot go. Such points may arise laa^B i when F(x, u) ceases to be continuous, or when FJ = 0. IMPLICIT FUJsCTIONS 291 440. 1. We proceed now to extend the theorem of 438 to embrace the system S of 436. To this end we generalize the lemma of 437 as follows : Lemma. Let fiC^i ■ • ' ^m^i •-Up)--- fp(x^ ■ ■ ■ x^u^ • • • Up), and i ,,« = !, 2, ...;,. be continuous in the domain D defined by 21; \^\- a^\<(T ■■■\x^ — a„,\p\l (^ Thus, as we have Kir ~ ^i|< |*^U — ^J + ^''■7 t = 1, 2, •••p. or using 2), l^tr— M0, m, \u^n — u^^\<€. n>m. (4 1 = 1,2, -p. For, as in 3), r \'^i2 —'^ui\<^Vi by 2) and 3), \u,a -u,^2\<^\ by 5), \'^i,i— '^i,3\< ^^V-< 6tC. These relations are analogous to the relations 10) in 437. The rest of the demonstration can now be conducted as in 437 to estab- IMPLICIT FUNCTIONS 293 lish not only the relation 4), but the remainder of the theorem in hand. 2. We can state 1 in a form less explicit, but easier to remem- ber, as follows : Let and „ „ he continuous in the domain of the point a^ ••• amh-^ •••hp. Let these p^ -\- p functions vanish at this point. Then the system of equations u^-h^ = f^(x^---Up) ••• Up-lp=fj,(x^--'Up) admits a unique system of solutions which is continuowi in the domain of the point a:j = a^ ••• a;^ = «^, and takes on the initial set of values u^ = b^--- 11^= b^. 441. We can now generalize 438 as follows : Let Fi(x^---x,„7i^---Up') ■■■ Fp(x^---x„,u^---Up'), and dF ^ .,. = 1,2, ...p. (1 be continuous in the domain D of the point V 5 •''1 ^^ ''l " ' ^m ^^ ^mi U\ = O^"- Up = Up. Let F^---Fp vanish at Q, while the derivatives 1) have the values d^ at Q. Let \d .-.d I i"ll "l?^ I ^p\ ' ' ' ^pp ^Q. 294 IMPLICIT FUNCTIONS Then the system of equations 18 satisfied hy a set of functions which are one-valued and continuous in a certain region 31, about the point a \ x^ = a-^"- Xjf^ = aj^ ; and at this point, these functions have the values u^ = b^---Up= bp. Furthermore, the system S admits no other set of p functions, con- tinuous at a and taking on the initial values b at that point. We replace the system /S by the equivalent system ^ii(«*i - ^i) + ••• + d^(Up - bp) = d^^Qui -bj)-\-'" + d^p(Up-bp)-F^=gj_ (2 + dpp(Up - bp) -Fp = gp. Since A t^ 0, we can solve this system for the differences Ui — bi, and get Wj _ 5j = e^^g^ + • • • + e^pgp =f^ (3 Up- bp = epig^ + ■■■-{- eppgp =fp. Obviously, the functions g, and hence the functions /, are con- tinuous in D. IMPLICir FUNCTIONS 295 So are the derivatives r^- For where Since the ^'s vanish at Q, so do the /'s. Since the derivatives 5) vanish at Q, so do also the derivatives 4). Obviously, therefore, the numbers a, t, of lemma 440 exist, such that 3/ < -, in D. We can therefore apply this lemma to the system 3). Since this system and the given system )S are equivalent, the theorem is proved. 442. 1. Let f(xi-x^,u)=0 (1 admit a solution u = b^ at the point x = a. In D(a^ 5), let f(xi • • • x^^u) have continuous first partial derivatives. Let fl^^ in D. Then 1) defines a. one-valued fu7iction u, in a certain domain A, of the point a, whose first partial derivatives in A are given hy ^ = -^. . = 1,2, ...r^i. (2 For, let a; be a point of A. Let x receive the increment Aa:t, while the other coordinates of x remain constant. Let the corre- sponding increment of u be Aw. Then f(x^---xi + ^Xf-Xj^u + /:i.u')—f(x-^--'X^u')=0, (3 by virtue of 1). Applying the Law of thfe Mean to 3), we have, setting x[ = Xi-\- 6Axi, u' = u + dAu, f'^Jix^ •■■x[--- xy^Ax, +f'uix^---x[--- x^u'}Au = ; 296 whence IMPLICIT FUNCTIONS Passing to the limit, we get 2). 2. For, by 2), df=^^dx,+... + ^Jx,^ + fju = 0. du ^0. (4 du = — T^ dx^ du dx^ du dx^. Multiplying by the common denominator, we have 4). 443. 1. Let the system F^(x^---x„,,Uj^---Up')=0 admit a solution u = b at the point x= a. Let the functions F-^---Fp have continuous first partial derivatives in L(^a, 6). Let (1 J= 5mj du^ dl\ dF, BUp du^ =^0, in D, Then 1) defines a system u-^-'-u^ of one-valued functions in a cer- tain domain A of the point a, whose first partial derivatives — ' in A are given by the system of equations " ai\_^5^awi_^_^3^5^^^ dx du, dx, dUp dx^ (2 dFp_^^J\,du,^_^d_Fpd_Up^^ dx^ 5?! + til l^ + ^i: Au^ ' Ax^ + - + t Am, ip A?/i Ax^ AFp = ct>p + y{rp,^ + y}rp,^ + - + fpp^^ A 1^2 Ax^ Ax. Am, ^=0 (3 0. Thus, AUl Ax^ ■^11 •••^IP "^21 •••^2P i^Pl---fpp Let Aa;^ = 0; the limit of the right side exists, since the partial derivatives of the F's ave continuous, and J^O. Hence the derivatives — ^ exist. Hence in the limit, the system 3) goes over into tlie system 2). 2. The determinant J is called the Jacobiati of the system 1). CHAPTER X INDETERMINATE FORMS Application of Taylor s Development in Finite Form 444. The object of the present chapter is to show how in cer- tain cases we may determine the limit of expressions of the type which, on replacing /(a;), g(x) by their limits, assume the forms — , — , • GO, GO — GO, 1°°, 0'^, GO**. Q' QO' ' 1 1 •> These are ordinarily called indeterminate forms. 445. Suppose by the aid of Taylor's development in finite form, or otherwise, we find that, in R = RD(a)^ f(x)= a(x — a')"^ + (f)(x)(^x — a)"*, m' >m. g(x) = ^(x — ay + '^(x) (x — a)^\ n' > n. where <^, yjr are limited in M, and a, ^4^0. g(x) /3 + (a; - a)'^ -"i/r ^ ^ Passing to the limit a; = a, we have 0, if w > w. i21im4^ = «=« g(x) a//3, if m = n, (T • cc, II m, f are limited in Z>(0). Thus, fW I + x(x') g{x) \^-x^{xy whose limit for x = is 1. 446. To find the limit of Ax^-gix-), (1 when / and g are infinite in the limit, we may sometimes find a development of /(a;), gix) in the form 0^ + (. -Tr- + ■ ■ + "0 + Hi- --»)+■■■+ (^ - ay^i^-), valid in -Z)(a) or RD(^a)., the function <^ being limited here. This method of finding the limit of 1) is best illustrated by an example. lim ( cosec x ) = 0. 1=0 \x / We have cosec x= = , rf)Cx)=i. sinx x{l -x2{x) l-x2,/,(x) ^ "^^ ^' i^w ? Hence cosec X = - {1 + x2^ (x)}. Therefore — cosec x = — xf(x) = 0. 300 INDETERMINATE FORMS 447. When the independent variable a; = + oo, we may set 1 u which converts the limit into B, lim, by 290. «=0 I Example. y = x{a^ — 1 ) t a > 0. lim y = log a. X=+oo For, ■ 1 where Hence 0X -qU - e« loga _ 1 + j^ log Qj ^. u'^^(jt), (p(u) = i log'^a. y = log a + U(p(u) = log a. 448. When the preceding methods are not convenient, we may often apply with success one of the following theorems. These rest on Cauchijs theorem. Let f(x), ,9'(^) ^^ continuous in 21 = (a, 5). Within 9t, let f (x) he finite or infinite and g' (x) finite and ^0, Then f(h-)-fia^ f\c-) g(b)-g(a}. Within B, let f^^-^^x) be finite or infinite, g^''-'^\x~) finite, and g' , g" ••• ^^"~^^9^ 0. Let /, g and their first n—\ deriva- tives vanish at a. Let f^'^\a') be finite or infiriite, while g"^(a) is finite and :#:0. Then For, by 449, f(x)^ f^-'\c} ^ c-a g(x) g^--'\e) g'^-'\c) - g'^-'^a^) ' But as x = a, so does c = a. Hence, passing to the limit x = a, we get 2). Example. Let f(x) = x^, for rational x. = 0, for irrational x. Let g(x) = sin x. Here f'(x) does not exist except at x = 0, where it = 0. Hence, by 2, j;„„&L) = r(0)^0 ^=0 g{x) g'(0) 1 a result which is obvious from other considerations. f(x') 4. In 1, we assume the existence oi X = R lim ,^ :^ , and then show that ^ ^"^^ 721ini4^=\. (1 THE FORM 0/0 303 That the limit on the left side of 1) can exist when \ does not is shown by the example in 3. It is also illustrated by the following : Let f(x) = x2 sin - , for x^ 0. = 0, for X = 0. Let g(x) = X. Then, f or x ^ 0, while Hence f'(x) = 2 X sin cos -: X X g'(x) = 1. x=o g'(x) does not exist. On the other hand, lim4^ = 0. 1=0 g{x) We observe that this result also follows from 2. 451. Suppose: 1°. /(a;), g(^x) are continuous in -D(+ oo); 2°. f (x) is finite or infinite in D ; 3°. g' (x) is finite and ^0 in D; 4°. /(+oo)=^(+oo) = 0. Let lim ,^ = \, X finite or infinite. x=+=o g {x) where x runs over only those values for which f (jc) is finite. Then'' lim 4^ = A-. We set 1 X = —' u Then D goes over into R = BD*(0'). * Cf. footnote, page 301. 304 LNDETERMINATE FORMS Let The functions <^, i/r not being defined for w = 0, we set (/,(0)=t(0) = 0. (1 Since /, g are continuous in i>, ^, t/t are continuous in i2, by virtue of 1) and 4*^. For points of I) at which /' (a;) is finite, ^ ^ -> dx du -^ ^ ^ Hence at the corresponding points u in JR, <^' (u) is finite. ■ From the relation A(^ _ A/ Aa; Alt Aa; Am' we see that when f (x) is definitely infinite in Z), ^' (u) is also infinite at the corresponding u point in R. Thus ^'(w) is finite or infinite in B, while -y^r' (u) is finite and ^ there. Then by 450, 1, if R lim , ,, { = X, \ finite or infinite. «=o ->/^ (w) -^ -^ u running over only those points for which <^' (u) is finite, i21im^=X. «=o ^W But Also R lim 7^ <^ = lim —~4-- (1 „=o '»/r('M) ^=+«> g{x) X = i2 Inn ^f^ = hm :\) { = hm ^^yf^. (2 Hence 1), 2) give the theorem. THE FORM c»/oo 305 The Form g- 452. Let f{a + 0), g{a + 0) he infinite. In R = RD{a) suppose that 1°. /(a;), g{x^ are continuous; 2°. f (x) is finite or infinite ; 3°. g' (x) is finite and =^0. f'(x) R lim ■ ; , = \, X ^mVe or infinite. .=a g'ix) X ranging over only those values for which f (x) is finite. Then * R ii,n 44 = ^• x=a g{x) Let a be small at pleasure ; we can take B so small that „, ^^ y(D = \ + o-'. \a'\ be small at pleasure. We can choose a + ?; < 5, such gix) ' gCx) are numerically 0. Choosing r] as before, we have for every x in (a*, a + t;), ^ = r' + if(l + /.)(l-T"). If we suppose t<1, and TJf sufficiently large, ^>iJf(l-T)-l>a, inD/, where Gr is as large as we please. 453. Xei/(+Qo), ^(+Go) he infinite. In D(+oo}, let • 1°. f{x~), g{x) he continuous; 2°. /'(a;) he finite or infinite; 3°. ^(a;) he finite and 4zO. lim -^ ) -^ = \ X iinite or infinite. !Z%gw We deduce this theorem from 452 in the same way as 451 was derived from 450. THE FORMS • oo, oo-oo, Qo, 1", oo" 307 Jhe Forms • oo, oo-oo, 0", 1", oo" 454. 1. Let/(a;)=0, g(x)=±cx^. Then f(x)g(x) is of the form • GO. Setting /^ = Y» tliis form is reduced to ^• 2. Let f{x) = ± GO, g(x) = ± go, the infinities having same signs. Then f{x) — g(x) is of the form oo — oo. Setting f-9 = this form is reduced to — 1_1 9 1 fg 3. Let f=0, g= 0. Then [/(2;)]^(^) is of the form O^. Let y=/^ /(x)>0. Then log^ = ^log/=-f- is of the form -• If \ogy = \ then lim y = lim [/(a;)]^^^^ = e\ The other forms 1", oo^ are treated in a similar manner. EXAMPLES 1. a;'*log(l — cosx), fx, x^O. (1 has the form • oo for x = 0. We may write it log(l — cos x) which has tne form ^. The conditions of 452 being satisfied, we differentiate numerator and denominator, getting as new quotient sinx 1 — cosx_ 1 x^+^ 2 308 INDETERMINATE FORMS This has the form -, for x = 0. Applying 450, 1, we get, differentiating once more. -2 fX.+ I xf^ ^ sec2* 2 whose limit for x = 0, is 0. Hence, i? lim x*^ log(l — cos x) = 0. 1=0 2. /ilimx«|logx|'^ = 0. a, /x>0. (2 x=0 I'or, ,, ,„ llogxl'" f(x) ' x''logx>" = ' ° ' =~^- ' ° ' x-« g{x) We apply 452. f'{x) ^ IX I log X 1^-1 ^'(x) a x-« If /u < 1, this expression =0. If /x > 1, we differentiate again, etc. 3. At first sight one might think that lim /(^ + ^) - 1 n since — ^^^- = 1. This is, however, not true in general. n For example, let /(x) = e'. Then fin + 1) _ e"+i _ /(n) - e" ~^' Hence the limit 3) is here e and not 1. Again, let Then I /C«+i)_e^ _,,,_,, which = + 00. Criticisms 455. 1. The treatment of indeterminate forms in many text- books is deplorable. We consider some of the objectionable points in detail. When /(a;), g(x) vanish at a;= a, the function g(x) is not defined at a. CRITICISMS 309 Some authors admit division by 0. From this standpoint the value of ^ at a is hidden because ^ takes on the indeterminate form — The true value, as such authors say, may often be found by a simple transformation, or by the method of limits. For example, if f(x)= x^ — a\ g(x) =zx—a, the true value of (f) may be found by removing the common factor a;— a in x"^ — a^ , ^ X — a X— a ^ X — a Thus (x + a^ As already remarked, division by is ruled out in modern analysis. First, because it is nowhere necessary ; and secondly, because of the difficulties and ambiguities it gives rise to. The expression 1) has then no value assigned to it for x = a. We may therefore, if we choose, agree that in all such cases (\> shall have the value lim 9i^) when this is finite. Some authors do this; in this case ^ has a true value at a. However, we shall make no such convention in this work. 2. In this connection let us. give an example of the so-called paradoxes which arise from division by 0. Let x=l; then ™2 i _ i Dividing both sides by a: — 1, we get a; -f 1 = 1, which gives, since x=l, o _ i It is easy to see where the trouble arises. 310 INDETERMINATE FORMS When e^O, we can always conclude from ac=bc (2 tnatj 7 yet a = o. (d If, however, 0. g' {x) X X has no limit for x= 0. Cf . 450. 457. Some writers, using the relation of Cauchy, conclude now that limc^ = ^> This is true if /'(x), g' (x) are continuous at a, and ^ (a)=^0. 458. Some writers, in order to evaluate lim 0, develop /(a:), g(pc) into infinite power series. The possibility of such a develop- ment is established only for a few simple cases in many text-books. For example, such books do not show that sec a:, tana;, e*'°^ can be developed into power series ; yet they give examples of indeterminate forms involving these functions. There is, however, no necessity of using infinite series ; all that is needed for such cases is Taylor's development in finite form. See 445, 446. 312 INDETERMINATE FORMS 459. To evaluate the form ^, some writers proceed thus 1 Hence lim c^C:.) = lim ^ = lim Kx^j^^ Dividing by lim 0(x), they get Hence f'r\ g'{x) This method assumes the existence of lim (^{x) ; that is, the existence of the very thing we are seeking is put in question. Suppose by this method we find that for example; what right have we to say that therefore lim 4^ = 1? None whatever, until by some subsidiary investigation, the existence of lim ^ is established. See 377, 3. Scale of Infinitesimals and Infinities 460. Consider the functions f(x) = log" X, g(x) = x^, a, /S > 0. Both increase indefinitely as x = + . We may ask which increases faster. SCALE OF INFINITESIMALS AND INFINITIES 313 The quotient is of the form ^. The conditions of 453 being satisfied, we consider ^' g\x~) ^ x^ ' IfO 1, we consider ^2 g'\x) /32 x^ Thusif0<«<2, ^2 = 0; hence ^ = 0. If a>2, we may continue this process. As the exponent a is diminished by unity each time, log x must have finally a negative or zero exponent. Thus in every case ^ = 0. 461. 1. Let/(a;), ^(a;) become infinite for a; = a, a being finite or infinite. If lim '^^ ^ is finite and =^ 0, we say ^ and g are of the same order infinite. If we say y is of lower order infinite than g. If lim '' ' is infinite, we say / is of higher order infinite than g. These three cases are denoted respectively by /(^)~^(a^). /(«)<5'(2^). /(a^)>5'(^)- We may also say more briefly, that /(a;) is infinitarily equals leSB than, greater than gix). 314 INDETERMINATE FORMS 2. Similar definitions hold when /(:r) = 0, ^(2;) = 0. If, for example, /.^ n we say /(a:) is infinitely small relative to g{x), or an infinitesimal of higher order than g(x). We may also say f(x) is infinitarily smaller than gipc). In symbols, f{x) 0, 2; = + 00. however large a, is and however small /3. 462. Let us consider now functions of the type For a: > 1, we have < log x 0. We have, then, The values of these iterated logarithms decrease very rapidly. For example, let a; =1,000,000,000 =109. Then ?ia: = 20.723, ^22^= 3.031, ?3a;= 1.108, ^42;= 0.103 ••. l^x = a negative number. Hence l^x does not exist. 463. 1. When x= + 00, we have, if a, a^, "2 "*■ ^^» re* > l-^^^x > l^'^x > l^'^^x ••• (S The sequence S may be called the logarithmic scale. SCALE OF INFINITESIMALS AND INFINITIES 316 To prove S, let Then w = + oo with x. We have lim — ^^^^ = hm _& = 0, 771 = 1, 2, ••. by 461, 3. 2. /f ofj, a^, ••• >1, a: =+00 ; l-f^x > l-^xl^^^x > l^xl^xl^^x > ••• This follows at once from 1. 464. Let a; = + 00, while a, «j, a2, ••• > 0. Then a;"< (e^)"i< (g''"^)»2< (e«''')"3< ... (2» The sequence T may be called the exponential scale. Let ^i^2-\ Ax~) = x% g{x)=e'^^\ ^ J{JS We apply 453. f(^^«^_ g^ {pc) ttje"!^ If now 0l, /^(a:) _ «(ft-l>°-2 g"{x^~ ttjVi^ If 1 < a< 2, this shows that ^ = 0, and so on. Hence , , To show ^ ^N ^ ,=a:x„ ,- let us set u = e^. Then .. (g^Vi „ M«h lim / = lim ^ „^ = 0, as just shown. This proves 1). The rest of the theorem follows now in the same way. 316 INDETERMINATE FORMS Order of Infinitesimals and Infinities 465. 1. Let a: = 0; then x is an infinitesimal j x^^ t? ••■ are also infinitesimals. Taking a; as a standard, we may say x^ is an infinitesimal of order w, n being a positive integer, and, in general, if x>Q, x*^ is an infinitesimal of order ytt, where ^l is any positive number. Then, if i^lim^ x=0 X'^ is finite and z^ 0, we say that fix) is an infinitesimal of order fi. Not every infinitesimal, however, has an order. For example, by 464, there is no number /u,, such that B lim — x=0 X'^ 1 is not 0. Hence e ' has no order. 2. On the other hand, an infinitesimal f^x) may not have an order /*, because x=0 X'^ either does not exist, or when it does it is infinite or zero. Thus X sin - E lim 1=0 x'^ does not exist. Hence ^ X sin - X is an infinitesimal without an order. 3. Obviously, similar remarks hold for infinities. 466. of 21. CHAPTER XI MAXIMA AND MINIMA ONE VARIABLE Definition. Geometric Orientation Let /(a;) be defined in 21 = (a, 5). Let c be an inner point / has a minimum at c. If A/ = /(:r)-/(c)<0ini)*(O, (2 / has a maximum at (a)=^0. Let Then f has no extreme at a, if n is odd. If n is even, it has a minimum, iff^"\a')>0; a maximum, if f"\a')<0. CRITERIA FOR AN EXTREME 319 For, under these conditions, we have A/=/(a + A) -/(a) = J^ /(») (a + ^A). n ! Since f^"\x) is continuous at a, sgn/(">(a + OK) = sgn/(»>(a) = a. If w is odd, sgn A/= a sgn Ji. As h can take on positive and negative values, A/ does not pre- serve one sign in D*(a). Hence, / has no extreme at a. If n is even, sgn A/= 0-. Thus in D*, A/>0, if/«(a)>0, <0, if /(«>(«)< 0. 2. iw 2t = (a, 5), let f (x) exists finite or infinite. The points within 21 at which f(x} has an extreme, lie among the zeros of For, suppose /(.r) has a maximum at c. Then for A > 0, /(c + 70 -/C^X 0, /(c - 70 -/(c) < 0. Thus, /l — A But when 7i = 0, • limi2 = limX=/'(c). (2 On the other hand, 1) shows that lim -B ^ 0, lim i ^ ; which with 2) shows that /'(c)= 0. 3. The reasoning in 2 also shows : If f{x) has an extreme at x= a, thpnf'(a)= 0, if f (cb) exists 320 MAXIMA AND MINIMA 469. Let f(x) he continuous in D(a). Let f'(x) he finite in D*(a~). Let Ef'(a^ = acc,Lf'{a')= — (rcc, a = ±l. Then f has a minimum at «, if cr = + 1 ; and a maximum, if (T=-l. To fix the ideas, let tr = + 1- ft=0 — fl Thus there exists a S > 0, such that /(a + A)-/(a)>0, \h\5*(a). Hence / lias a minimum at a. 470. Let'f(x) he continuous in I)(cl). In D*(a), let f (x) he finite or infinite, and never vanish throughout any interval of it. In RD*(ci), letf'(x) he positive when not zero ; in LB* (a'), let f'(x) he negative when not zero. Then f{x) has a minimum at a. If these signs are reversed, f has a maximum. For, using 403, we see that f(x) is an increasing function in RD(a), and a decreasing function in LD{a). Hence / has a minimum at a. EXAMPLES 471. - fix) = « + xi In 388, we saw i?/'(0)=+oo, Z/(0) = -oo. Hence, by 469, / has a minimum at 0, a result which may be seen directly. 1 472. /(x)=^^, forx:5fcO, = 0, for X = 0. This function was considered in 366. Applying 470, we see that / has a minimum at the origin, a result that may be seen directly. CRITICISM 321 473. Let /(x) = e '\ for xi=Q, = 0, for x = 0. This function is Cauchy's function. We see directly that it has a minimum at the origin. The same result is obtained by 470. Criticism 474. Some writers confound the terms the function has a maximum or minimum at a point, with the terms maximum or minimum of a function in an interval. These two terms may or may not mean the same thing. For example, let f(x)=smx, 31[=(0, 2 7r). q Then / has a maximum at -, and a minimum at -^. These are also the maximum and minimum of/ in 21. On the other hand, if we take ^ = (0, — ] as one interval, / has neither a maximum nor a minimum at any point in Sd ; yet its maximum in ^ is 1, and its minimum in ^ is 0. 475. The following example also illustrates this point. Find the greatest and least distance 8 between a fixed point A within a circle and any point P on the circle. Let the circle be and the coordinates of^lbea, 0; a>0. Then h = -yj {X — cCf- + y^ = Vrt^ -{-r^ —"l ax, and ,-, do — a . Q dx ^ci^ ^r^-2ax Hence S is a decreasing function in 2l = (— r, r), and has no maximum or minimum at any point in 51. It has, however, a maximum and a minimum in 21, viz. 322 MAXIMA AND MINIMA SEVERAL VARIABLES Definite and Indefinite Forms 476. 1. Let f(x-^ ••■ x„,) be defined over a region, of which a is a point. ^^ A/ = /(:r)-/(a)>0, in D*Ca% f has a minimum at a. If A/<0, ini>*(a), / has a maxim.um at a. The theorem of 468 may be generalized thus : Let the partial derivatives of f(x-^ ■ ■ • x^i) of order n-\-l he con- tinuous in I>(ji). Let the partial derivatives of order in I)*(a^, f has a minimum ; if it is < 0, it has a maximum at a. If d"f(a) has both signs in D*(jx), f has no extreme at a. Let iCj = a^ + A J . . . a;^ = a^ + A^. Then A/ = -1- d-fQa-) + _^ d-^'fia + eh-), n\ n + W by 434. Let 7;j ••• t;^ be the direction cosines [244, 4] of the line L join- ing a and x. Then h-^ = rT}^---hjn = rr]^, where Then A/ = f'H(ri^ + r'^+^Kirj), since d^f d"'^]f are homogeneous in A^ ••• A„j. Since the derivatives of order n-\-l are continuous in D^(a'), there exists a positive number (r, such that \K\5(a). (1 DEFINITE AND INDEFINITE FORMS . 323 If now (^y(a)> in D^Qa), ^ is > on the sphere to which 7] is restricted. Then, by 355, 2, there exists a X > 0, such that Then, by virtue of 1), we can choose 8' <8 so small that A/>0, ini),*(a). Hence / has a minimum at a. Similar reasoning shows that if d^f(a) < 0, / has a maximum. Consider now the case that d"f(a} has both signs in D*(^a). Suppose that it is positive at a + A and nega- tive at a + k. Let j&, M be the lines joining these points with a. Let r], k be their direction cosines. ^®* R(iv) = A, A>0. b:(k}=b, b for all points on L between a and a -f- i^A, a excluded. Similar reasoning shows that A/< for all points on iHf sufficiently neUr a, the point a excluded. Thus in any domain of a, however small. A/ has opposite signs. Hence in this case / has no extreme at a. Let n be odd. Then IT being homogeneous, Hence d^f^a) has opposite signs in every domain of a. Hence when n is odd, f has no extreme at a. 324 MAXIMA AND MINIMA 2. Let f(x^ ••• a:„j) have partial derivatives of the first order ^ finite or infinite^ in the region R. The points of i2, at which f has an extreme^ satisfy the system of equatio7is , „ .^ 1^=0 ... ^=0. The demonstration is analogous to that of 468, 2. 477. 1. We have just seen that the sign of c^"/(a) plays a decisive r81e in questions of maxima and minima. But as already observed, d"f is a homogeneous integral rational function of h^ Agi ••• h^ of degree n. Such functions are subjects of study in algebra and the theory of numbers, where they are often called forms. A form ^(^x^ ••. x^~) which has always one sign, except at the origin where it necessarily vanishes, is called definite. Such a form is a^%^+ ■■+ajxj, (1 the a's being not all 0. If the sign of a definite form is positive, it is called a positive definite form ; if negative, it is a negative definite form. Thus 1) is a positive definite form, while 4 •'^l "'m •'^m is an example of a negative definite form. If can take on both signs, it is called indefinite. Thus X^+-'- + xJ is an indefinite form. There is a class of forms which vanish at points besides the origin and yet, when not 0, have always one 'sign. They are called semidefinite forms. Such a form is, for example, {a^x^+ - + a^x^y, which is positive when not 0. Consider the quadratic form F = Ax"^ + 2 Bxy + Cy^. DEFINITE AND INDEFINITE FORMS 326 If A=^0, we can write it F= i j (Ax + Byy + (AC - B2)y^ If the determinant D = AC-Bi is > 0, J?' does not vanish except at the origin, and is therefore a positive definite form if ^ > 0, and a negative definite form if ^ < 0. If Z> < 0, J?" is an indefinite form. F=^lAx + By\ . Hence F vanishes on the line Ax + By = 0, but has otherwise one sign. Thus, in this case, F is semidefinite. 2. The theorem of 476 may now be stated as follows : If d"f(^a) is an indefinite form, f has no extreme at a. If it is a 'positive definite form^ f has a minimum ; if it is a negative definite form^ f has a maximum at a. 478. When n = 2, i.e. when not all partial derivatives of the second order are at a, d'^f(cC) becomes a quadratic form., d^f(a')= ^a^jiji^; i, /c= 1, 2, ••• w. (1 where j-t, ^ n «c.=/ x,x,(«l "O' and hence The determinant A — ^11 ^12 ■ • «l7n ^m — ^/nl ^ni2 * ^mm is called the determinant of the form 1). Let Am_i be obtained by deleting the last row and column in A^ ; let A^_2 be obtained by deleting the last two rows and columns in A, etc.; finally, let Aq= 1. In algebra the following theorem is proved : In order that the form .2/.^.^« ' (2 be a positive definite form, it is necessary and sufficient that the signs of ... „ ^ Ao, Ai, ... A, (3 326 MAXIMA AND MINIMA are all positive. For 2) to be a definite negative form, it is neces- sary and sufficient that the signs in 3) are alternately positive and negative. Applying this result to the theorem in 476, we have : Let the partial derivatives of the third order he continuous in D(a), and let those of the second order not all vanish at a. Let all the first derivatives vanish at a. Let r=0, 1, Ao=l. m. A.= ! If the signs in the sequence \. Aj, ••• A^ are all positive, f has a minimum. If the signs in this sequence are alternately positive and negative, f has a maximum at a. Semidefinite Forms 479. Up to the present, the case that d"f(a) i-s a semidefinite form has not been treated. It is, however, easy to show that in this case / may or may not have an extreme at a. Ex. 1. Here Hence, /(xy) = x^-Qxy'^ + 8yi = (x-2 y'^){x - 4y2) a = (0,0). A(0) = 0, A(0) = 0; /".<0) = 2, /%(0) = 0, /'V(0) = 0. d2/(0) = h-^. We have here a semidefinite form. That / has not an extreme at the origin is obvious. For, if P is the parabola a; = 2 2/2, and Q the parabola cc = 4 2/2^ f>0 we see that /< between these parabolas, and > in the rest of the plane, points on the parabolas excepted, as in the figure. CRITICISM 327 Ex.2. f{xy) = 2/2 + x^y + x* x2\2 , 3 = (..f)%f.. a a = (0, 0). Obviously, from 1), / has a minimum at the origin. Here /x'(0) = 0, /;(0) = o. /i;i(0) = 0, /i^(0)=0, /;i(0) = 2. Hence, d2/(0) = k\ We have here a semidefinite form. 480. It is beyond the scope of this work to do more than show that the semidefinite case is ambiguous and requires further investigation. We refer the reader for a detailed treatment of this case to Stolz, Grrundziige^ Vol. 1, p, 211 seq. ; Jordan, Gour%.^ Vol. 1, p. 380 seq.\ Scheeffer, Math. Ann., Vol. 35, p. 541; V. Dantscher, Math. Ann., Vol. 42, p. 89. Criticism 481. The partial derivatives of order n being continuous in D(a), we have seen that A/= ^/(a) + |j (^y(a) + - + ^1; <^"/(« + ^A) = ^1 + n + - + T,- The terms T^, T^, ••• are polynomials in Ap h^, •■•h^ of 1°, 2°, ••• degree, whose coefficients, except the last, are constant. Letting Aj, ^2, ••• be infinitesimals of the 1° order, ^^ if ^0 is thus an in- finitesimal of rth order. The assumption is now made by many authors that if r <.s, then T^ is infinitely/ small compared with T^. When there is only one variable h, this is indeed true ; it is not, however, always true when there are two or more variables As an example, consider the form 328 MAXIMA AND MINIMA Let the increments Aj, h^ be related by the equation K^^Jh\ (1 i.e. let the point (A^ h^) approach the origin along the parabola 1). Then which-shows that T^, T^, instead of being infinitely small compared with 7^2' ^^6 in fact numerically 6 and 8 times larger than T^. 482. Let us see now how this erroneous assumption regarding infinitesimals, when applied to the semidefinite case, leads to a false result.* For simplicity we take only two variables x, y. = 1 lAJfi + 2Bhk + CB\ +... 2! = ^2+^3+- Let the determinant AC — &=0. Then T^ is a semidefinite form. To fix the ideas let ^ =5^ ; then 2^1 ^2 = A l^A + Bk\^ by 477, 1. Thus A/ has the sign of J., except for the points (A, ^) on the line L, Ax + By = 0. (i For points on i, A/ becomes For points on the line i, on opposite sides of the origin, T^ takes on opposite signs. As the sign of A/ at these points depends on the sign of T^ (making use of the above erroneous assumption), it is thus necessary that jPg = for points on i, if / is to have an extreme at a, 6. If ^g = for these points, * Cf . TocUiunter, Differential Calculus ; Desmartes, Cours d' Analyse. RELATIVE EXTREMES 329 for these points. If now T^i^O on i, / has an extreme * at (a, 6), if T^ has the same sign as A, for points of L, the origin excluded. That this result is wrong may be shown by applying it to which we considered in 479, Ex. 1. Here ^2 = 7^2, T^ = -6hk^ T^ = 8k\ A = 2. The line L is, in this case, the ?/-axis. For points on X, T^ = 0; while T^ > 0, the origin excepted. Thus T^ has the same sign as A. We should have therefore an extreme at the origin, if the above reasoning were correct. But as we already saw in 479, / has no extreme at the origin. 483. Another error which is sometimes made is the following. It is assumed that the function /(a;, y) has an extreme at the point R when and only when / has an extreme along every right line through JR.. That this view is incorrect is seen by the function /(^W) = (a: - 2 /)(a; - 4 /), given in 479, Ex. 1. As the figure shows, a point *S' moving along any line L toward the origin 72, finally remains in a region for which A/> 0, the origin of course excluded. If, therefore, this view were correct, / would have a minimum at M, whi'ch we know is not true. Relative Extremes 484. 1,. Let us consider the problem of finding the points of maxima and minima of a function ^ * According to the above erroneous hypothesis. (1 330 MAXIMA AND MINIMA where the variables u^ ••• Up are one-valued functions of x^ ••• a;^, defined over a region M and satisfying the system . • • • (2 Such points of maxima and minima are called points of relative extreme, to distinguish them from the case when the variables Xj^ ■•• x^, u^ ••• Up are all independent. Let the point (a^j • • • Up) run over the region T when Qx^ • • • a;„) runs over M. Let/, ^j ••• ! + ••• + /*P<^p- We observe that, considering x^ ••■ Up as independent variables, 1^ = 0, ...^ = 0, 1^ = 0,... 1^ = (8 dXj ox^ dMj dUp are precisely the equations 5), 7). 485. To determine whether a point of *S' is a point of extreme, it is often necessary to consider the second and even higher differ- entials of Fl^x^ ••• Xjn). Here it is sometimes convenient to make use of the fact that where the differential on the right is calculated supposing h '" "^mi ■**! to be independent variables. 332 MAXIMA AND MINIMA For, let us denote g considered as a composite function of x by G-i^x^ ••• x^^. Then F{x^-- x^)=a(x^- x^\ since the <^'s vanish now by 484, 2. • Hence But, by 433, 2), = (^V.by484, 8). 486. Example. Let us find the shortest distance from the point P = {axa^az)^ to the plane = biXx + 62^:2 + ^'3X3 + 60 = 0. (1 Let to = 52 = S(a;, - a,)2 i = l, 2, 3. L = f{XiX2Xz). The points of minimum value are the same for w and 5. We have ••• 333 334 INTEGRATION Over each interval (a^, a^+i) ^^^ have two rectangles and „ _ ,, Let Then /Sn contains F, while s„ is contained in F, Let us now divide each of the intervals (a, ftj), («i, ^^a)' •••(«n-i'^) , into two equal parts. We get two new sums S2n and iS'g^. In this way we may continue without end. Let us now give n the values 1, 2, 4, ... We get two limited univariant sequences Each sequence has a limit by 109. These limits are, moreover, the same. For *S'„ — s„ is obviously the area of the shaded region in the figure. b — a Hence Evidently Hence 'S'n - S„ = (/3 - «) ^ lim(*S'„-O=0. lim *S'„ = lim s„. As in the case of the circle, the common limit is, by definition, the area of F. 3. Now T ^ — a Hence » — a s„ = n rm=——f(^rn)' f/(a)+/(ai)+-+/(«^«-i)|; ANALYTICAL DEFINITION OF AN INTEGRAL 335 and therefore, setting for uniformity, a^ = a, ^ = lim2/(0^- (2 But as the reader knows, the expression on the right of 2) is the integral ^j I f(x)dx. 488. Example. Let us find the limit A of 487, 2) for the function /(x)=c^. c>0. 6 — a Set then and Hence n a„ = a + m5, m = 0, 1, ••• n — 1. s„ = 5 . c«{l + c« + c2« + ... + c("-i)«} Now Also, by 311, From 1), 2) we have = 5.c«l^^ = (c^-c»)-l— (1 1 _ c« ^ ^ cs - 1 ^ lim 5 = 0. lim— i— = -i-. (2 5=0 C* — 1 log C ^ = lim s„ <^ ~ ^ logc Analytical Definition of an Integral 489. 1. Let f(x) be a limited function, defined over the inter- H — \- val 21 = (a, 6), a < 6. , ^ ^ ^ ^ ^ , a ttj ttj Qj a-n-i Let us divide 21 into n sub-intervals K = («m-l«m). m = 1, 2, ... W by interpolating at pleasure the points For uniformity of notation, we set a = a.Q, h = a^. 336 INTEGRATION This set of points 1) produces a division of 51, which we denote by Since no confusion can arise, let S,„ denote also the length of the interval S^. The greatest of these lengths we call the norm of D and denote it by 8. In each 3^, let us take a point |^ at pleasure, and build the sum ^8 = 2;/(l™)(«^-«m-i) = 2/(IJC (2 In passing, let us note that the sums S,^, s„ of 487 are special cases of 2). Let now 8=0. If ^T^ converges to a limit J, which is independ- ent of the choice of the points a^, |^, we write J= lim V/(^^)S^ = ffCx^dx. We say f(^x) is integrahle from a to b, and call J the integral of fix) from a to h. f(x) is called the integrand; a, h are respec- tively the lower and upper limits of integration. We also write J= ffdx. The symbol J is a long S^ the first letter of the word sum. 2. To fix the ideas, we have taken ab, the 3's would be all negative. Evidently, whether a is greater or less than b, if ffCx^ldx (3 exists, then '^'' £f(^')d^ (4 exists and 3), 4) have the same numerical value, but are of op- posite sign. UPPER AND LOWER INTEGRALS 337 Without loss of generality, we may therefore, in our discussion, take a< J so that the S's are > 0. 3. Obviously the symbol has no sense when a = b. In this case we shall assign to it the value 0. 4. Letf(x) he integrahle in % = (a, 6), and Then I Cf^xWlMih-a). (5 For, or, - M(h -a) 0, such that -MKfix^KM. or - M(h -a} 0, there exists a 8q, such that S^-S,S'o<'S'^<'^o + f- (2 Let 7/ J, 772, ••• rj^ be the intervals of A. Let 11' u' is' be the intervals of D lying wholly in 77^, t = 1, 2, • • • v ; let a;. Si, ... be the other intervals of D. UPPER AND LOWER INTEGRALS 339 We take now 8^ so small that for all 8 < 8q. This is evidently possible, since A has only v inter- vals, V being fixed. I.et JM:,= Max/, in S,„ Then or. M[ = Max/, in 8[. <^^ + |, by3). (4 Hence, from 2), 4), Sji-S^^faj>m^. Hence Summing, we get By hypothesis, lim Sj) = lim Sj) = L, say. S=0 6=0 Hence lim Jg = L. It is necessary. For, since the integral J" exists, e>0, S,>0, 1-^-2/^)8 I <| (1 for any division D of norm S < 8^- Let 7}^ be any other point in the subinterval B^ belonging to the division D just mentioned. We have also, as in 1), - (2 |J--E/(OS.|<^. CRITERIA FOR INTEGRABILITY 341 Subtracting 1), 2), we have 1 2/(105.- 2/(7; J8 J <|, (3 for any division of norm 8 < 8q. On the other hand, in each S. there are points f^, t/^, such that 4(6 — a) 4(6 — a) Multiplying these inequalities by 8^ and adding, we have Hence S,= ^mA>^f(ivJK-l- S^ - S, < S/(fOS, - ^fCvjB^ + |- This gives, using 3), Hence S-S<€; hence S=S. 494. We can state the theorem of 493 a little differently by introducing the following definitions. The difference between the maximum and minimum of a function /{x) in an interval 31, is called the oscillation of f in 21. It cannot ever be negative. Let D be any division of % into subintervals S^, of length 8^. Let (Ok be the oscillation of/ in 8^. The sum is called the oscillatory sum of /for the division D. We have n^f= 2 (M^ - mj 8^ = ^MA - 2mA ' = Sj) — Sp, (1 342 INTEGRATION 495. In order that the limited function f(x) he integrahle in 21, ii is necessary and sufficient that limn/=0. (1 For, by 494, 1), Qf^iS/) — Sj). By 493, /(a:) is integrable when and only when lim xS'^, = lim >S'^, or when and only when 5=0 which is 1). 496. if /(a;) is integrable in 51, it is integrable in any partial interval Sd of^. Let' 2l = (a, 5), « = («, yS). c I I 6 Since lim 2a)^S^ = (1 6=0 D for any system of divisions whose norm 8=0, let us consider only such divisions involving the points a, /3. Let D^ be the division of ^, produced by D. Then Z)j D since the first sum contains only a part of the intervals S^, and ©^ is Passing to the limit in 2), we have, by 1), lim 2&)^8^ = 0. S=0 D Hence, /(a;) is integrable in ^. 497. In order that the limited function f(pc) be integrable in 51, it is necessary and sufficient that, for each e > 0, there exists at least one division D for which _ „ That this condition is necessary follows at once from 495. CRITERIA FOR INTEGRABILITY 343 It is sufficient. For, by 494, for the division D ' "s ^J ' But then, as the figure shows, ~** ° or, by 491, But then, by 87, 5, Therefore, by 493, f(x) is integrable. 498. In order that the limited function f(x) he integrable in 51, it is necessary and sufficient that for any fair of positive numbers co, cr, there exists a division D of 21, such that the sum of the subinter- vals* of D in which the oscillation of f(x) is >&), is cozJJ^. Hence ^ ^ It is sufficient. For, having taken e > small at pleasure, take O" = ■, CO ^ , (z 2{M-my 2(b-ay ^ where M= Max /, m = Min /, in 51. * For brevity, instead of sum of the lengths of the subintervals. 344 INTEGRATION Then, by hypothesis, there exists a division I) for which SD, < a. This, with 2), gives < (M- m)(T + (o(h — a) = I + i = e. There is, therefore, at least one division D for which Then, by 497, / is integrable in %. Classes of Limited Integrahle Functions 499. If f(x) is continuous in the interval 51, it is integrahle in 51. For, since / is continuous in 5t, it is uniformly continuous. Hence, by 353, we can divide 51 into subintervals of length S>0, such that the oscillation of / in each interval is < &>, an arbitrarily small positive number. There is thus no subinterval in which the oscillation >&>. Therefore, by 498, / is integrahle in 51- 500. If f(x') is limited in the interval 51 = (a, 5) and has only a finite number of points of discontinuity a^, a^, ■•• a^, it is integrahle in 5t. Let CD, (T be any pair of positive numbers. On either side of the points a^ mark the points a'^, a'J , k=1, 2, • •• s, as in the figure; but such that the ' ' , ' — 4? irH — 1 h J total length of these little intervals is <(t. Since / is continuous in (a, aj), we can divide it into subinter- vals such that the oscillation of / in each of them is < *». The same is true of the intervals (a", a'2), {a'^l ^ ag), ••• But this set of subintervals in 51 gives a division of 51 for which the sum of the intervals in which the oscillation is >« is to is as small as we please. As in 500, the totality of these little sub- intervals furnishes a division of 21 for which the sum of the inter- vals in which the oscillation is >cd is <0 there exists a division D for which Then, by 497, / is integrable. 346 INTEGRATION To fix the ideas, suppose f(x) is increasing. Let us divide 21 into equal intervals of length e 8< Then /W-/(«) (1 i:ii>/=s[i/K)-/(«)i+i/(s)-/K)i+ - +i/(S)-/K-i)n 0 and < Jf in 31- Let D be any division of %. Let h^ be one of the subintervals. Let 0,. 0^ 0, be the oscillations of A(a:), /(a;), g(^x) in S^. Let F, /, be the maximum and minimum of fQx) and g(ix) respectively in Then 0,^Fa-fg = FCa-g) + g(:F-n = FO,-\-gOfft)2> ■•• be positive and =0. By 498 there exists a division such that the sura of the intervals in which the oscilla- tion of /is >&)j is <(T. Thus, if a- be taken less than ^, there is at least one subinterval within ^, call it ^j, in which the oscillation is <&)j. Similarly, there is an interval ^^ within :35ii hi which the oscillation of/ is be a division of 21 of norm 8; let Sj, h^., ••• be the sub- intervals of 21 corresponding to this division. Let ft)^ denote the oscillation of /(a;) in h^. Let us form the sum ft) = Max (O]) is finite for all possible divisions i), we say f(x) is a function with limited variation., or that fQc) has a limited variation in 21. We call 5 the variation of f(x) in 21. If S is infinite, we say f(x) has unlimited variation in 21- If f(x) is unlimited in 21, it cannot be a function with limited variation in 21- 350 INTEGRATION 510. Let i) = (aja2"0 be a division of 31. Let us form a new division A by interpolating a point a between a^_i, a^. Then h^ falls into two intervals 8[, h[' in A. Let M,, M[, M[' be the maximum of / in S^, 8[, S/', and let w,, m[, m[', be the minimum of / in these intervals. Then the term in Q)^ is replaced by c»[ + ft,;' = (M[ - m[) + (iM[' - ml') in ft)^. Now at least one of the M[, M[' equals M^ ; and at least one of the m[', m[ equals m^. To fix the ideas, let Then M[ = M,, m[' = w, ft,^ = (ft,[ + ft,[') _ ft,^ = ilif [' _ ^[ ^ 0. (1 511. 1. Let fQc) he a limited monotone function in 21. It has limited variation in 21. To fix the ideas, let it be monotone increasing. Then «/> = l/(«i) -/(«)! + l/(«2) -/K)l + - + \f(h^ -/(«„-i)l Thus, whatever division D is employed, fOp has the same value. Hence S=/(J)-/(a). 2. Let 2l = 2ti + 2(2-+2l^. Letf(x) he monotone and limited in each interval 21^. Thenf has limited variation in 21. For, we get the maximum value of co^ when D embraces all the end points of the intervals 2l«. In fact, let i> be a division which does not include one of these end points, say a, which lies in the FUNCTIONS WITH LIMITED VARIATION 351 interval 5,. Let A be a division formed by adding a to D. Then, by 510, 1), If, on the other hand, i) is a division including all the end points of 2lj, Slg, ••• we cannot increase (o^ by adding other points to i>, as we saw in 1. Thus the variation oif(x) in % is the sum of the variations in each 5l«. As these latter are j&nite by 1, / is of limited variation in %. . -^ 512. 1. It is easy to construct functions having an infinite number of oscillations in 21, which are of limited or unlimited variation. Let J > 1, and in 51 = (0, 5) take the aggregate 1 i 1 i J^i 2' 3' t' Let the line OL make the angle 45° with the a;-axis. Between each pair of 6 points ^ take a point a^. Let /(a;) have the graph formed of the heavy lines in the figure. ^'' n (^ 1 ^\ X* =-,«„_!, -, ••• «!, 11- \n 11 — 1 J Then All 1\ „„= 2(1 + 1 + 1+... + !). As we shall see later, the limit of the expression on the right is infinite. Hence f(x) is of unlimited variation. 2. To form a function having limited variation, take the parabola, instead of the right line OL in the last example. Ax)=o 352 INTEGRATION Then / 1 1 1 «.= 2(l + - + - + ...+ l But as we shall see, the limit of the right side is here finite. Hence /(a;) is of limited variation. 3. Similar considerations show that y = x sin - , a; ^ ; y — ^ for re = ^ X has unlimited variation in (0, 6) ; while y = x^ sin -, a; ^ ; y = ^ for x = has limited variation in (0, J). 513. If f(x) has limited variation in 21, it is integrahle in 21. We apply the criterion of 498. Let then, oo, eo. Then, for any division whatever, where a is the total variation of / in 21- Let then v» is <.vB, ^^23 be the sura of the intervals of D containing points of 21, ^, S, 5, respectively. Let SC^, ;^^, ^^^ be the sum of the intervals which lie wholly in 21, ^, (5, respectively. Then i^<^^ + e,,^ + f >2l. But I = 2t = Cont2l, B = ;^=Cont«. Hence i = ^= Cont 21- Cont «S, which gives 1). 516. 1. By the aid of the auxiliary function /(a;) introduced in 514, 1, the criteria for integrability which we deduced in 495, 497 give at once criteria that A have a content. Thus 495 gives : In order that A have content, it is necessary and sufficient that the sum of those intervals containing hoth points of A and points not in A, converge to zero, as the norm 8 of D converges to zero. 2. From 497 we have : In order that A have content, it is necessary and sufficient that for each positive numher e there exists a division D of %, such that the sum of the intervals in which hoth points of A and not of A occur, is be arbitrarily small. " ^ 2 1 Let us define the division D as follows. Inclose each of the points 1 1 1 .. 1 ' 2' 3' n-l within intervals of length 3^' where e The remaining points of A fall in the interval ("■i-^y »=r Then the sum of the intervals containing both points in A and not in A is 2, the second sum in 2) is 0. Since in _Z>j, we have now Now, by hypothesis, / is integrable in ^, and On the other hand, lim|.9(fj8, = 0, 6=0 ^ by 516 and 520. Hence, passing to the limit, S= 0, in 3), we have 1). 2. The reasoning in 1 gives as corollary : If f(x) is limited m ^, and g{x) is integrable in 21, then f(x) is integrable in ^, and f gQx)dx= i f(x)dx. This is at once evident, on passing to the limit in 3). 3. Let /(a:) be limited in % = (a, b). Let A be a discrete aggre- gate in 51. Let ^ = 2t — A. Let /(a;) be integrable in ^. Then f(x) is integrable in 21, and Jfdx = I fdx. a c/SB' The demonstration is similar to 1, omitting the system D^. 522. 1. Let f(x~) be a limited integrable function ivith respect to the measurable aggregate iB, lying in the interval 2l = («, ^). Let D = (ay, ^2' ■■■ ^n-\) ^^ ^ division of % of norm 8. Let GENERALIZED DEFINITION OF AN INTEGRAL 359 he resulting intervals formed of one or several contiguous intervals of -Z), lying in ^. Then ( fdx = \imXrydx. (1 Let us introduce the auxiliary function g{x) of 521. Then, by 521, Now the division D breaks 51 up into the intervals (a, aj), (ap a^, (a^, ag), ••• (a„_i, 5). Letting i>j, i>2, A have the same meaning as in 521, we have f gdx = T f'^V^-^' = 2 f'^ V<^-^ + 2 I '^V^^; + 2 ( '^V^2;. »/a V*^«' ^^^''i ^"^"t A ^«i But while 2 f'^V^^ = o» i), »/ a^ since ^ = in the intervals of D^. Thus, j^dx = xjydx + 2X"'^'^'^'^- ^^ Now, if we have, by 489, 4, 1 2 C'^'gdx < if 2 («^+i - «.) = ^^' But since ^ is measurable. lim A = 0. Hence the second terra on the right of 2) has the limit 0. Hence, passing to the limit in 2), we get 1). 360 INTEGRATION 2. The preceding reasoning gives the corollary : If f(x) is limited in ^ and integrahle in each of the intervals (««' /5«)' ^^^ 2 I y^^ *^ convergent as S = 0, then Jfdx = lini V ( fdx. This is evident on passing to the limit in 2). 3. Letf(x) he limited in 31= (a, 5). Let A be a discrete aggre- gate in 31. Let^ = %- A. Then Jfdx = lim ^ I fdx^ provided the limit on the right is finite. For, by 2, lim V I "/(ia; = Cfdx. S=0 ^»^'='k '^S But, by 521, 3, Jfdx= i fdx. CHAPTER XIII PROPER INTEGRALS First Properties 523. In the last chapter the integrand f(x)^ as well as the interval of integration 21, were limited. Integrals for which this is the case are called proper integrals, in contradistinction to those in which either f(x) or 21 is unlimited. These latter are called improper integrals. In this chapter we consider only proper integrals. We wish to establish their more elementary properties. In 21 = (a, 5), we shall take a<6, unless the contrary is stated. All the functions employed as integrands are supposed to be limited and integrable in 21- 524. For the sake of completeness, we begin by stating the three following properties already established respectively in 489, 2; 489, 4 ; 504, viz. : f{x)dx=— ^^f{x)dx, a^b. (1 \f^f{^x)dx\f^gdx. (2 For, hCx-)=f(x}-g(x')%0, in SI. Hence, by 1), f^hdx=:f^fdx-f^gdxsO, which gives 2). FIRST PROPERTIES 363 527. 1. We saw in 508 that, if /(a:) is integrable in 21, it must have points of continuity c, in any subinterval of 21. This fact leads us to state the following theorem : Letf(x)^^ in 21. Iffis continuous at c, andf(^c')>0, then //' 'dx>0. To fix the ideas, suppose c is an inner point. Then by 351, 2, there exists an interval (c', e") about c, in which /(a;) > p > 0. Hence by 526, But £fdx^O, £"fdx^p(c"-c'} = ^(0i then I fdx> I qdx. 3. By means of the preceding inequalities, we can often estimate approximatel}^ the value of an integral with little labor, as the following examples show. Ex.1. .b

2. (1 For, if < X < 1, 1 1 Hence * which gives 1). Ex. 2. Vl — X" Vl — x2 I dx<,\ — ^^ri^r: < 1 — = = arc sin i = - , Jo J Vl _ rn Jo VI _ .r.2 6 xe-'^< r'e-«'dtt0. (2 For by 413, 2, 2 €" = 1 + z+—e^', 0<^<1, * In order to illustrate these and a few immediately following theorems we assume the elementary properties of indefinite integrals, which are treated in 536 seq. 364 PROPER INTEGRALS Hence e^>l + s, z>0. Thus _ ^2 ^ „_„2 ^1 -c n ^ ^^ e-»: 0, mb follows now at once. The above is called the first theorem of the mean. We give now some special cases of it. FIRST THEOREM OF THE MEAN 367 532. Letf(x) he integrable in 21 = (a, h'). Let m = Mean/(a;), in %. Then ^b jj(x)dx = m(h-a~). a0, S>0, Z(^+^lziiM=/(^) + ,r, |^|<,, for |A|a. It is now assumed that e = with S. Hence passing to the limit, S=0, 1) gives 538, 1). The objection to this demonstration lies in the tacit assumption that the difference quotient converges uniformly to the derivative. Cf. 404. In other words, that it is possible to divide the interval (a, ^) into subintervals 7ij, h^, ••• h^ such that e^, e^-, ■•• e„ are all < cr, a positive number, small at pleasure. As elementary text- books say nothing of uniform convergence, the above reasoning is incomplete. Change of Variable 540. 1. Let f(x) he limited and integrahle in 51 = (a, 5), a^h. Let u = <^(x) (1 he a univariant function in 51 having a continuous derivative 0'(a;)^O. Let ^ = (a, yS) he the image of 21 afforded hy 1). Let X = yfr(u) 372 PROPER INTEGRALS he the inverse function of (f). Tlieii, if fl^jr^u^lylr' (u^ is integrable in ^, ^J(x)dx = £f\_-^(u)-\-^\u)du. (2 By 358, the correspondence between the two intervals 21, ^ is uniform. Let -£'(%!, ^2, •••) be a division of ^, of norm S. Let Aa;^ in 51 correspond to Aw^ in ^. By the Law of the Mean, Aa;« = '\/r'(77^)Aw^, 7]^ lying in Au^. Let ^« in 21 correspond to rj^ in ^. Then Ifa.^Ax^ = 2/[^(^0]^'(^<)Aw.. (3 are limited, and integrable by hypothesis, we have 2) by passing to the limit in 3). 2. If the conditions of 1 are not satisfied in the intervals 21, ^, it may be possible to divide them up into subintervals, in each of which these conditions hold. 541. 1. Let us evaluate We set X = tan u = i/'(m), or u = arc tan x = 1 + tan V Jo 2 J- = log 2 C'^*dv=- log 2. J = 7r/8 log 2. (2 2. That we should not affect a change of variable in a definite integral, without due precaution, is illustrated by the following example. Let XVw^^^-£rf^=[-''*g]!..=i- (3 Let us change the variable, setting x = - = ^(u). u Then a = -l, 6 = 1; a = -l, /3 = L ^\AKn)WWau = - £ ^, = - 1- (4 The two integrals 3), 4) are not equal. The reason for this is that the function u = (p(x)= - X of 540 does not have a continuous derivative in 2l=(— 1, 1). Indeed, it is not even defined throughout %. 542. Let X = -v/r(w) have a continuous derivative in S& = («, yS), «^/3. Let 51 be the image of ^. Let f(x) he limited and inte- grahle in 91, and let f[^(u)^y^'(u) he integrahle in Sd- Then jjix)dx =£f\:s^(u)W(i'^^du. a = ^(«), h = ^(yS). (1 1. Let us note first the difference between this theorem and that of 540. In 540 i|r(w) is univariant, and 91, ^ are in uniform correspondence. In the present theorem, i/r may have any number of oscillations in ^. Furthermore, the intervals 21 and (a, 5) may not be the same. 374 PROPER INTEGRALS Example. Let a; = >/'(?<) = sin m, S3=(0, ^-v). Then the image of S3 is the interval 91 = ( — 1, 1). On the other hand, a = sin = 0, ft = sin ^- tt = J. Thus (a, 6) = (0, ^) is different from 21. Let /(x) = X. Then /dx = i "xdx — l; \ f{ypu)\j/'udu= \ sin « cos « d« = }. Thus the two integrals are equal, as the theorem requires. 2. To prove the formula 1), consider We shall show that F(u) is a constant in ^. As it is at a, ^=0 throughout ^. To this end we show -r,, . . ^ . n« F'(u)= 0, m ^. Then, by 400, 2, .^(w) is a constant in ^, and therefore 0. At any point u of -33, we have /(^)c^:r-J ^(w)^w. (2 There are two cases : 1°. yjr'C^u^ T^ 0. Then, by 403, ■\lr(u) is univariant in V(u). We can thus apply 540. Hence AF=0, in VCu'), and therefore tt/ n F' = 0, at M. 2°. '\jr'(^u)= 0. Let us apply the theorem of the mean 532 to each integral in 2). Then AF = ^Ayfr - ^Am, where ^ = Mean /(a:), ^ = Mean ^(«*) in Ayfr, Au respectively. Thus Au Au CHANGE OF VARIABLE 375 Am ^ ^ ^ lim ^ = 0, since /['^(w)] is limited in 51, and limi/r'(w) = ',^'(w)=0, tl=U -»/r' bei9g continuous. Hence F' (u)= 0, also in this case. 543. ltetf(x) he limited in 21 = (a, 6), a^5. Xet u = (f)(x^ have a continuous derivative <^' (x) ^ in 21. iei ^ = («, yS) 5e the image of%; a=z(fi(^a}, /8=^(6), Jjet x = -^(u) be the inverse function of (f). Then _ _ jjdx=£f(if(u)^ir'(u)du, (1 j/dx = J[j^(i/r(w)) Vr' (u-ydu. (2 Let us prove 1) ; the demonstration of 2) is similar. Since <^ is univariant, the intervals 21 and © stand in uniform correspond- ence by 358. To fix the ideas let ^ be an increasing function. Then by 881, yjr' {u^ >0, and continuous. Let ^= (wj, u^, •••) be a division of ® of norm B into subintervals Aw^, to which cor- responds a division D= (rc^, x^, •••) of 21 of norm d into intervals Ax^. Let L^ = Max /(a;), in Aa:^. = Max/(-v|r(M)), in Aw^. M^ = Max/(i/r(w))-«/r'(w), in Au^. \ = Min i/r' (m), yi4^ = Max '«/r'(w), in Aw^. l'=Max|/|, in 21. We have to show that S,, = S^Aa:,, /S^ = ^M^Au^ have the same limits. 376 PROPER INTEGRALS Since ^'(2;) and yjr'^u) are continuous, they are uniformly con- tinuous. Hence d and 8 converge to simultaneously. For this reason for any 8 < some Sg, fi,-\<.=^ [, uniformly in «. By the Law of the Mean, Ax^ = -«/r'(vJAw„ v^ in Aw,. Hence But, obviously, Max/ Min i/r' < Max/i/r' ^ Max/ Max t/t' ; if Max/>0, while the signs are reversed if it is < 0. Thus in either case M^ lies between L^\^ and L^/x^. Also, L^yfr'Cy^^ lies between these same bounds. Hence /3-a\ Hence ,^|^e|EAwJ r. ^ r. I p — « I 544. Let X = ^(u) have a continuous derivative in SQ = (a, /3), a ^ /3. Let i/r' vanish over a discrete aggregate A, hut otherwise let it have one sign. Let % = (a, J) he the image of ^, a = t/^(«), h == i/^(/3). J/ owe 0/ ^Ae integrals eadsts, the other does, a7id hoth are then equal. To fix the ideas let JJ exist. By 403 the correspondence between 31, ^ is uniform. Let us effect a division, of norm S, of ^. Let the norm of the corresponding division of 31 be 77. Let ^^ be those intervals con- taining no points of A, while ^2 = ^ — ^1 i^ ^^® complement of ^j. Let 3li, 3I2 correspond to ^j, ^2 respectively. seco:nd theorem of the mean 377 Now But, by 543, while, since A is discrete, lim f =0. 6=0 -/S^ Hence 1) gives J= lim I = lim I 33 «=o »^2t^ ,,=0 »/2lj = X- by 522, 3. A similar reasoning holds, if we assume that X exists. Second Theorem of the Mean 545. Let f(x) he limited and integrable in 31 = (a, 5). Let gix) he limited and monotone in 2t. Then jjgdx = g{a + 0)£fdx + g(h - ^')£fdx, a<^KL')K>rnA, (2 and also r From 2), 3) we have Hence r fQjK=Xfdx+cr^, (4 Multiply 4) by ^(f«) ; and letting «; = 1, 2, ••• w, let us sum the n resulting equations. We get Now /^ /»& rf> or more briefly, ^ ^ ~ Jk-\ Jk Hence letting « = 1, 2, • • • w, we get Adding, we have, SECOND THEOREM OF THE MEAN 379 Since g is monotone increasing, Let 9)Z be the maximum of the integral I fdx^ and m its minimum as X ranges over 21. Then and adding, Thus 7) gives X [ ^(^0 - ^(^«-i) }X-/^^ = ® { ^^^"^ - ^^^1^ } ' ^^ '^^'®''® m<©<91W. (9 Thus 5), 6), 8) give In this equation let 8 = 0. The limit of the left side is fjgdx. ^^'° lim gC^,) = g(a + 0) ; lim gQ,-) = g(h - 0). ^'* IK^)l %)a «/a c/f we get 1) at once from 11). 546. If g{x) is not monotone, the formula 1) of 545 may not be true, as the following example shows. Let f{x) = x'^, g{%) = cos X. Then \ x^cosxdx^O, by 527, L since the integrand is never negative and is in general positive. On the other hand, g(a + 0) = cos - 7r/2 = ; g(b-0)= cos ir/2 = 0. Hence the right side of 1), 545, is zero. The formula 1) is thus untrue in this case. INDEFINITE INTEGRALS Primitive Functions 547. 1. The theorem of 538 is of great importance in evaluating integrals. For, to find the value of = rfdx, (1 %/ a f(x) being limited and integrable in 21 = (a, x), we have only to seek a function F(x) which is one-valued in 21 and has f(x) as derivative. Then, as we saw, J=F{x)-F{a). (2 PRIMITIVE FUNCTIONS 381 Let G-(x) be any other function which is one-valued in 51 and has /(a;) as derivative. Then J= G^x') - 6^(a). (3 Comparing 2), 3), we have a0. log a /dx 1.x n a'^ + x^ a a /dx . X = arc sin -, a^O. Va^ — x^ * I sin xdx = — cos a;. I cos xdx= sin a:. I tan xdx= — log [ cos a;|. I cot xdx= log 1 sin x\. I tan a; sec xdx = sec a;. I sec^ xdx = tan x. 549. Not every limited integrable function in 51 = (a, 6) has a primitive, as we now show. Let /(a;) be continuous in 21; let F(x)= I fdx, a 0. Here M = 1 — 2 2/ cos X + y2 = (?/ — cos x)2 + sin^x 5. 0, and hence m = only for the points whose coordinates are X = W47r, y = (- I)"*. (2 388 PROPER INTEGRALS Continuity 561. Let 7] be an arbitrary but fixed value of ^z in ^ = (a, /3). Let us denote the line y = ^ by ^. Let 4^{x) be defined over 21= (a, 6). If for each e > 0, there exists a S > 0, such that \f(x,'n^K)-^(x)\(^x) along the line H, or with respect to the line H. We denote this by p a 1 a b or lim/(a;, y') = (^(a;), y=ri f(x, 2/) = j>(x). uniformly ; uniformly along H. If in the relation 1), only positive values of h are considered, we say /(a;, y') converges on the right uniformly, etc. If only negative values of h are considered, /(a;^/) converges on the left uniformly, etc. \if(xy^ converges uniformly to /(a;, rf) with respect to the line _ff, we shall say f(xy) is a uniformly continuous function of y with respect to the line H, or along the line H. lif(xy') is a uniformly continuous function of y with respect to each line ^ = ?; in ^ = (a^S), we shall ssij f(^xy) is a uniformly con- tinuous function of y in ^. 562. 1. Letf(x, «/) be regular in any R = (a57/3), a<7Qx) uniformly along the line y = cc. Let 4>(x) he limited and integrahle in 21. 1 hen ^j -J ^j i^lim I f(xy~)dx= I R\\mf{xy^dx= ) <^(x)dx. For, let A = j/ix, a + h~)dx - jj>(x^dx = Jjfix, a + A) - (^Qx')\dx. (2 (1 CONTINUITY - 389 We have to show that e>0, 8>0, lA| 0, there exists a S > 0, such that \f(ix,a + h)-cf>(x-)\<-L- (4 o — a for each 00. Jo-' Joa;2 + w2 °« i?limj'=-. (6 y=0 2 As 5), 6) have different values, the relation 1) does not hold here. Obviously /(x, y) does not converge uniformly to in any interval containing the origin. 563. 1. As corollaries of 562 we have : Let f(x, if) he regular in R = (aba^~). Let it he uniformly con- tinuous in y, along the line y = r). Then is a continuous function of y at 7]. ^^V^^- 2. Let f(x, 3/) he regular in R = (^aha/B}. Let it he a uniformly continuous function of y in ^. Then JQy^ is continuous in Sd- 390 PROPEli INTEGRALS 3. If f(x, ^) be continuous in B,Qaba/3}, JQy^ is continuous in « = («,y9). This follows at once from 352. Example. In 538 we proved the relation ^^ -^ Jo 1 + X2tan2x 2 1 + |X| for all values of X. It required, however, a separate integration to establish it for X = ± 1. By the aid of 3, we may prove the correctness of 1) for these values without any calculation. Consider, to fix the ideas, X = 1. Since the integrand of 1) is obviously a continuous function of x, X in the band B = (0, ir/2, 1 — S, 1 + 5), the integral is a continuous function of X at 1. Hence 1) holds for X = 1. 564. The results of 562, 563 may be generalized as follows : Let A be a discrete point aggregate in 51. We can divide 51 into two systems of intervals, (5 and 3D, such that (S contains no point of A, and the total length d of the intervals T) is as small as we please. We shall say/(aj, y) converges uniformly to (j)Qx} along the line «/ = 77, except at the points A, when, for each e > and any (5, there exists a S > 0, such that 1/(2;, 77 4-A)-<^(2;)|<6 for each < [ A| < S and every x in (5. The terms, /(a;, ?/) converges on the right, or on the left uniformly, except for the points A, need no special explanation. Also the meaning of the term /(a;, 3/) is uniformly continuous along the line y — t], except for the points A, is obvious. 565. Let f(x, y') he regular in the rectangle R(a, b, a, yS). Let f converge uniformly to <^(x) along the line y — 'r], except for the points of a discrete aggregate A. Let <^{x) be limited and integrable in 51 = («, J). Then f{x,y')dx=\ \\XQ.f{x,y^dx= \ <^(x)dx, a<'^<^« fix, 7} + h^dx — \ji{x^dx. Let CONTINUITY 391 We must show that 6>0, S>0, \I>\<€, 0<\h\ in 21, \(x')\, \f(x,^)\ small at pleasure, and then fixing it, we choose the system ® such that its length d< ' 4.M Then D = (\Kx, r) + h:^-(x)\dx+ flfCx, ^ + A) - «^(a:) I dx. ence , ^ , , ^ , But '©1*^2' IX On the other hand, by hypothesis, for each < | A | < S, and every x in (5. Hence IXNl Hence 2) gives which proves 1). 566. As corollary we have : Let f(x^ y) he regular in the rectangle R= (a, 5, a, yS). Let it be uniformly eoiitinuous in y along the line y = ri., except for the points of a discrete aggregate A. Then is continuous at ij^ « < ^ < /3- 392 PROPER INTEGRALS Differentiation 567. 1. 1°. Letf(x,y')^f'y(x,y)heregulm'inR = (aha^'). 2°. Letf'y he uniformly continuous in y along the line y = r], ex- cept for the points of a discrete aggregate. Let pt, Then c^ J' On) =j/yi^^ Vy^, ab. Obviously, f/dx=-jjdx. (4 PRELIMINARY DEFINITIONS 401 For, to fix the ideas, suppose fix) regular except at «, and let the integral on the right be convergent. Then, if a0, g>0, I (^dx- Cfdx\0, S>0, L/- P|<1, a0. I" But Hence lim?-li^ = l, by 299, 2. «=o X \ ^=1. 576. Letfix) he integrahle in 5t= (a, 6), awe? regular except at a, 6. igi c he any point within 21. Then fdx=jjdx+j/dx. (1 For, since J" is convergent, e>0, S>0, |j--rir!0, let there exist a F*(a) sueh that (ix-ay\f(ix)\0. Ex. 2. is convergent. For, \ X ^ sm- dx Jo X iJlima;'* . |x~^sin-| =0, M>f 580. Let f(x) he regular in % = (a, 5) except at a. In V*{a) let f(x) have one sign a, while Then ix-ayf(x)>M>0. J= i fdx = a- CO. For, let a I dx = M\og \Ja'^ \—Jo. X — a °a—a = + 00, when a = a. I fdx = 0- • CO. Example. /•! ^^j. Hence Jo 1 _ a;2 + 09. For, for X near 1. (i-.y(x)=^>l. 408 IMPROPER INTEGRALS. INTEGRAND INFINITE 581. Let f(x) he regular in % = (a, 5), except at a. ^^^ \=\im(x-a^f(ix) x=a exist. If f(x) is integrahle^ \ must he 0. Let us prove the theorem by showing that the contrary leads to a contradiction. To fix the ideas suppose \ > 0. Then, for each /i such that 0 fi, in F/. Then the singular integral, iJf>0, yu>0, Jdx = + co, by 580. Hence / is not integrable in %. 582. The criteria of 579, 580 admit a simple geometric inter- pretation. Consider the family of curves H^l (x~ayy = M, in the vicinity RV*(a). The curve H^ is a hyperbola. If ft 1, H^ lies above H^ Further- more, if l>ft>)u,', H^ lies above H^.. The curves H^ all cut each other at the point a; = a + 1. As we are only interested in these curves in the immediate vicinity of the point x = a^ the point a-\-l lies beyond the range of the figure. The jx tests may now be stated as follows : If in some F'*(a), |/(aj)| remains below some H^ which lies below Hy, fix) is integrable. If, on the other hand, f(x) has one sign in F'*(a), and | f(x)\ remains above H^, the corresponding integral is infinite. CRITERIA FOR CONVERGENCE 409 583. Ex. 1. ^^ ri dx Jo , a/1-x2 The only singular point is x = 1. Let us apply the n test at this point. Since /(x) = - 11 1 Vl - x2 Vl - a; vT+x we see that j ^ ilim(l -x)2/(a;) = — . We may therefore take ij. — \, and J is convergent. 584. Ex. 2. ^r (?X Vx2 _ 1 . 1 - K2a;2 0 0. lim x'^ log X = 0, lim x^ log gr(x) = 0. Z=0 2=0 lim x^ log sin x = 0. Thus in the /x test, we can take for /x any positive number < 1. Hence J is convergent. 410 IMPROPER INTEGRALS. INTEGRAND INFINITE 586. Ex.4. J= f^^^dx, a>0. , The singular point of the integrand /(x) is x = 0. In its vicinity F*(0), /(x) has one sign a- = + 1. Then, by 579, J is convergent for yu, < 1 ; and, by 580, it is divergent for m ^ !• 587. Ex. 5 d - \ ■ J^ psinx^j. ^^o_ Jo y-u. The only singular point is x = 0. In F*(0), the integrand has one sign 0, X > 1, s, let there exist a V*(a), such that in it ix-a-)- ?i-J- . ?2-l- - ?,_i-i- • ts-^ ' \Kx-)\a sufficiently near a, and s = 1, 2, ••• \-l DX \-K 1,1 7A 1 x-a r^_ay _j_i I- _ x—a x—a x—a Integrating, we have, for a < a' < «" < a + S, p' dx ^ 1 r^i-A 1 _ ^i-A 1 1 Ja , . ; 1 7A 1 X — 1| a" — a * a' — aj (x — a)l^ •■• I, — ^ -• X — a X — a < e, for 3 sufficiently small. Thus the singular integral of |/(a^)| at a is evanescent; for J^'\f(ix^\dxM>0. X — a X — a Then /»& J= 1 fdx = cr • 00. From 389, 4), for s = 1, 2, •••, and x>a sufficiently near a, -»''"■ ^^Ta = z — m — n- ' (x—ay^— — '■■ Is X — a X — a Integrating, ^ i— = 4+1 ' a a— a = + 00, when a =i a. Hence ^ t/ =s O- • OO. 590. The logarithmic tests 588, 589 admit a simple geometric interpretation. Consider the family of curves (J Os,k; y = i \ — 1— ' >'>i; (x — a)L ^ ^ ^x — a D X — a (x — a)L .7 1 ' and in RV*(a'). It is shown readily that any C curve finally lies constantly below any D curve. 412 IMPROPER INTEGRALS. INTEGRAND INFINITE For a given \, the (7 curves rise as s increases; while the D curves sink as in the figure. The logarithmic tests may now be stated as follows : If |/(a;) I finally remains below some C curve, f{x} is integrable. On the other hand, if /{x} preserves one sign near a, and |/(a:) | remains above some D curve, the corresponding integral is infinite. Properties of Improper Integrals 591. In the following, as heretofore in this chapter, we shall suppose that the integrands have but a finite number of singular points in the intervals considered. When /(a;) has more than one singular point in 51 = (a, 5), we can break 21 into partial intervals, such that /(a;) has a singular point only at one end of each such interval. For example, if the points a, Cj, c^ are the singular points of /(a:) in 21, we have, by 576, 577, / being integrable, where aj, a^ are points lying between ^ ' 1 1 ^ the singular points. On account of this property, we may simplify the form of our demonstration often, by supposing 21 to have but one singular point, which for convenience we shall take at the lower end of the interval. 592. Let /i(2;), ■■•/„(2;) be integrable in (a, 5). Then (^i/i + • • • + c„f„')dx = Cj J^ f^dx H h c^J^ f„dx. (1 Suppose /, •••/„ limited except at a. PROPERTIES OF IMPROPER INTEGRALS 413 Then, if a ^J(x)dx^M(h-a'), ag(x). Then J-'b r*b Jdx>jjdx. (1 Suppose /, g are limited except at a. If a<.afgdx, by 526, 2. Passing to the limit a=a, we have 1). 414 IMPROPER INTEGRALS. INTEGRAND INFINITE 596. Let f(x), g(x) he integrahle in (a, h}. Except possibly at the singular points, let f(x')^g(x'). At a point of continuity c of these functions^ letf{c} >g(c)' Then j fdx> I gdx. (1 Suppose /, g are limited except at a. Let a I gdx; a %y a by 595, Adding, we have 1). 597. Let f(x) be absolutely integrable in (a, 6). Then f(pc) is integrable in (a, 5}, and \£fdx\<£\fcx-)\dx. (1 In 578 we saw that /(a;) is integrable in (a, 5). Suppose /(a;) is limited, except at a. Let a0, S>0, \i"fgd d < e, h — h- Hence, since (7 is a constant, ^= |-((7+ K)=^=f(x). dx dx dx 2. Letf(x) he integrahle in 21 = (a, 5). Iff is continuous at x^ -^ r*x+h lim- I f(x)dx=f(x), a; in 21. This is a corollary of 1. 605. /w 21 = (a, 5) let /(a;) 5e integrahle. Let it he continuous except at certain points c^ ••■ c^, where f(x) may he unlimited. PROPERTIES OF IMPROPER INTEGRALS 419 If F(x) is a one-valued continuous function in 21, having f(^x) as derivative, except at the points c, £fix)dx = F(b) - ^(a). (1 Suppose /(a;) is continuous except at a. Let a' (a;) ^ 0, in 21. Let ^ = («, /3) 6e /Ag image of 31, «w^ Zg^ ^'6 ^Ag inverse function of (f). If either J:c=j^K^)dx, or -^„ = J^/[l/r(M)]l/r'(M)(?W z's convergent^ the other is, and J^ = J„. By hypothesis the points of 51, ^ stand in 1 to 1 correspondence. To fix the ideas, let / be regular except at a. Let fydx+d,(^)\g(:^-0-)-g(a+0}U as we have just seen in 3). Passing to the limit, we have, as before, jjgdx = g(a + ^~)jjdx + ^,\g(h - 0)-^(a + 0)|. This may be transformed as before, giving 1) also for this case. Let us suppose finally, that the singular points of / are ^V ^2' '" '^s' Then ^ ^ ^j If a or h are singular points, the first or last integral may be discarded. By the preceding, J"' = g(^a 4- 0)j^Jdx + ^ (ci - 0)pfdx -j^}+K^:-0)|j^ -jj. SECOND THEOREM OF THE MEAN 42^ Similarly, x;=^(^.+<'){i;'-X}+^«^^-«){X'-X} Adding all these equations, we get, setting e^ = a, c^+j = b : J= gia + 0)jr'' + ^ [g(c^ + 0) - gic^ - 0) | £^ /c=l +i;W.+i-o)-^(e,+o)lJ' ^g(a + 0)r + S+T. (5 Now m, 9}J denoting the extremes of the integral 3), mi\g 0, arbitrarily small are called the left and right hand singular integrals relative to the lines a; = a^, 4 = 1, 2 ••• r. 5. In 609 we made the formal requirement that fixy') should be defined at every point of R. It usually happens in practice that / is not defined at its points of infinite discontinuity. ; Such is the case in such integrals as -\/xy ^ (x^ + y^y ^ It is, however, easy to satisfy the above requirement in all the cases we shall consider; for, by 598. the value of X/(^' y)'^' is not affected by a change of the value of / at points lying on the lines x = a^--- subject to the restrictions of that theorem. 6. This fact may also be used to advantage sometimes to sim- plify /(a;, y) by changing its value at points lying on these lines. 611. 1. Let f(xy') be regular in R = (aba^'), /3 finite or infinite, except on x = a^, ••• If the singular integrals relative to these lines be uniformly evanescent in ^, we say J = j^f(x,y}dx is uniformly convergent in ^. 2. If J is uniformly convergent in the intervals :33i, ••• ^mi it is obviously uniformly convergent in their sum. 3. If J is the sum of several uniformly convergent integrals in ®, it is itself uniformly convergent in ^. 426 IMPROPER INTEGRALS. INTEGRAND INFINITE 4. Letf(xy^ he in general regular with respect to x in Z2 = (a6a/3), ^finite. If J is uniformly convergent in ^, it is limited in -©. For simplicity suppose x=b \s the only singular line. Then fdx\ < 0-, uniformly in ^. But in the rectangle (^ab'a/3), \f(xg)\dx, by 526, 2. But <^ being integrable in 21', we can take 6g so near h that the last integral is (x) = log (1 — cos x) is absolutely integrable in (0, a'), using the /^-test. Now in 454, Ex. 1, we saw that J? lim x'^ log (1 - cos x) = 0, 0S^ is uniformly evanescent not only in 58', but in 33. The same reasoning may be applied to the singular integral relative to the line X = w. Here the only point of infinite discontinuity is y = — 1. Hence, by 611, 2, the integral J is uniformly convergent in S3. 428 IMPROPER INTEGRALS. INTEGRAND INFINITE 614. Example. Let us consider the integral J-\ xv-^ I log a;| Mx, n % 0. (1 We show first that it is convergent only for y > 0. For, let y>0. Applying the /x-test, we have lim xi^ - x^~i I log X I " = lim x^ | log x | ", X > x=0 = 0, by 454, Ex. 2, for properly chosen < m < 1- Hence, by 579, J is convergent. Let 2/^0. Then lim a; . x^-i | log x | » = lim x^ ] log x | ", X^O = +00. Hence, by 580, J is divergent. Let 0< a < ^. We show that J is uniformly convergent in 35 =(«, jS). In the first place we note that the integrand is continuous in B —(0, 1, a, /3), except on the line x = 0, where it has points of infinite discontinuity. We have, therefore, only to show that the singular integral S relative to this line is uniformly evanes- cent. To this end we use 612. Now x3'-i|logx|"< xa-i|logx|", 2/>a. But we have just seen that 0(x)=: X«-l|l0gx|". (2 is integrable. Hence S is uniformly evanescent in 55. 615. Let f(xy) he regular in M = (aJa/3), /3 finite or infinite, except on x=h. The singular integral i Jixy^dx, b' 0, a S > such that IX (^(x)dx =) \f(xy)-K^')\dx. We wish to show that € > 0, (r, I i> I < €, for any y>G-. Now . , , D=J^ \f(xy') - ^(x) \ dx + j^Jdx - J^, <^dx, b' < b, 430 IMPROPER INTEGRALS. INTEGRAND INFINITE Now by 1°, 3°, the last two integrals are nnmerically < e/3, if h' is sufficiently near 6, for any y. On the other hand, if Gr is suffi- ciently large, we have for any y>G-, \f{xy^-(\>(x')\< for every x in (a, J'), by virtue of 2°. Hence |i)j|a, which establishes 1). 2. Letf(xy^ he regular in R= (^aba^'), ^ Jitiite or infinite, except on the lines x = a^, •■■ x= a^. 1°. Let the singular integrals relative to these lines be uniformly evanescent in SQ = («, /8). 2°. Let 7], finite or infinite, lie in ^, and lini/(2:, ?/) = ^(po) uniformly y=r) in % = (ab), except possibly at x = a^, ■■■ x= a^. Then ^^ j = lim I f(xy^dx exists. 3°. Let ^(x) be integrable in 21- Then f(xy~)dx= I \\m.f(^xy)dx= | <^(x)dx. We need only show that j exists, since the rest follows by 1. Let us suppose, to fix the ideas, that x = b i^ the only singular line, and that 77 = 00. Then I)=r\fix,y'^-fix,y"y,dx = £\A^^ y)-/(^, y")\dx + £fix, y'^dx-£f(x, y"^dx = L>, + D^ + I)^. CONTINUITY 431 But by 1°, there exists a h' such that |i>2l, |i>3| 7, and x in (a, 6'). Thus , ^ , |i>J<6/2. Hence |X)|7; and the limit y exists. 617. 1. In 561 we have defined the term/(.'r, 3/) as a uniformly continuous function of y in ^. It may happen that /(a;, y + h) converges to f(xy^ for each y in Sd and any x in 21, but that the uniform convergence breaks down at points lying on the lines X = a-^^ ••■ x = a^. In this case, we shall say/" is a regularly con- tinuous function of y in ^. If /(a;, y + K) converges uniforml}^ to f(xy) except on x=ay, •••, where it may not even converge to /(a;, ?/), we shall say that f(xy') is a semi-uniformly continuous function of y. In both cases, we can inclose the lines x = a^, ••• in little bands of width small at pleasure but fixed, such that the convergence is uniform in ^, when x ranges over 31, excluding values which fall in the above bands. 2. It may happen that /(a;, ?/) is a regularly or a semi-uniformly continuous function oi y in ^ except on the lines y = a^^ ... y = a^. We shall say in this case that / is in general regularly or semi- uniformly continuous in y. 3. We wish to make a remark here which will sometimes per- mit us to simplify the form of a demonstration without loss of generality. In questions of uniform convergence or uniform con- tinuity, the uniformity may break down at points lying on certain lines x = ay, ••• a;=a^. In this case we may count such lines as 432 IMPROPER INTEGRALS. INTEGRAND INFINITE singular lines. When y8 is finite, and no points of infinite discon- tinuity lie on these lines, their singular integrals are obviously uniformly evanescent in ^. 618. 1. As corollaries of 616 we have, using 611, 4: Let f{xy') he in general regular with respect to x in R =(a6a/3), /S finite. Let f(xy^ he a semi-uniformly continuous function of y, except at Let JW = i K^y')dx he uniformly convergent in ;53 = («/S). Then J is limited in Sb and continuous, except possibly at a^, • • • «,„. 2. Let fi^xy) be continuous in R, except on x = a^, ••• x = a^. Let 1) be uniformly convergent in ^. Then J is coyitiiiuous in ^. 619. Ex. 1. The integral J = \ log (1 — 2 ?/ cos X + y'^)dx Jo is a continuous function of y in any interval i8 =(a^). For the integrand is contin- uous in (0, TT, «, /3), except on the lines a; = 0, x = tt. In 613 we saw J is uniformly convergent in 58. Hence, by 618, 2, J is continuous. Ex. 2. The integral J =\ a:^'-! I log 2/ |"dx is continuous in (a, /3). < « < ^. This follows from 618, 2, and 614. Integration 620. 1. Up to the present we have been considering the case when the singular integrals Sc =jjf(xy)dx relative to a line x= e are uniformly evanescent. INTEGRATION 438 For the purpose of integrating ^(^) = ) Kxy)dx %J a with respect to the parameter y over a finite interval ^ =(«, /3), we can take a slightly more general case. As before, let /(a;?/) be in general regular in R = (^aba^'). To fix the ideas, let us consider the left-hand singular integral at c. Suppose now that for each e > there exists a S > such that for any y in Sd* = («, /S*), for every e' in (c— 8, 0, a S > 0, such that \j "^dyjfda I ady. Hence 434 IMPROPER INTEGRALS. INTEGRAND INFINITE I-' r/3" -/3' '1 a 821. Let f(xy^ he in general regular with respect to x^ in R = (aha^^^ /3 finite. 1°. Let f he in general a semi-uniformly continuous function of y in ^ = («, /3). 2° Let r* he normally convergent in ^. Then J is integrahle in ®. To fix the ideas let x = h be the only singular line. Since the singular integral is normal in ^, it is uniformly evanescent in any («7), 7 < /3, by 620, 2. Hence, by 618, 1, t/is integrahle in («, 7). To show that J is integrable in ^, we have only to show that T=) Jdy, /3-v<^'<^"<^, 6' 6 converges to 9 as 77 = 0. T=\ dyj fdx=j J +J J =T,+ T,. (1 But, by 620, 4, there exists a h' such that |7'2|0, there exists an 17 >0, such that I Tl < e for any pair of values yS'^" in (/3 — 77, yS*). INVERSION 435 Inversion 622. 1. Let f(xy^ he in general regular with respect to x^ in R = (aba^). 1°. Let the singular integrals he normal. 2°. Let fixy^ he in general a semi-miiformly continuous function of y in Sd. 3°. Let inversion of the order of integration he permissible for ayiy rectangle in R not emhracing the singular lines. Then K= \ dy \ fdx, ^— ) ^^ } f'^y exist., and are equal. For simplicity, let us suppose a; = 6 is the only singular line. By 1°, 2°, and 621, the integral K exists. ^J '^ /»« ff fb' fp j dyj^fdx = j^ dxj fdy. h-S0. Hence f or < a < j3. We can, by 622, 2, invert the order of integration, since \ "xy-hlx, < a is uniformly evanescent in (a, ^) by 614. Thus 1) gives, inverting, log ^ = (\lx (^xy-'^dy = C ^^'^ ~ ^""^ dx- ' (2 a Jo Ja Jo log X For a = 1, this gives f' ^^~'-^ dx = log^. (3 JO log X ^ If we set here /3 = 2, it gives r^^dx = log2. (4 Jo logx 624. We give now an example where it is not permitted to invert the order of integration. We have for all points different from the origin, x^ + y'^ (x^ + y'^y D y ^ x-^-y2 *x2 + y'^ (x2 + 2/2-)a Thus Jo ^Jo(x2 + 2/2)2 Jo ■'[x^+lJ^So Jo 1+2/2 4' Jo Jo(x^ + y-2yi " Jo \x-^ + yij> Jol+a;2 4 Hence A, B are both convergent ; but they are not equal. INVERSION 437 625. 1. Up to the present we have supposed that the points of infinite discontinuity of the integrand f^xy) He on certain lines parallel to the «/-axis. We consider now a more general case. Let us suppose that these points of infinite discontinuity do not lie only on a finite number of lines parallel to the z/-axis, but that it is necessary to employ in addition a finite number of lines par- allel to the a;-axis. To fix the ideas, let these lines be x = a^, ••• x = a,.', y=a^, .-' y=a,. (1 If f(xy') is otherwise regular, i.e. if properties 2°, 3'' of 610, 1 hold, we shall say f(xy^ is regular in R except on the lines 1), or that it is in general regular with respect to x, y. 2. Similarly we extend the term simply regular, viz. : If /(xy') is continuous in R except on a finite number of lines parallel to each axis, say the lines 1), where it may have points of finite or infinite discontinuity ; if, moreover, it enjoys properties 2°, 3° of 610, 1, we shall say f(xy^ is simply regular in R except on the lines 1), or that it is simply irregular with respect to x., y. 3. The lines y = a^., ••• are also called singular lines, and the integrals r>y are singular integrals relative to the lines y = a^, tw = 1, 2 ••• s. 4. In accordance with the present assumptions, we should modify the definition of normal singular integrals given in 620, 1, so as to allow o-(«/) to have singular points at a^, ••• a^. As an example, consider V(x-&)(^-/3) Here every point on the lines x = b, y = ^ are points of infinite discontinuity. If E =(«, b, a, ^), a there exists an t; > 0, such that ^^" is numerically < e for any pair of numbers /3'/3" in (5 = (^ — 97, /3). T=XdySjdx=X£^X,X=T,+ T,. (1 But by 2° and 620, 4, , ^ , ^ /^ if h' is taken sufficiently near h. Suppose b' so chosen and then fixed. On account of 4°, ^ ^ T*' r^\ By virtue of 3° we can take 7; > sufficiently small, so that ^^ I 2(6 — a) INVERSION 439 "^"^" I^il0 is taken, we may take S>0 so small that irX'h^' by 620, 4. (3 From 1), 2), 3) we have I C^' C^ I - Jrke, b-So. The singular lines are aj — 0, 2/ = 0. The singular integral relative to a; = 0, Jo , dx is normal in 53 = (0, /3). For, Vy*^" Va; setting J" Vx y/y DIFFERENTIATION 441 Here, for any e > 0, there exists a 5 > such that e' < e, for any < a < S. On the other hand, a- is integrable in 53. The singular integral j,^ r° dy relative to y = is uniforiuly evanescent in any (a, 6), a > 0. For, ~Vrt^" y/y But for any e > there exists an Uo > 0, such that pi^^eVa, 0<«iix'), \D,\<2Jjdx<'-, But Also, by 2= Hence provided h' is taken sutBciently near h. On the other hand,/^ being uniformly continuous in 21' = (a, 6'), we can take h so small that \f'yix, v) -fy{^. y~) I < 2(5^' i^ 21'. Then lAK^A Hence |Z)| < e, for any | A| < 8 ; and 1) follows at once from 2). 630. Example. As an application of 629, let us consider the integral I a;*' -1 log" xdx^ n ^ 0, integral, wliich was taken up in 614. The integrand f(xy) = a^-i log" X is not defined for x = 0. Let us give it the value 0, when x = 0. Then /^(x, ?/) =:xs'-Uog"+ix forrc>0 = for X = 0. DIFFERENTIATION 448 Then / and/y are simply regular in R = (0, 1, a, /3), «>0 except on the line X = 0. Moreover, fy is a uniformly continuous function of ?/ in 58 = («, /3), except possibly on the line x = 0. It is therefore a regularly continuous function of y ni ^. Thus condition 1° of 629 is fulfilled. Condition 2° is also satisfied, as 614, 2) shows. Hence if we set J =j;x.-idx, we get, by 629, , ^ -i dy or differentiating n times, dy T /^i -= I xJ'-Uog«xdx. (1 » Jo But, by 575, Ex. 3, , J = -- Hence y ^=(_1)«_!LL. (2 ayn yn+1 Comparing 1), 2), we get rx3'-ilog''xrfx=(-l)"-^, y>0. (3 Jo 2/»+i 631. 1. By using double integrals, we can obtain more general conditions than those given in 629, for differentiating under the integral sign in \J a For example, the following. Let f(x^ y')^ f'y(x^ y^ he in general regular with respect to x in Ji = (aba0). 1°. Let f'y{xy^ he a semi-uniformly continuous functioyi of y in « = («, /3). 2°. Let he uniformly convergent in S8. 3°. Let Xy+fi /»6 dyj/I^Cx, y)dx, \h\<8 admit inversion. ^ Then , f^=j/lCxy}dx, in«. (1 444 IMPROPER INTEGRALS. INTEGRAND INFINITE For = i J^ dxj^ f'yix, y)dy, by 605, 610, 6 1 ry+h = U Ky)'iy- (2 But by 1°, 2°, and 618, 1, g(^y} is continuous in SB. Thus on passing to the limit h = 0, in 2), we get 1), using 537, 2. 2. As a corollary of 1 we have : Let f(xy')^ fy(xy') be regular in Il = (aba^'), except on the lines x= ay, ••' x = a^. Letfy(xy') he continuous hi R, except on these lines. Let he uniformly convergent in ^ = (a, ^S). d C C CHAPTER XV IMPROPER INTEGRALS. INTERVAL OF INTEGRATION INFINITE Definitions 632. 1. If f(x) has no points of infinite discontinuity in 21= (a, 00 ), and is integrable in any partial interval (a, 5), of 21, A'e shall say that f(x) is regular in 21. If on the contrary, /(a:) has a finite number of points of infinite discontinuity Cj, Cg, ••• in 21, but is integrable in any partial interval (a, J), we shall say f(x} is in general regular in 21- The points Cj, c^^ ••• are singular points. Let /(a;) be in general regular in 21- Let us consider lim j f{x)dx^ (1 a=cc»' a which we denote more shortly by I f{x)dx, ^ a (2 and which is called the integral of f(x) from a to + oo, or the integral of f{x) in 21. If the limit 1) is finite, we say the integral 2) is convergent or that /(re) is integrable in 21. If the limit 1) is infinite, the inte- gral 2) is divergent. If the limit 1) does not exist, i.e. if it is neither finite nor definitely infinite, the integral 2) does not exist. If |/(a;) I is integrable in 2t, /(a^) is absolutely integrable in 21, and the integral 2) is absolutely/ convergent. 2. We make a remark here which will often enable us to sim- plify our demonstrations, without loss of generality. 445 446 INFINITE INTERVAL OF INTEGRATION If /(a;) is in general regular in 51= (a, oo), we can take h so large that /(a;) is regular in (5, oo). But the integral j f{x)dx *y a has been treated in Chapter XIV. We have thus only to consider I f{x)dx, in which the integrand is regular. We may therefore often as- sume in our demonstration, without loss of generality, that fix) is regular in 51- Ex. 1. For, Ex. 2. Eor, Ex. 3. For, does not exist. C^ dx \ — = 1 ; it IS convergent. iC=ooJl X? \ X] J— = + CO ; it is divergent. 1 X lim \ — = lira log x = + oo . x=xJ'^ X I cos X dx does not exist. Jo lim \ cos X dx = lim sin x X — 00 */ 633. 1. G-eneral a-iterion for convergence. Letf(x) he regular in %— (a, cx)). For to he convergent, it is necessary/ a7id sufficient that, for each e > 0, there exists a G->0, such that £fdx\a, (3 the singular integral. The integrals 2) and 3) are obviously equivalent. 3. Letf(x) he regular in % = (a, oo). Iff(x) is absolutely inte- grahle in 21, it is also integrahle in %. For by hypothesis \ \f(ix')\dxo, ^>o, \j- r|a. Subtracting, we get ■ -te|<6. I *J a. I ^ \J a. I a DEFINITIONS 449 Hence, by 633, J fdx is convergent. Similarly, I fdx is convergent. Therefore when J is convergent, Jy, J^ are con- vergent. Conversely, if J^, J2 ^^^ convergent, J is convergent. For, let a be fixed and u arbitrarily small, Hence (3^. Therefore lini j = Jj + t^2- Hence, when Jj, J2 '^'"6 convergent, J is, and 3. As a result of the foregoing, we see that the integrals Cfdx. J fdx (3 do not differ essentially from C/dx. For convenience, we shall therefore study only this last; the results we obtain are then readily extended to the integrals 3). 450 INFINITE INTERVAL OF INTEGRATION Tests for Convergence 635. 1. The /* tests for convergence. Let f(x) be regular in 51 = (a, oo). If there exists a /m > 1, such that x>^\f(x')\<]yL M>0, inVCcc); f(x) is absolutely integrable in St. For let (7 < a < /3 lie in V. Then, by 526, 2, < e, if Cr is taken sufficiently large. 2. Let f{x) be regular in % = (a, oo). If for some /jl>1 lim x'^\f(^x^)\ is finite., fi^^ *'s absolutely integrable in 21. 3. As corollary of 2 we have : In % = (a, oo), a > 0, let f(x) = ^-^^^ or "^ ^ , X.>1; t'>0, where g is limited in 21 and integrable in any (a, 6). TAg/i /(^c) is absolutely integrable in 21- 636. Test for divergence. Let f(x') be regular in 2t = (a, oo). In F^(oo) let f have one sign o-, and axf{x)>M, M>0. Then ^^ J= \ fdx = cr • oo. For, let aM T— = iJf lofT - = +Q0, when a;= + cc. TESTS FOR CONVERGENCE 451 637. Logarithmic test for convergence. Let f(x) he regular in 31 = (a, oo). If there exists a /i>l, and an s, such that xl-^xl^x ■ ■ ■ l^_ixl/x Ifix') I < iHf, m Fi^QO ), /(a;) is absolutely integrable in 21. We have, by 389, 2), for x>0 sufficiently large, DM-'^x = l-f. Hence, if < (r < « < /3, ^a xl^x---l,>^x /L4 — lU/-'a ^/~^/3j when (r = + 00. Now I fdx\<\ \f\dx iHf > 0. Then ^^ \ fdx = a ■ cc. *'^^' /^X /»a /^X •>^u *^a ^ a. j afdx>MJ j"" J = Ml,,,~ since by 389, 1), for sufficiently large a; > 0, As X lim ls+i- = + Q0> our theorem is establisherl. 452 INFINITE INTERVAL OF INTEGRATION 639. Ex. 1. A quarter period of Jacobi's function sn(u, k) is r ^^ — ■ o<.0, > (l + x)«+" is convergent when tt>0; and divergent, when m^O. Consider the point x = oo We have •^^^'^ ~ (1 + x)»+'' ~ x^+i * / l\«+v' TESTS FOR CONVERGENCE 453 Applying now the tests of 635, 636, we see that r f{x)dx, M > 0. converges for v>0, and diverges for v^O. Thus the integral 1) has a finite value for every «, u>0. The function so de- fined is called the Eulerian integral of the first kind or the Beta function. 642. Ex.4. The Gamma function, r(u) = ij e-^x»-^dx. (1 The point x = is a singular point if m < 1. Applying the tests 579, 580 at this point, we see that f a>0. Jo , is convergent when ii > 0, and divergent when m .^ 0. On the other hand, applying the ix test of 635, 3, we see that r is convergent for any u. The integral 1) therefore defines a function of u for all ?/ > 0. It is called the Eulerian Integral of the second kind or the Gamma function. 643. 1. Let f{x) he regular and integrahle in any partial interval (a, A), of%= (a, oo). Let fh^'ydx be limited zw 21. In ^(oo), let g(x) he monotone, and ^(oo) = 0. Then f{x)g(x) is integrahle iti 21. We apply the criterion of 633. Let (r<«f^fdx+g{^ - 0)f^fdx, «< ^<.i3. We can take Gr so large that are numerically as small as we please. As the integrals on the right are numerically less than some fixed number, we have I r^ I fgdx Q-. Hence ^ is integrahle in %. 454 INFmiTE l^^TEKVAL OF INTEGRATION 2. Letf^x) he regular and integrahle in % = (a, oo). Let g(x) he limited and monotone in 21. Thenfg is integrahle in 51. For, by 545, Jjgdx = g(a + 0)fjdx + g(/3 - 0')ffdx, (1 Let \g(x-)\7. Then the right side of 1) is numerically < e. Hence fg is integrahle. 3. Let f(x^ he regular and absolutely integrahle in 51= (a, oo). Let g(x) he limited and integrahle in 51. Then f(x)g{x) is ahsolutely integrahle in 51. We have only to show by 633 that e>0, 6?>0, j]fg\dx Cr. Now g(^x} being limited, we have ' |^(a:)|a. M This in 2) gives 1). 644. Let f(x) he in general regular in 51 = (a, oo), hut not inte- grahle in 5t. Let )Kx)dx (1 TESTS FOR CONVERGENCE 455 he limited in 51. Let g(x') he monotone in % and ^(oo)=(r^0. Then fj{x)gix)dx (2 18 not convergent. For, if 2) were convergent, we would have I C^ e>0, h, I fgdx b. \%/ a. But this is impossible. For £fgdx = gCa + 0}£fdx + ^(/3 - Q-yf^'/dx. (3 Let the integral 1) be numerically < M. We can take h so great that \g(x)-a\<(7, x>b. where a- is small at pleasure. We can therefore write 3) f%dx = (G-\- <^')f[fdx + ( a + a"yfydx. \a'l |o-" | < Hence Jf'^^'-^l < — TTTi — < ^'' ^ small at pleasure, which states that /(a;) is integrable in %. 645. Ex. 1, For what values of fx does j^ pcosx^^ Jo ^f. converge ? Set j^^C''92^ax, j,= C 92^dx; a>0. Jo ^M Ja x'^ The integral Jo is convergent by 643, 1, provided. /u > 0. For ix30, it obvioualy does not converge. The integral Ji is convergent, as we saw, 586, only when /u0. Jo j-M Ja. y^fx. In 587, we saw Ji is convergent only when m < 2. By 643, 1, J2 is convergent when ;u > 0. When ,u ^ it obviously does not con- verge. Hence J is convergent when, and only when, 0 ^L_ f sjn y c?y = -2_. Jo {y + (m - l)ir}'^ (witt)" Jo (mTr)'* Let Then Hence T- ^ 2 r 1 , 1 , ,11. A„> — J ^ =00, when n = 00, as the reader probably knows, or as will be shown later. Properties of Integrals 647. In Chapter XIV we established the properties of the improper integrals, ^^ fdx. by a passage to the limit. We propose now to develop the properties of improper integrals, the interval of integration being infinite, by a similar method. In many cases the reasoning is so similar to that employed to prove the corresponding theorems in Chapter XIV, that we shall not repeat it, referring the reader to the demonstrations given in that chapter. 648. 1. Let /(x) he integrahle in (a, oo). Then fdx = -jydx. PROPERTIES OF INTEGRALS 457 2. LetfQc) he integrahle in (a, oo). Then I fdx= I fdx->r I fdx, a i gdx. a %/a 2. Letf(x)^ ^(^) he integrahle in (a, oo). Except possihly at the singular points, let f(x') ^g{x^. At a point c of continuity of these functions, letf(c)>g(^c). Then £fdx>J^ gdx. 3. Let f{x) ^0 be integrahle in (a, oo). At a point c of continuity offletf(ic)>0. Then j^f{x)dx>0. 4. Let fix) he absolutely integrahle in (a, oo). Then \fjd.\<£\fid.. 5. Let ^oo j=lfd. he convergent. We may change the value of f(x) over a limited dis- crete aggregate, ivithout altering the value of J, provided the new values off are limited. 650. Let f(x) be integrahle in 21 = (a, oo). Then Jr»ao fdx a< x. X is a continuous limited function of x in 31. 458 INFINITE INTERVAL OF INTEGRATION For, by 648, 2, ^^ ^^ ^„ 1=1+1, c>x. *^x *^x *^c But j fdx is a continuous function of x in (a, c), by 603. As fdx is a constant, J(x) is continuous in 21. J is limited in %. In fact, for each e > 0, there exists a c such that I ff'^- 'a^ < e. But being continuous in the limited interval (a, c), is limited. Hence J is limited in 21. 651. Let f(x) he integrahle in 21 = (a, oo). Then for any point x of %^ at which fix") is continuous. For, ii e>x, ^ ^ J(x)=i =1 +( =K(x-)+0, ^x *^x *^c C being a constant. By 604, 1, dK j.^ >. ax Hence dJ^d{K±0)^dK^,,.. dx dx dx 652. In 21 = (a, oo), let f{x^ he continuous excepting possibly at certain points c^--- c^, where it may he unlimited. Let it he inte- grahle in any (a, ?»). THEOREMS OF THE MEAN 459 Let F(x) he one-valued and continuous in 21 ; having f{x) as derivative except at the points c. Then f fdx = F(+oo-)-F(a-), (1 where F( + oo) is finite or infinite. For, by 605, , I fdx^^Fih-y-FQa-), however large h is. Passing to the hmit, we get 1). Theorems of the Mean 653. First Theorem of the Mean. In% = (a, oo) let gQic) he inte- grable and limited. Let f(x) he integrahle^ and non-negative in %. Then rfgdx=mrfdx, (1 where 2)^ is a mean value of g in 21. For, by 602, m\ fdx<\ fgdx^M) fdx, a - 0) rV^a; + ^( + 00) Cfdx ^h ^b = ^(« + ^)fydx + p(6 - 0) - ^(a + 0) |/7^a; + {K + ^)-^(^-0)|jr7(^rc = ^(a + 0) J /c?x + ^7+ V. (4 Let X, /i be the minimum and maximum of fdx in 21. Then obviously, fdx < /*, J ^00 I fdx0. (1 Let us set z =il/(u) = e-". Here = 0, 6 = 1; a=+oo, /3 = 0. In 58, is continuous and always negative. Then, by 655, /-o J'a = - ( - 1)" I e-'^yu'^du (2 is convergent, since 1) is. Hence 1), 2) give r e-^^M" dtt = -^i^ , 2/ > 0. (3 yn+l 657. 1. Stake's Integrals. Let us consider the convergence of the integral J'= I X &\n {x^ — xy)dx,, (1 which comes up in the theory of the Rainbow. Let us set u = x^ -^ xy = x(x2 — ?/) = (^(x). (2 The graph of this is a curve which crosses the axes at the points x = 0, x = ± \/y, if ?/ > ; and at the point x = 0, if y = 0. To fix the ideas, let us suppose ?/ > ; the case when y ^0 may be treated in a similar manner. Supposing, therefore, y>0, the graph of 2)shows that as x rangesover 2I=(Vy, ao), u ranges over i8 = (0, oo), the correspondence between the points of 31 and 58 being uniform. Thus the relation defines a one-valued inverse function x = f (m) in SB. Let us write 1) j= (''+ r, Jo Jy/p and denote the latter integral by J^. The corresponding integral in m is ^ J"„ = I g(u) sin u du, setting 9(u) = "" 3x2-?/ We can now apply 648, 1. For, x = + oo as w = + oo. Hence g(u) is a monotone decreasing function, for any positive y, and gf(ao) = 0. Thus Ju is convergent. Hence Jx is ; and therefore the integral J is convergent. 2. The same considerations show a fortiori, that ir= r cos (x^ — xy)dx (3 is convergent for any y. 464 INFINITE INTERVAL OF INTEGRATION 3. In connection with these integrals, occurs another integral L= i x"^ cos (x^ — xy)dx, (4 which, it is important to show, is not convergent. In fact, effecting the change of variable defined by 2), in Lx= \ X- cos (x3 — xy)dx, supposing to fix the ideas that y > 0, we get r°° x' C^ i„ = 1 cos udu= \ Mil) cos u du. Jo 3x2-2/ Jo ^ ■' Here k{u) is a monotone function, and Thus Lu is divergent, by 644. Hence ix is. Therefore L is divergent. INTEGRALS DEPENDING ON A PARAMETER Uniform Convergence 658. 1. Let f(x^ y) be defined at each point of the rectangle B, = (aooa/8), /3 finite or infinite. Let % = (a, oo), :53 = («, /5). We shall say f{xy~) is regular in R when : 1°. f(xy') has no point of infinite discontinuity in R. 2°. fQcy') is integrable in 51 for each y in ^. At times Ave shall need to integrate f(xy^ with respect to y. In this case we shall also suppose : 3°. f(xy') is integrable in ^ for each x in 31. 2. If f(xy^ is regular in R, except that it may have points of infinite discontinuity on certain lines x = a^^ ■■■ x= a^, we shall saj^ f{xy^ is regular in R except on the lines x= a^, ••• or that it is in general regular with respect to x. 3. Let us suppose that the points of infinite discontinuity of f(xy^ do not lie all on a finite number of lines parallel to the y-axis, but that it is necessary to employ in addition a finite num- ber of lines parallel to the a:;-axis. To fix the ideas, let these lines he x= ay, ••• x= a^\ y = a^, •■■ y = ag. If f(xy^ is otherwise regu- lar in jB, i.e. if it enjoys properties 2°, 3° of 658, we shall say UNIFORM CONVERGENCE 465 f(xy^ is in general regular with respect to x, y, or f(xy^ is regular except on the lines x= a^^ ••• y = a^, ••• 4o Let/(a;«/) be continuous at each point of R except on certain lines rr = aj, • • • a; = a^ ; y = a^^ -•• y = a^. On the lines x= a-^^ ••• it may have points of infinite discontinuity ; on the lines y=a^, • • • it may have finite discontinuities. If f{xy^ is otherwise regular in a, we shall say it is simply regular with respect to x except on the lines x = a^, ■•• or that it is simply irregular tvith respect to x. 5. Let f(xy') be continuous at each point of R except on the lines 2;= a J, ••• x=a,,; y = a.^, ••• y=a^. As in 3, let us suppose that all the points of infinite discontinuity cannot be brought on the lines x — a^^ •■■ \jetf(xy^ be otherwise regular. We shall say fQxy) is simply irregular with respect to x, y^ or that it is simply regular except on the lines x= a-^^ ••• y = a^^ • • • 6. The lines x = a^^ ••■ y = a^, •■■ on which are grouped the points of infinite discontinuities oif(xy^^ are called singular lines. To each of these belong right and left hand singular integrals as in Chapter XIV. Cf. 666. 7. The integral f{xy')dx, x^Gr, (1 where Gr is large at pleasure, is called the singular integral relative to the line x = cci. If for each e>0 there exists a Gr, such that 1) is numerically < e for any y in ^ and every x'^Gr, we say 1) is uniformly evanes- cent in ^. 8. If the singular integrals relative to the lines x=a^, ••- x = a,., as well as the singular integral relative to the line iC = oo are uni- formly evanescent in ^, we say the integral J = £f(x,y~)dx (2 is uniformly convergent in ^. If the uniform convergence of J breaks down at certain points Yj, ••• 7i in ^, we shall say J is in general uniformly convergent in ^. Cf. 666. 466 INFINITE INTERVAL OF INTEGRATION 9. As in Chapters XIII, XIV, we wish now to study the inte- g-ral 2) with respect to continuity, differentiation, and integra- tion. We may often simplify our demonstrations without loss of generality by observing that we may write fdx = I fdx + J fdx = t/j + J^. Here we may take h so large that none of the lines a; = aj, -• fall in (JooKyS). The integral J^ has been treated in Chapter XIV. 10. In this article we have considered f(xy) chiefij^ with respect to X. Evidently we may interchange x and ^, which will give us similar definitions with respect to y. We wish also to note that all the following theorems apply to the integral on interchanging x and y. 659. Let f(xy^ he regular in R = (acca^^, y3 fi7iite or infinite. Let ^(x) he integrahle in 51, and . 1/(2^3/) I <(^). in R. Then ^^ j /(^, y)^^ (1 is uniformly convergent in ^. For , ,„ ,„ I fdx\< I \f\dx, by 528. < f Mx, by 526, 2. Since

h. Hence 1) is uniformly convergent in ^. UNIFORM CONVERGKNCE 467 660. 1. Letf(xy^ he regular in R= {acca^), ^finite or infinite. Let fix, i/) = (t){x')g(x, y), where., 1°, ^ is absolutely integrahle in 51. 2°, g(xy^ is limited in R and integrahle in any (a, h},for each y in SQ. Then ffixy^dx (1 is uniformly convergent in ^. For, g being limited in i2, let \g{xy)\ 0, a 5 such that jj'\4>ix')\dx<^. bgdx I b' I \*yb' \dx, by 529. <6, by 2). Hence 1) is uniformly convergent in Sd- 2. As corollary of 1, we have, by 635 : In R= (^acca^^; a > 0, ^finite or infinite, let where g is limited in R, and integrahle in any (a, 5) for each y in ^. Then 1 fi^y^dx is uniformly convergent in ^. 468 INFINITE INTERVAL OF INTEGRATION 661. 1. Let f(xy) he regular in R= (aooayS), ^ finite or infinite. where 1°, <^(x) is integrahle in 21. 2°, g(xy^ is limited in M and a monotone function of x for any y in ^. Then ^ao 1 f(xy^dx is uniformly convergent in ^. For, b}^ the Second Theorem of the Mean, 545, r (l)gdx = g(h' + 0, 2/) P4>dx^g(h"-Q,y^ C cf>dx, hb^. UNIFORM CONVERGENCE Hence for any y in 48, 469 <€. 662. Integration by Parts. Letf{xy^ he regular in J?=(aQ0a/3), ^finite or infinite. The integral ^= ( Axy')dx is uniformly convergent in SQ, if J A^' there exists a b such that ^(^, y') < 5, and any y in ^. Hence 1) and 2) give \ffdo. <€. 663. Examples. 1. The integral •« = X' g_X^_l ^y.^ (1 defining the Gamma function, considered in 642, is uniformly convergent in g3 = (a, ^), «>0. For, consider the singular integral relative to x = 0. We have, since < x < 1, x3'-i < xi-i, in S. Hence , „ i ^ i Thus, by 612, the singular integral relative to x = is uniformly evanescent in S5. Consider next the singular integral relative to x = co. We have, since x > 1, e-xa3,-i^5f_ in SB. ^ e^ Hence this singular integral is uniformly evanescent in 33. Thus 1) is uniformly convergent in 33. 470 INFINITE INTERVAL OF INTEGRATION This is uniformly convergent in any 33 = («, qo), which does not contain the point ?/ = 0, as may be seen by 662. For, integrating by parts, Jx X L xy Jj Jx x^y _ COS xy p cos xy ^^ ^2 xi/ J^ x'^y To fix the ideas, suppose a > ; then I COS xy I ^ 1 I xy I ax This shows that the first term on the right of 2) is uniformly evanescent in S3. The second term is uniformly evanescent by 660, 1, as is seen, setting . / N 1 / N cos x?/ x^ y For later use, let us note that y=x xy i/=x y Jx x? lim J'=0. y=« xy y=« y Hence r^ - sin Xx dx (3 is uniformly convergent in 53 = (0, oo), by 661. For, in the first place, the inte- grand /(xy) is continuous in i2 = (0, oo, 0, co). For, the only possible points of discontinuity lie on the line x = 0. But, the Law of the Mean gives, e~'y =\ -xy + '^^^ e-O'v. 2! Hence for x ^ 0, ^^^^^ ^ ^ ^.^ ^^ _ ^^2^-0x3, gin Xx, < ^ < L This shows that / is continuous at each point on the y axis, if we give to / the value at these points. This fact established, we can apply 661 by setting ^(a;)=5HL2^, g(^xy)=\-e-'y. Then (/> is integrable in (0, 00 ) by 646 ; while g is obviously 'limited in S, and a monotone increasing function of x for each y in S. Hence 3) is uniformly conver- gent in S3. UNIFORM CONVERGENCE 471 4, r"" sin XM cos Xa; , .. \ dx (4 Jo X is uniformly convergent in 33 = (0, qd) except at y = | \ |. For, in tlie first place the integrand /(x, y) is continuous in B = (OfloOco) if we give to /the value y at the point (0, y). For, the Law of the Mean gives Hence for x^O, Thus sin xy = xy ^sinOxy, 0 , cos (y — X) cos (y + X) ,„ Jsmx,cosXxdx=-^^iLi__yL_Z. (6 Let \y-\\, \y + \\><7. (7 Then 6) shows that each of the integrals in 5) is numerically <2/4/e(r; then l-B|0. For . . - X sm x?/ _ sm xy 1 1 + X2 ~ X ', . 1' We have now only to apply 661, 2, using the result obtained in Ex. 2. 472 INFINITE INTERVAL OF INTEGRATION ^- f * ^^^-y sin (x3 - xy) dx. (8 We assign the value to the integrand, for a; = 0. To show that this integral is uniformly convergent in any 33 = (a/3)) let us use the method of 662. If we set M = — , dv = (3x^ — y) sin (x^ — xy) ; Sx /"» _ r cos(a;3 — x?/)n°° r'°cos(x^ — xy) Jx ~\_ 3 X J.c Jx Sx^ cos (x^ — xy) f"^ cos (ic^ — xy) f 8 X J-r 3 x^ dx. Here both terms are uniformly evanescent in 33 by 660, 2. Hence 8) is uniformly convergent in 33. 7. I cos (x^ — xy)dx. (9 We can write '3x^ — y „ y C^ COS (x^ — x?/) i cos (x* — xy)dx — \ — .^ cos (x^ — x?/)c?x + o l X' dx. The second integral on the right is uniformly evanescent by 660, 2. The first integral is also uniformly evanescent. For integrating by parts, J'=° 3 x2 — w ^ „ ^ ^ sin (x^ — xw) 2 f » sin (3 x^ — xy) , , ^^'C0S(XB-X,)dx = W^ + 3i XB ^ ^- Here both terms on the right are obviously uniformly evanescent. 664. lieif(xi/y be regular in R= (aaoa/3), yS finite or infinite. Let ^00 converge uniformly in ^, except possibly at «j, ••• a^. To establish the uniform convergence of «/ throughout ^, we have only to show that J is uniformly convergent in each of the little intervals That is, we have only to show that for each e > 0, and for some S > 0, there exists a b^ such that for any i/ in ^^ and every b' >b^, k = 1, 2 "• m. UNIFORM CONVERGENCE 473 665. Examples. 1- j^ ^-smysmxy^^ is uniformly convergent in 53 = (0, oo). For, in the first place, the integrand f{xy) is continuous in i2 = (0, oo, 0, oo) if we set /(O, y) =y sin y. We have therefore only to consider the singular integral relative to a; = oo, in the intervals i8i =(0, 5), $82 = (5, 00). Now as in 663, 2, we have for y>0, -^ p sin y sin xy ^^ _ sin y cos xy _ ^^^^ p cos xy Jx X XV J.C X^ll The reasoning of 663, 2 shows that iTis uniformly evanescent in 352. As to S81, we note first that ^ = f or y = 0. Also that sin y 1 , ; I / 1 ^ — ^=1 + 7;', h K'7) y 17 being as small as we choose, if 5 is taken small enough. Hence for any ?/ in S3i, X Jjo X^ which shows that ^is uniformly evanescent in S3i. r^Hi^dx, x>o (1 Jo weAi y& is uniformly convergent in 33 = (0, /3). For, the integrand /(a;?/) is continuous in jB = (0 co /3), if we set Let us consider therefore the singular integral relative to x=oo. We set 35i = (0, 5), 332 = (5, oo). Obviously 1) is uniformly convergent in 332. To show the same for 33i, we note that sin xy = xy + TX^y^, 1 t |< 1, by the Law of the Mean. Hence |/(«y)(<^ + ^', m(0, Qo,0,S). Thus, by 659, the integral 1) is uniformly convergent in Si. 474 INFINITE INTERVAL OF INTEGRATION Continuity 666. 1. Let f{xy) he regular in R — (acC) a/3), ^finite or infinite,, except on the lines x = a^ ■■• x = a^. ±. Let /^« J=jJ(xy^dx he uniformly convergent in ^. 2°. Let Vim f(^xy) = (f)(x^,, rj finite or infinite uniformly in any (a, 5), except possihly on the lines x = a^ ••• Then y=]im ) f{x, y^dx exists. (1 3°. Let 4){x} he integrahle in any (a, J). Then \\xiiJ=\\m I f{xy^dx= I ^(x)dx. (2 By virtue of 616 we may assume that/ is regular in jB, and that f== uniformly in any (a, 6). To fix the ideas let rj = ao. We show first that j exists; i.e. for each e>0 there exists a 7 such that D= f \f(x,y'}-f(x,y")ldx is numerically < e for any pair of numbers y', y" >y. Now i>= rV(^/) -fCxy"}\dx-hrf(xy'yx- ff(xy")dx = i)l + 1)2 + 2)3. By 1°, there exists a h such that lAJ, |i)3|7. CONTINUITY 475 Hence l/(-,y)-/(.,y")i<2(jr^. for any ?/', y" >7, and x in (a, h'). Thus I T^ I /o /--( |i)i|0 there exists a 5^, such that I ^j I \j — I <^c?3; hQ. From 1), there exists a 7 such that j=jj{xy^dx + e\ |e'|<| (6 for any ?/ > 7. From 1°, there exists a h^ such that jy(ixy^dx=jjixy^dx + e'\ \e"\<± (7 for any h > b^, and any y m ^. From 2°, we can take 7 large enough so that also f(x,y) = ct>{x} + g(xy'), \g\<-^L^ (8 for any x in (a, 6), and any y > 7. Hence, by 3°, ., jy(a:z/y:r=J^(/,(:r)(i:r + e'", l6'"I<| (9 for an}^ h > J^, and any ?/ > 7. From 6), 7), 8), 9) we have for any b > 5q. But this is 5). 2. The reader should note that the lines aj= a^, ••• on which the uniform convergence of f(xy') to , 0, oo). We also saw that the singular integral relative to sc = oo is uniformly evanescent in i8= (0, oo). Now . , T„ ^/ N sin Xx hm /(a;, y) = 9=00 X uniformly in 21 =(0, oo) except at x = 0. The line x = is therefore a singular line by 617, 3. But \ sm Xx c?x < \ dx < e, < a < 5 J« X I IJo X I if 5 is taken sufficiently small. This singular integral is therefore uniformly eva- nescent. Hence J is uniformly convergent in 58. Thus all the conditions of 666 are satisfied. From 1) we may deduce the following relations : psinjxxcos^^^^ (2 Jo X 0 0, a — ^ < 0, we have from 1), X 2' '^ sin(a-^)x^^^_x .g '0x2 ^ But sin (a + /3)x + sin (« — ;3)x = 2 sin ax cos /3x, (7 sin (a + /3)x — sin (« — ^)x = 2 cos ax sin /3x. (8 Adding and subtracting 5), 6) and using 7), 8), we get 2), 3). CONTINUITY 477 668. That the relation Km j f(x,y')dx= ) \\mf(x,y^dx (1 is not always true is shown by the following example : From f e-^ sin X dx = - i::!l55^^+x!iE^, J 1+ W2 we have for ?/ > 0, /♦^o ^jj^ ^ ^ r?i^dx = -J— . (2 Jo (xy 1 + m2 gxy 1+2/' y=o Jo e^y On the other hand, P2?lim515^dx= (""sinxdx Jo v=o e="J' Jo 70 3,=o e=^J' does not even exist. Thus the relation 1) does not hold 669. 1. Let f{xy) he regular in M = (^acca^^ except possibly on the lines x= a^^ ••• x= a^. Let f(xy^ he a uniformly continuous function of y in ® except possihly on the lines x = a^^ ••• Let %/ a he uniformly convergent in ^. Then J is continuous in ^. For, f{x, y -{- h') converges uniformly to /(a;, y), h = in ^ except on the lines x = a^, • • • We have, therefore, only to apply 666. 2. Letf(xy^ he in general regular ivith respect to x in R= (^acca^^. Let f he in general a semi-uniformly continuous function of y in ^. Let J'iy')= ) Kxy^dx he uniformly convergent in ^. Then 'lis limited in ^, and in general a continuous function of y. 478 INFINITE INTERVAL OF INTEGRATION For, we can take h so large that <€ for any y in >&. On the other hand, is limited in ^ by 618, 1. Hence J is limited in iB. That J is in general continuous in ^ follows from 1. 3. In this connection let us note the following theorem whose demonstration is obvious. Let f{xy') he regular in R = {ayDa/3'), /3 finite or infinite, except on the lines x = a^ ■•• ; y = a^^ ■•■ Let 1 K^y~)dy he uniformly convergent in any (a, 5) except at a^ a^ ••• Then the points of infinite discontinuity of fixy^dy, y in «. must lie on the lines x^ ay, ••• 670. 1. Letfixy') he regular in R= (aooayS), except on x^a^-" ; y = H- 1°. Let Sj'y converge uniformly in any (a^ b) except at x— a-^ 2°. Let ^:c ^y 4>(y)= ) dx\ fdy converge uniformly in '^. Then cf) is continuous in ^. This is a direct application of 666, 1, where g(xy) = I f(xy)dy takes the place of/ in that theorem. rNTEGRATION AND mTERSlON 479 In fact, by 669, 3, g{xy) has no points of infinite discontinuity except on 2; = ai •••; and is therefore by 2°, regular in R except on these lines. Also g{x^ J/ 4- ^) converges uniformly in any (a, 6) to g{x^ if) as ^= 0, except at rc = a^ ••• ; since V is uniformly evanescent in (a, 6) by 1°. Thus applying 'o^^^ 1, we have lim 0(?/ + li) = lim | g(x. y + li)dx = I lim g{x, y + Ti)dx = J[ dxjydy=^<^iy^. That is, <^(«/) is continuous at ?/. 2. As a corollary of 1 we have : Letf(xy') he in gerieral regular tvith respect to xin R= («ooa/3). converge uniformly in Sd- Tlien (f) is continuous in SS' Integration and Inversion 671. 1. Let f(xy') he in general regular with respect to x in R= (^aooa^). Let f he in general a semi-uniformly continuous function of y in SQ. Let i(y)= j fC^y~) 0, a 5q such that \rfdx<-^, (3 Wh H — ft /3-a for any y in ^, and every h > h^. Thus 2), 3) give for any y in ^, j dy \ fdx- j dy \ fdx\< I ^ ^ <6. (4 But by 2°, Hence J ^2/ /*=° /'ft /*y| J -J J k^' ^>^o> (5 , a ^a ^a ^ a. \ which proves the theorem. • 2. Let f(xy') he simply irregular with respect to x in R= (aooa/3)< Let jjdx he uniformly convergent in ^. mTEGRATIOX AND INVERSION 481 Then ^^ J dyj fdx, J dxj^ fdy (6 are convergent and equal. For, bj 671, 2, the integral on the left of 6) exists. Moreover, condition 2° of 1 is fulfilled, by 622, 2. 3. As corollary of 1, we have : Let f(xy^ he in general regular in R = (aQoa/3) with respect to x. Let Cfdx he uniformly convergent^ and integrahle in ^. Let I dy \ fdx^ h arbitrarily large, admit inversion in ^. Then ^^ ^^ I dx\ f{xy')dy is uniformly convergent in ^. This follows at once from 5), since this inequality holds for any y in ^. 4. From the relation 4) we have also the following corollary, setting y = (3. Let f{xy^ he in general regular in R = (aoDa/3) with respect to x. Let Cfdx he uniformly convergent., and integrahle in ^. Let Jdy I fdx., h arhitrarily large, admit inversion. Then lim } dyl fdx = \ dy \ fdx. 6=00 *^a ^a »'a *^a 482 INFINITE INTERVAL OF INTEGRATION 5. As a special case of 4 we have, 622, 2 and 671, 2 : Let f(xy) he simply irregular with respect to x in R = (aa^'), except on the lines^ x^ay, ■■ x = a^\ y = a-^,---y = a,. I dy I fdx he coiivergent^ and admit inversion in any interval (\, fi), which does not embrace a^ ••• a^. J dxj fdy he a continuous function of y in ^. Let ^= 1 dy j fdx, L= \ dx) fdy. Then K is convergent, and K= L. Foi- simplicity, let 3/ = 7 be the only singular y-line ; a < 7 < yS. Then by definition, dy\ fdx=\xm\ I +limj j . (1 a ^'a u=y ^a. *^ a v=y ^^ *' *'« a^ /»=<= /»» eince i is by 2°, convergent. INTEGRATION AND INVERSION 483 Now by 2°, Hence from 2), 4), lim J" r=rr; (6 also from 3), 5) lim rr==rr-rr=rr. o Hence from 1), 6), 7) we have */(i tJa J a *^a ^a ^y *^a *^a 674. 1. Let f(xy) he simply regular in Ii = (acca^')^ except on the lines x = a-^, ■•• y = a.^... 1°. Let ) K^y^dx converge uniformly in ®, except on y = a^ ••• 2°. Let converge uniformly in any (a, 5), except on x—a^ ••• 3°. Let J ^aj f(xy~)dy converge uniformly in ^. -Br=J (^yj fdx, L=J dxj fdy. Then K is convergent and K= L. This follows from 673. For, in the first place, condition 1° of 673 is satisfied, by 672, 2. Secondly, condition 2° of 673 is fulfilled, by 670, 1. 484 INFINITE INTERVAL OF INTEGRATION 2. As a corollary of 1, we have : Let fixy') he simply irregular with respect to x in ^ = (aQoa/3). Let ^00 I f{xy~)dx converge in general uniformly in ^. Let j <^^j^A^y)dy converge uniformly in ®. Then J <^yj fdx=J dxj fdy. 675. For y>0, we have from 668, 2), r^^^^^ax^^—. (1 The integral on the left does not exist for j/ = 0. Let us therefore set sill \x = 0, 2/ = 0. Then integrating 1), from to y we get |>j;>x = j;^^d, = arctg^, X:^0. (2 We may invert the order of integration in 2) by 674, 2. For, / is continuous in i? = (OQoOy), except on the line ?/ = 0, and limited in J2. It is therefore simply irregular with respect to %. The integral 1) obviously converges uniformly in 33 except at 2/ = 0. The integral J^dxJ^/fZ^ = j^ —^sinXx^x (3 is uniformly convergent in 33 by 663, 3. Hence the integrals on the left in 2), 3) are equal, and /•« 1 _ g-xy \ y t sm Xx dx — arc tg - , X ^ 0. Jo X ^ X 676. We saw in 667 that ^^ ^.^ ^^, f 0, y = 0, li f =° sm xw , " ,, Jo X ! -, 2/>0. Hence, integrating between and 1 , we get INTEGRATION AND INVERSION 485 "We can invert the order of integration by 674, 2. For, in the first place, the integrand, not being defined in 2), we can make it continuous in B = (0x01), giving it the value y at the points (0, y). Secondly, the integral 1) is uniformly conver- gent in 33, except at y = 0, by 663, 2. Finally, p^ r.sinxy p l-cosxy ^^ Jo Jo X ^ Jo X2 is uniformly convergent in 53, since 1 1 — cos xy I 2 I ^ | — x2' We can therefore inveit in 2), which gives IT f" cZx fi . , f " 1 — cos X J -=l — I sm xy dy = \ ax 2 Jo X Jo " " Jo x2 ^2psin2x/2^^^psin^j£d« .3 setting X = 2 M. Thus 3) gives /*°° sin^x dx _v ,. Jo x2 ~ 2 ' ^ 677. That the order of integration can not always be inverted is shown by the following examples. Ex. 1. Let us consider I dx j co&xydy=\ — dx (1 TT ~ 2' The integral obtained by inverting the order of integration, viz., i dy i cos xy dx does not exist, since 1 cos xy dx does not. Inversion in the order of integration in 1) is therefore not permissible. Ex. 2. Let — -= = 0(m), u = xy. 1 + xV 1 + M* Then Let /(x,2/)=0'OO=^f -if- y dx X dy 486 INFINITE INTERVAL OF INTEGRATION AVe have also where lA («) = arc tg u^. Thus and hence ^ 0(m) ^ 3f a; 5a; Hence =irg-=.-[^«]:=!- On the other hand Thus it is not permissible to invert the order of integration in ]-;..!;/(., i/)^.=f ^x|;| ^ .,. (2 678. 1. Letf{xy^ he regular in _B = (aQoaQo), except on the lines x = a-^, •••; y = ai, ••• 1°. Let ^« ^y 5e uniformly/ convergent in Sd- he uniformly convergent in any (a, 5), except at a^, a^-, •••; awe? inte- grahle in (a, 6). Then ^^ ^^ I t(?a: I fdy exists^ lim I t?a; I /(?y = I (^j: 1 fdy. (1 This is a direct application of (SQQ, 1 ; the function fixy^dy a taking the place oi f(xy^ in that theorem. For, in the first place, g has no points of infinite discontinuity, except on the lines x = a^ INTEGRATION AND INVERSION 487 •••, by 669, 3. Moreover, g{xy) is integrable in 31, by 1°. Hence gi-ry^ is regular in i?, except on the lines x = a^, ••• Secondly, I g{xy)dx is uniformly convergent in ^, by 1°. Finally lim gixy') = I fdy = ^(x), uniformly in any (a, 6), except on the lines x = a^^ •••; moreover (^ is integrable in (a, 5). Thus all the conditions of 666, 1 are satisfied, and the present theorem is established. 2, As a corollary of 1 we have : Letf(xy^ he simply irregular with respect to y in R =(aQt)aQo). Let I dx\ fdy be uniformly convergent in -53. Let be uniformly convergent in any (a, 6). Then dx I /ii/ = ) dx\ fdy. , For 2) is integrable in any (a, 6), by 671, 2 ; on interchanging a; and y in that theorem. 679. If the conditions of 678 are not satisfied, the relation lim \ dx\ fdy = \ dx I fdy (1 may be untrue. Consider, for example, J=\ dxi cosxydy, a>0. J» X 488 INFINITE INTERVAL OF INTEGRATION Here limj"=0, by 663, Ex. 2. On the other hand, the integral I da; t cos xy dy Ja Jo. does not even exist, since I cos %y dy does not. Thus the relation 1) in this case is not true. 680. 1. Let f(xy^ he simply regular in i2 = (aQoaao), except on the lines x= a-^, •••', y — (f-y, ••• 1°; Let ^^ I fdx he uniformly convergent in any (a, /3) except at «j, ♦•• jjdy he uniformly convergent in any (a, 5) except at a-^ •••; moreover let it he integrahle in (a, J). . 3°. Let ^a= ^y j dx j fdy he uniformly convergent in Sd- Then ■'■''''-'If' ptn ^CXJ ^00 ^00 I dy j fdx, I dx I fdy are convergent and equal. For, by 674, 1, ) dy) fdx = i dx i fdy. . But by 678, 1, we may pass to the limit /3 = oo, which proves the theorem. 2. LetfQcy^ he simply regular with respect to y in R = (aoc aco), except on the lines y= a^ ••• Let INTEGRATION AND INVERSION 480 he uniformly convergent in any («, /S) except at Oj, ••• Let ^^ he uniformly convergent in any (a, 6). Let I dx I fdy he uniformly convergent in ^. Then are convergent, and equal This follows as in 1, by 674, 1, and 678, 2. 3. Let f(x, 2/)>0 ^^ simply regular in Ii = (^accacc^ except on the lines a; = 0. Then Jo " Jo xet^y JO Jo Ja ^ 2 ixe^i^ "We can invert the order of integration in 1) by 680, 2. For, in the first place . , . sin xy cos Xx f(xy) = is simply regular in Jr! = (OcoOoo), if we set Secondly, C ^-, r°^ sin xw cos Xx J \ fdx = e-y-y \ dx Jo-' Jo X is uniformly convergent in 39 = (0, ») except for ?/ = X by 663, Ex. 4. Thirdly, Cfdy = cos Xx P ^'" ^y dy for x > Jo*^ ^ Jo xei-y \ —dy for X = Jo eMy ^ is uniformly convergent in any (0, 6) by 66.5, Ex. 2. Finally, r= Cdx {jdy = Cdx (^J^^.^im^ay Jo Jo-' ^ Jo Jo xei^y " is uniformly convergent in 33. For, Hence fysinxy _r _ /xsinxy + xcosxy"]!' Jo et^y 2/-|_-e >^y ^12^7^2 Jo' f^cosXx .. „„ ., r^'sinxw cos Xx , ^=Jo ^^^,(.'^-e->^^'^o^xy)dx-,.e-.y)^ __l.__d^ mTEGRATION AND INVERSION 491 Fi, Y2 are uniformly convergent in 35 by 659. For, cos Xx ,, . 2 I sin xy cos Xx I X ;u'-2 + x"-^| — M^ + k' Thus all the conditions of 680, 2 are satisfied. Inverting therefore in 1), we get fl^ + X- Comparing with 1), we get ^ C ' cos Xx , C sm xy , K = I dx \ =- dv Jo X Jo e>^» f cos Xx , r fjL Bin xy + X cos xtf~| Jo X L /x2 + x2 J /i^ -f x'^ Jo cos Xx , f cos Xx , W ^ = rt ^ A /-o \ -r, 5 dx = ^- X > 0, /A > 0. (2 2. Let us integrate 2) with respect to X. We get fA r- cos Xx , T f A ^ ,^ = 2^(1 -e-V). (3 We can invert the order of integration in the integral on the left, by 674, 2. For, r'" cos Xx Jo /i^ M^ + X^ is uniformly convergent by 659, since dx cos Xx I 1 \fj:-^ + x;^i — ii^ + z^ la the second place, C dx C^ ^ 7^ f smXx , I -3 5 I cos \xd\= \ —r-i, sv dx Jo /i2 + ^2 Jo Jo x(ai2 + x2) converges uniformly in any interval (0, /3) by 661, 2 and 663, Ex. 2. Inverting therefore in 3) , we get f sin XW , TT ,, - ^ n = rw jo ^0^^" = V^'-'""'^' ^>0,2/>0. 492 INFINITE INTERVAL OF INTEGRATION 682. Let us evaluate ^co ^,, * J= j % (1 Jo g«2' '^ which is convergent by 635, 3. We change the variable, setting u-xy, y>0. (2 Then ^„ ^ j_ C y dx^ Jo gff'' Multiplying by e-»* and integrating, v?e get ' dy r°" 7 r* y ^* jC%=CclyC Ja e» Ja Jo ^=(1+1=) This relation is true for any a > 0, by 2). Passing to the limit a = 0, we have, since the limits exist, ' dy j.^ ("^ , C" y dx ^ c^ dy -TO C^ n C" y dx -"•Jo J = ^=io^^^jo ifc^)- (3 "We may invert the order of integration in the integral on the right by 680, 3. For, in the first place, the integrand is regular and continuous in B = (OcoOco). S^^^^^^y' p ydx Jo e»'(i+^'> is uniformly convergent in any (a, /3) , a > by 659, since y ^ P Thirdly, ^ ydy r is uniformly convergent in 21 = (Oco) by 659, since y _^y_, _2^ 5^ ..2* Finally, Jo Jo ej^(i+x==) Jo L 2(1 + x''')Jo dx IT 2J0 1 + a;2 4 is convergent. Thus all the conditions of 680, 3 being fulfilled, we can invert in 3), which gives J^ = L = ir/4. Hence J=±-y/^/2. Here we must take the positive sign, since the integral 1) is positive by 649, 3. ITpiicp. finally, .- r°" dx _ V X ^ J Jo I^~~2~* DIFFERENTIATION 493 Differentiation 683. 1. Let f(xy')^ f\j{xy^ he in general regular with respect to X, y in 11= (aye «/3 ) . 1°. Fo)' each x in 91 let f he continuous in y, while f'y is in general eo7iti7iuous in y. 2°. For each y in ^, let I dx I f'ydy^ h arhitrarily large^ admit inversion. Then ay *^a dy &=« »^a *^a provided the derivative on either side exists. For, by 605, Hence and therefore J fAy =f(x, y') -fix, «). 1 fdx =1 dx \ fydy -\- I /(a;, a)dx X'j rb r-b dy I f'ydx+ I f{x, ft)dx; a. *^a *^a I fdx=\\m I dy I f'ydx+ I /(a;, «)(fa Differentiating, we get 1), since the last term on the right is a constant. 2. As corollary of 1 we have : Letf{xy) he regular in i2 = (fflco«/3) except on the lines a; = a^, ••• and continuous with respect to y for each x in 21. Let f'y he regular in R except on the lines a; = a^, ••• and uniformly continuous in y, except on these liiies. Let I fydx he uniformly convergent in ^. Then ^ -3- 1 .f(xy^dx = I fydx. 494 INFINITE INTERVAL OF INTEGRATION For, condition 2° of 1 is fulfilled by 622, 2. Hence by 1), 3- I f(x^}dx = — Yim I di/ I fi,dx = -T- f'^^ f/^^a:, by 672, 4, and 671, 2, = Cfydx, by 669, 1. 684. 1. When ^» \fydX is not convergent, the following theorem may serve. Let f{xy^ he in general regular in R= (^aooajS), and continuous with respect to y for each x in %. Let fy he simply irregular with respect to x in R. 1°. Let , I f'ydx^ h arbitrarily large, he uniformly convergent in ^. 2°. For any b, let Jr»6 /»6 fydx= I g(xy}dx-{-h(h, y), where 3°. g(xy^ is simply irregular in R with respect to x and J/»ao I gixy^dx a IS uniformly convergent in ^. 4°. Then lim I h(b^ y^dy = 0. ^ =3- ) fQ>^y^dx= I gQcy^dx. Q dy^a *^a DIFFERENTIATION 495 2. As corollary we have : Letf{xif) he regular in R= Qaco a^'), and continuous with respect to y for each x in %. Letf'y he continuous in R. For any h, let Jr'b /*b I f'ydx= J g{xy)dx+ h(h, y), where g is regular in M, and I gi^y^dx is uniformly convergent in ^ : also lim I A(5, y^dy = 0. Then r, ^^^ ^oo ^Jy(^y)^^=J^ 9(S^y')dx. For, condition 1° of 683 is obviously satisfied, while condition 2° is fulfilled by 672, 2. Hence J"' = -- lim I dy ) f'ydx. dy 6=« *^a '^a But fy fb fy fh fy y dyy f'ydx = j dyj^ gdx + J^ hdy, by 2°. Hence by 4°, ^ ^y ^b J' = — lim I dy I gdx dy ^a. ^a =:^ rdxCgdx, by 3% dy^a ^a = I gdx, which is 1). EXAMPLES 685. 1. Let We show that j=r'-}i^dx. (1 Jo are^ dJ^ rcosxy^^^ 2/ arbitrary, (2 dy Jo e=^ using 683, 2. For, in the first place, the integrand f{xy) is continuous in B — (Ox>a8), if we set /(O, y) = y. 496 INFINITE INTERVAL OF INTEGRATION Obviously J is convergent in 33, by 635, 3. Secondly, f, _ cos xy e* is continuous in B ; and pcosx,^^ is uniformly convergent in 33, by 660, 2. Thus 683, 2 gives 2). By means of 2) we can evaluate 1). For, obviously, Jo e^ 1 + 2/2 Hence integrating 2), we get J 1 + 2/2 Since J" = 0, for y = 0, we have C = 0. Hence r!!E^dx = arctgy. (3 Jo xe^ 2. From this integral we can also show that /•« sin X2/ T, Jo X 2 a result obtained in 667, by the aid of 675. For, set x = ^*, 2/>0, y in 3), we get I • e ydu — arc tg m. k«» Jo M We now apply 666, 1, letting y = cc. This is permissible, since sin u _'i . sin u ., , e y— , uniformly u u in (0, co) except for u = 0. The integrand /(rt, y) is continuous in i? = (Ocoaco), if we set /(O, y) = 1. The only singular line is therefore u = 0. Obviously the singular integral for this line, as well as for the line m = oo, is uniformly evanescent, by 615 and 659. Hence passing to the limit, y = oo in 5),we get Jo M 2 If we set u = xy, y> 0, we get 4) . 686. Let ^'^ 1 — cos xy diffp:rentiation 497 ^p l-cosxy ^^ (1 Jo xe^ ^ Applying 683, 2, we get dJ_ r'^siii.xy y aJ C^siaxu ^ y d^ = }o ^^^^ = m^' y arbitrary. (2 In fact, the integrand /(x, y) is continuous in B — (Ocx)a/3), if we set /(O, 2/) = 0, while J is convergent, by 635, 3. Moreover _ sin xy Jy- ex is continuous in i?, and r" sin xy , i -d% Jo &" is uniformly convergent in S, by 660, 2. This establishes 2). As in 685, we can use 2) to evaluate 1). For, integrating 2), we get J 1 + V' 2 + y Here C = 0, since J = for y — 0, by 1). Thus /'=° 1 — cos xy ^ 1 , ,, „- 687, Let us evaluate Fourier''s Integral j^C- coB2xy ^^ Jo gxi' ^ Using 683, 2, we get ^=_2r ^^i^2^^a; = ^. (2 - dy JO e^2 "^ For, the integral 1) is convergent by 635, 2 ; while the integral 2) is uniformly convergent in any («/3), by 660, 2. In 2), let us integrate by parts, setting u = sin 2xy, dv = — 2 xe-'*dx. Then K= uv\ — i V Jo Jo du = — 2y\ g-*^ cos 2 x?/ c?x = — 2 yJ". This in 2) gives, since J"=jtO, ^=~2ydy. 498 INFINITE INTEKVAL OF INTEGRATION Hence log J=-y^+0. (3 To determine C, take y — 0. Then C = log-Vw/2, (4 by 682, 4). Hence 1), 3), 4) give Jo e^2 2 688. In 681, 2) we found rcosxy^^ _jn_ Q y^a>0. (1 We can differentiate under the integral sign, by 683, 2. For, denoting the inte- grand hy f(xy), we have ^ , ^ X sin xu which is continuous in E = (Ox>a(3). Also ' sin xy dx j;/,cte=-j'; 1+^ is uniformly convergent in 93, by 661, 2. Hence, differentiating 1), we get Jo ;^2 + a;2 2 689. In 682, 4), let us replace x by xa/a, ?/>0. We get r e-!'^VZa; = — 2/-5, ?/>a>0. (1 We can differentiate under the integral sign, by 683, 2, getting Cx^e-y^^dx = — 2/-f . (2 In fact, the integral on the left of 2) is uniformly convergent in Sd =(«, jS), since /).2 0.2 ^,<-^, in 33. '\Ve may obviously differentiate 1) n times, which gives rx^-e-y^^-dx = y^ . ^ . 5 ... 2-^^^li 2/-^, 2/>0. (3 Jo 222 2 -^ - ' •'^ ^ DIFFEKEJSTIATION 499 690. FresneVs Integrals. Let us start with the relation 689, 1), /.CO /~ j ey^-dx — ^^^y-i, y>0. (1 Let /(^2')=^' forx>0; = 0, for x = 0. Then since the integral on the right is convergent, by 646. We can invert the order of integration here, by 680, 1. For, f(xy) is continuous in i? = (OcoOoo), except on the line x = 0. It has, moreover, no point of infinite discontinuity in B. The integral 1 fdx= \ — -^dx Jt) ■ Jo e'^'y is uniformly convergent in any (0, ^) except at ?/ = 0. The integral is uniformly convergent in any (0, 6), except at x = 0. Finally, is uniformly convergent in 33. For r. rx^sin^^^cos^y, ^3 Jo L {\ + x*)e''y Jo Hence ^^p_gx__p x^siny + cos?/ ^^^ -^ Ju 1 + x* Jo (l + x*)e''* Here I'l is uniformly convergent in i8, since it is independent of y. Likewise T^ is uniformly convergent, since its integrand is numerically < l + x2 l + x* Thus all the conditions of 680, 1 being fulfilled, we can invert the order of inte- gration in 2), which gives ^ ^ J^^^dx^Jdy =r^, (4 Jo l + x* as is seen from .3), on passing to the limit y = oo. But p dX ^ TT 1 TT Jo 1+a-t 4sin7r/4 2\/2 500 INFINITE INTERVAL OF INTEGRATION This by 2), 4), gives r^j^dy = ^. (5 > V?/ \/2 If instead of multiplying 1) by sin y, we had multiplied by cosy, we would have got by the same reasoning _ r^J^dy = ^. (6 Jo y/y V2 The integrals 5), 6) are known as FresneVs integrals. They occur in the Theory of Light. If we set y = x^, these integrals give 1 sin x'^dx = \ cos x^dx = \\/-k /I. 691. 1. Let us show that Stoke's Integral B = \ cos (a;^ — xy)3x (1 satisfies the relation ^ ^ , ^ + 12/^=0. (2 This fact will enable us to compute S by means of an infinite series. We have in the first place, ^= f Ik sin (x3 - £cy)dx (3 dy Jo by 683, 2, since the integral 3) is uniformly convergent in any 33 = (a, /3). In fact, using the transformation of the variable employed in 657 u — a;(x2 — ?/), (4 \ x sm (x^ — xy)dx= \ — — , Jb Jc Sx^ — y where b, c are corresponding values in 4). But xsmu smu xu _ ^(„)^(„, y). 3x2 — 2/ y 3x2 — 2/ We can now apply 661, 1, replacing x in that theorem by u. Thus there exists a Co such that \ C xsmudul ^ „ = „ I Jc 3 x2 — 2/ 1 But then the relation 4) shows that there exists a B such that I X sin (x^ — xy)dx < e, Jft I for any h ^B, and for any y in ^. Hence the integral 3) is uniformly convergent. ELEMENTARY PROPERTIES OF B(u, v), T(u) 601 To find the second derivative of S^ we cannot apply 683 to the integral 3). For j x^ cos (x^ — xy)clx is not even convergent, as we saw 657. We may, however, apply 684, 2. In fact, I a;2 cos (x^ — xy)dx = \ ^^- — V ^Qg ^^.s _ xy)dx + | j cos (x^ — xy)dx = i [sin (x3 - xy)-]l + r = 1 sin (63 _ hy) + Y. But .00 i cos (x^ — xy)dx is uniformly convergent, as we saw 663, 7. On the other hand, Psin (63 _ by)dy =cos(63-6y)-cos(6«-6«)^ Ja 6 which i as 6 = CO. Thus 684, 2 gives — r X sin (x3 — xy)dx = ^^ \ cos (x^ — xy)dx. (5 dy JO 3 Jo From 1), 3), 5) we have 2). 2. Before leaving this subject, let us show the uniform convergence of the inte- gral 3), by another method. From the identity 09 09 9 3.- 3x2-y y 3x2-y y^ 3x33x3 9x3' we have \ x sin (x^ — xy)dx = \ ^ ~ ^ sin (x^ — xy)dx + ^ \ ^ ~ ^ sin (x^ — xy)dx Jb Ji 3 X 3 Js 3 x3 t r sin(x3-x;/) ^^^y T y Obviously T3 is uniformly convergent by 660, 2. That Ti is uniformly convergent, was shown in 663, 6. That T2 is uniformly convergent, follows from 661, 2 ; since Ti is uniformly convergent. Elementary Properties of B[u, v), T{u) 692. 1. In 641 we saw B(w, -y) = r f'^^"" (1 is a one-valued function whose domain of definition is the first quadrant in the w-, v-plane, points on the w-, v-axes excepted. 502 INFINITE INTERVAL OF INTEGRATION In 642 we saw ^^ r(M)=J e-'^x'^-Hx (2 is a one-valued function whose domain of definition is the positive half of the w-axis, the origin excepted. We wish to deduce here a few of the elementary properties of these functions. 2. By a change of variable, the integrals 1), 2) take on various forms. Thus in 1) set x= - — 1-^ We get />i B(w, z;) = j y-\\-yy-Hy. (3 If we set here -. y = \-z, we get /»i B(m, v) = i 2^-1(1 - zY-^dz. (4 In 3) let us set y = sin^ ; we get B(w, v^ = 2 f'sin^"-! cos^''-^ dO. (5 If we set 11/ X = log \/y in 2), we get l\u)= ] log( 3. We establish now a few relations for the B functions. In the first place the comparison of 3), 4) gives B(w, V) = B(v, m), (7 which shows that B is symmetric in both its arguments. As addition formulae we have the three following 8), 9), 10), B(m + 1, vj + BCw, z; + l)= B(w, v) (8 For, pi B(w, w)= I x'^-Hl - xy-\\ - X + x)dx = f x^^^l- xy-^dx + Cx^'Xl - xydx, which is 8). i;B(w + l, v) = mB(w, ?j+ 1). (9 [ logn dy. . (6 ELEMENTARY PROPERTIES OF B(m, v), T{u) 503 For, B(m + 1, v) = r a:«(l - xy-'^dx; integrating by parts, = [ - ^^ — ^7 + - C^'^'K^ - xYdx L V Jo v^o u which is 9). From 8), 9) we have = -B(w, i; + l), B(w, v) = — ' — B(u, V -\-l) = — ^!— B(w + 1, v}. (10 We can show now that B(w, n) = B(?i, w) is a rational function of u, viz. : Tj/ i\ 1 / i'11 B(w, 1) = 1/u. (11 B(m, 7l) = 11 2 w-1 w w + 1 u + 2 u-\-n — l (12 For, B(w, 1)= rx"-^dx = which proves 11). From this we get 12), using 10). 4. We establish now a few relations for the V function. For, integrating by parts, x''e-'dx=\ -e-^x"" +u) e-'x^'-'^dx |_ Jo •^0 (13 Hx. We observe next that For, = 1. r(i) = i. r(l)= f e-^dx= -e- From 13), 14), we get T (u -\- n)= u(u -\-V) ••• (u + n- l)r(w); and this gives „ , . ^ „ ^ r(w)=l-2-3 •••7i-l = n-l!, on replacing n by w — 1 and u by 1. (14 (15 (16 504 INFINITE INTERVAL OF INTEGRATION A formula occasionally useful is 1 1 r°7 It is obtained from 2) by replacing a;, by ax. 5. The r function is continuous for any w>0. This follows from 669, 1 and 663, Ex. 1. The derivative is given by T'(u)= 1 e-'^x^'-^logxdx, u>0. (18 fc/0 This follows from 683, 2. Similarly J ■•00 e-''x''-nog^xdx, w>0. (19 We can now get a good idea of the graph of r(w). In fact, the expression 2) shows that r(M) > for all w > 0. From r(«)=X +X we see that T>^^ -r^^ ^ . It lim 1 (w)= +00. «=0 From 13) we see that lim r(M) = + Qo. w=H-oo From 19) we see that r"(w)>0, and hence the graph of r(w) is concave. Since r(l) = r(2), the curve has a minimum between 1 and 2. ^^'^^^^^''' 1.46163... 6. We establish now the important relation connecting the B and r functions, t^^ ^t-,^ n B(«,.) = I$^. (20 From 17) we have (1 + i- — = :f7-^ re-''^y>^x''+''-'dx. ELEMENTARY PROPERTIES OF B(u, v), T(u) 505 Hence by 1) B(w, v) = f y'\ ^y =,—^ — .^ Cdy rV+*'-y-^e-(i+^>^c?a:. (21 We may invert the order of integration, by 680, 3. For, in the first place f{xy)= ^(1+y)^ is continuous in i2 = (OooOoo), except on the lines a; = 0, ?/ = 0. Secondly, is uniformly convergent in any (a, /3), a > 0, by 663, Ex. 1. Thirdly, is uniformly convergent in any (a, 5), a > 0. Finally, exists. For in ^_ ^^1% seta;?/=f, a:>0. Then Hence for a > 0, (^rc ( fdy = r(M) I e-^2;''-ic?a;. But _ lim 1 e-^a;''-i(^a: = r(w). 0=0 '^a Hence X= lira X, = r(w)r(v). (22 a=0 Thus all the conditions of 680, 3 being fulfilled, we have L=K. From 21), 22), we have 18). CHAPTER XVI MULTIPLE PROPER INTEGRALS Notation 693. 1. In Chapters XII and XIII the theory of proper inte- grals of functions of one variable was developed. We now take up the corresponding theory with reference to functions of several variables. 2. We begin by explaining a notation which we shall system- atically employ in the following, and which is similar to that used in the earlier chapters. Let 21 be a limited point aggregate in an w-way space 9?^. Let f(xy, '•• a;^), or as we shall often write it, /(a;), be a limited func- tion defined over 21. Let us effect a rectangular division D of space of norm d. To simplify matters, we shall suppose d is not taken larger than some arbitrarily large but fixed number. Those cells which contain points of 21, as well as their volumes, will be denoted by d^, d^, •••, or by a similar notation. Let M,^ m^, be the maximum and minimum of /(a;) in d^. We shall set It sometimes happens that we are considering points of two or more aggregates 21, ^, ••• Then we shall write where the subscript indicates that the sums 1) are taken over the aggregates 2t, ^, ••• respectively. 3. We shall denote the maximum and minimum of / in 21 by M and m respectively. The greater of \M\ and \m\ we shall denote by F^ so that l/(2^r-^JI<^. in 21. 506 UPPER AND LOWER INTEGRALS 507 4. The oscillation of /(a;^, ••• a;,„) in the cell d^ is The sum _ is the oscillator^/ sum of /for the division D. Ujjper and Loiver Integrals 694. The sums S^, S^ form a limited aggregate, D representing any division of norm , or what is the same, the divi- sion I) on A. Then For, let d^ be one of the cells of B which is subdivided, on super- imposing A. Let , , denote the cells of LJ in d^ containing points of 2t. Then, to the term M^d^ in S^, corresponds the term in Sp. But "^""^^ ^Md<^Md a division A, such that ;^<^^>^+e/2. (2 , Let D be an arbitrary division. Let us superimpose A on 2), forming a division E. The division E is formed by interpolating certain points, let us say at most /i points in each of the segments Sj, • • • (5^. The inter- polation of one of these points may be interpreted as passing a plane parallel to one of the sides of (S;. Its effect is to subdivide certain of the cells of (S. The volume of the cells so affected is Hence the superimposition of A on i), being equivalent to pass- ing at most w/i. planes parallel to the sides of (5, affects cells of S belonging to the original division i), whose volume V so large that is positive. Let M^, iV^ be respectively the maxima of / and g in the cell d^. Obviously, We have seen in 696. 1 that lim 2iV^c?^, lim 2cc?^ exist. Hence lim Sj) = lim '2M^d^ = lim S(iV; - c)c?. = lim '^Nfil^ — lim %cd^ exists, and is finite. To show that i[^ ^ exists and is finite, we introduce the auxiliary function and determine c?>0 so large that h is always negative in 51. 698. The limits S, S, whose existence was established in 697, are called the lotver and upjjer integrals of fi^x-^--- x,^ over the field 21. They are denoted respectively by J^/(«i ••• ^m^d% =J^/(2:^ ... x„,^dx^ ••■dx^\ I_ 1 (1 J g/(a^i • • • ^™) ^51 = Jgj/(«i •••x^^dx^--- dx^. When the lower and upper integrals 1) are equal, we denote their common value by J^/(-^i •■■Xm)d%= J^i^i ■ ■ ■ x,rddx^ -dx^', (2 Hence As UPPER AND LOWER INTEGRALS 511 it is called the integral of f over the field %. In this case f(x-^ ••• a;,„) is said to he integrahle in 5t. We also say the inte- gral 2) exists. The integrals 1), 2) are called m-tuple or multiple integrals. 699. 1. Let f(x-^--- Xj„') he limited and integrahle in the limited field %. Let L he any rectangular division of norm d, and f^ any point of 2t in the cell d^. Then Mm y.fQ:)d = ffd%. (1 d=o *>'2l Conversely, if this limit exists, however the D's and ^'s he chosen, the ujyper arid lower integrals of f are equal, and f is integrahle. For r>i.0, there exists a division D for which the oscillatory sum 5. In order that the limited function f(x^ ••• 2;,,^) be integrable in the limited field 21, it is necessary and sufficient that, for each pair of positive numbers to, a there exists a division D, such that the sum of the cells of D in which the oscillation off is > oj, is < - only on the lines « = !, ^, 1, |, J, I, I, I, I, f, •••, the denominators of the fractions being ^q. On each of these lines the oscillation in any little interval is >-. On all other lines the oscillation 1 ^ is <-. Obviously there exists for each o- a division for which the sum of the squares in which the oscillation is >- is <(7 ; and the integral is zero. 2. Let % embrace the points x, y of the square (0101) for which x is rational. Let ' /(x, y) = -, for X = — ; m, n relative prime. Then / is integrable in %, as the above example shows. Content of Point Aggregates 702. 1. We extend now the notion of content, etc., considered in 514 seq., to limited aggregates in di^' Let us effect a rectan- gular division of space of norm B. Let dp 0,2^ wg, ••• CONTEXT OF POINT AGGREGATES 518 be those cells containing at least one point of the limited aggre- gate 21 ; while d[, d'^, d', - denote those cells, all of whose points lie in 21. Then the limits M = lim S<, n = lim -Zdl (1 6=0 £=0 exist, and are finite. For, let us introduce the auxiliary function f(x^ •• a^™)' ^^'hose value shall be in 9?„j, except at the points of 2t, where its value is 1. Then, using the notation and results of the previous articles, we have : 2t^ = 2itfX = 2(^«, 2l^=SmX = 2< But by 697, lim ia, lim 21^ 5=0 5=0 ist, and are finite. 2. The numbers 21, 21 are called the upper and lower content of 21 We have thus : 21 =ffd% 2t=J/c?2l. When 2t = 2t, their common value is called the content of 2t We denote it by Cont 21, or when no ambiguity arises, by 2t. To be more explicit it is often convenient to set i = Coht 21, i = Cont 21. A limited aggregate having content is measurable. Thus, when 2t is measurable. Cont 2t = ffdU The content of a measurable aggregate in ^^ is called its area; ill 9^3 the content is called volume. We shall also use the term volume in this sense, when w> 3. : 614 MULTIPLE PROPER INTEGRALS 3. As immediate consequence of the reasoning of 1, we have: Let ^ be a partial aggregate of 21. Then 703. By the aid of the auxiliary function employed in 702 we can state at once criteria in order that 21 is measurable. 1. For 21 to he measurable^ it is necessary and sufficient that the sum of the cells contairiing both points of 21, and points not in 21 con- verge to 0, as the norm of the division = 0. This follows from 700, 2. 2. In order that 21 be measurable, it is necessary and sufficient that for each e > 0, there exists a division such that the sum of the cells embracing both points of 21 and not of % is < e. This follows from 700, 4. Frontier Points 704. 1. The frontier ^ of any aggregate 21 is complete. For, let j? be a limiting point of ^. Then in any I)^*(p'), there are points of ^. If f is such a point, there are points not belonging to 21 in any D^*{f). We may take t] so small that B^ lies in D^. Hence jt? is a frontier point of 21. 2. Let 2t and SQ be two point aggregates. Let D = Dist (x, y') = V(2;i - y^^ + ••• ^ {x^- y^f be the distance between a point a; of 2t and a point y oi ^. Let S be the minimum of J), as x runs over 21, and y runs over ^. Then 8^0, and is finite. We say h is the distance of 21 from ^, and "^^'^^ S=Dist(2[,«). In certain cases, 21 may reduce to a single point a. 3. If %, Sd are limited and complete, there is a point a in 21, and a point b in SS-, such that Dist (a, b) = Dist (21, «). If Dist (21, ^) > 0, the two points a, b are frontier points. DISCRETE AGGREGATES 515 For, we may regard x^--- a:^, y^--- ym, as the coordinates of a point z in a 2 m-way space 9?2to- We form an aggregate (S whose points z are obtained by associating with each x of 31, every y of SQ. Then the domain of definition of Dist (a;, y) in 2, considered as a function of 2w variables, is precisely S. To represent (5 we may employ 2 m axes, as in 231. Obviously <^ is limited and com- plete, since 21 and ^ are. Then by 269, 2, there exists a point (a^ ••• a^, 6^ •-• 6^) in S, at which I) takes on its minimum value. Then are the points whose existence was to be proved. The points a, h ure frontier points of 21 and ^ respectively. For, if they were inner points, the distance between D^(a) and D^ih^ equals -^.^^ (a, 5) - 2 8 < Dist (a, 5). 4. Let ^ be a partial aggregate of 21. If the distance between the frontiers of 21 and ® is not 0, we say 53 is an inner partial aggregate of 21 ; also 21 is an outer aggregate of Sb. Discrete Aggregates 705. 1. Definition. An aggregate of content is discrete. Obviously, if Cont2l=0, 21 is discrete. 2. Every limited point aggregate of the first species is discrete. Let 21 embrace at first, only a finite number of points, say n points. Let us effect a cubical division of space of norm m I — such that the points of 21 lie within their respective cells. Then the sum of the cells containing the points 21 is ^v^ < nS"' < €. Thus 21 is discrete, and the theorem is true for aggregates of order 0. Let us therefore assume the theorem is true for aggre- gates of order n — 1 and show it is true for order n. 516 MULTIPLE PROPER INTEGRALS By 265, 21^"^ embraces only a finite number of points, say ^1' ^2 '" ^s' We can, as just seen, inclose these within cells of total volume p, it is discrete. For, let 21 lie in a cube (7, of volume O, in 9?^, so large that none of the points of 21 come indefinitely near the sides of C. Then the upper content of 21, relative to 9?p, is < C. We can effect a division D of dim of norm d such that the points of 21 lie within the cells of L>. Then the volume of all the cells containing points of 21 is less than ^7^ „ Cd'^-p, which converges to 0, with d. DISCRETE AGGREGATES 517 708. 1. Let y=f(^x^, ■•• x^ be defined over an aggregate 51. Let x = (x^, •■■ a;„j), x' =(x-^-\-h-^, ••• x^^ + h^^ be two points of 21. The increment that / receives when x passes to x' we have denoted by A/. Let us set Ax = Dist (x, x'} = VV + ... + AJ, and call ^ . A/ Ax the total difference quotient of /. The point a^ may or may not be restricted to remain near x ; if so, it will be stated. 2. Let the limited functions have limited total difference quotients in the limited discrete aggre- gate %. Then Sdi the y-image of 21, is also discrete. For, let us effect a cubical division of the rc-space of norm d. Since the difference quotients are limited in 21, there exists a fixed Gr, such that \Af\ OU have limited total difference quotients in the limited aggregate 31, except at points of a discrete aggregate A. In the cells of any cubical division of norm d < c?^, let at least m of these difference quotients remain limited. Then the image ^ of % is discrete. For, consider one of the cells c?^, containing a point of A. At least m of the coordinates of a point y remain in intervals of length m have limited total difference quotients in the limited aggregate 21- Let _ have limited variation in 51. As x = {x-^ ••• x^ ranges over 51, let y — (^j ••• ^„) range over ^. Then ^ is discrete. For, let us effect a division of the 2;-space of norm d. Then Vv '" Vn-x remain in intervals of length <_dGr as x ranges over the points of 51 in one of the cells d^. Thus if to^ is the oscillation of /„ in c?^, the point y remains in a cube of volume when X ranges in d^. Thus the upper content of ^ is Kd^a^-^io, by 709, 2). As this converges to as c? = 0, ^ is discrete. Proijerties of Content 711. 1. Let 5t be a limited aggregate. With the points of 51 let us form the partial aggregates %^, ••• 51^, such that the aggre- gate of the common points, or of the common frontier points, of any two of these aggregates is discrete. We shall say that we have divided 51 hito the unmixed aggregates 5li, ••• 5tr Also, 5t is the union of 51^, ••• 51^. 2. Let the limited aggregate 51 he divided in the unmixed aggre- gates %^. 5I2' ••• "^s- Then i = 5lj + ...+f,; 5l = ii + -+i,. For, let i) be a rectangular division of norm 8. Let ^^ be the volume of all those cells of D which contain points of more than one of the aggregates 51^, ••• 51^. Let 51^,/) be the volume of those cells containing points of 51^, t = 1, 2, ••• s. Then 620 MULTIPLE PROPER INTEGRALS Now, by hypothesis, _ lim ^o = 0- 6=0 Hence passing to the limit in 1), we get The other half of the theorem is similarly proved. 3. If the aggregate 51 can he divided into the measurable unmixed aggregates ^l^, ••• St^, it is measurable, and Cont 21 = Cont Ij + - + Cont 21,. This follows as corollary of 2. 4. Let 2lp ••• %s^e limited aggregates whose frontiers are discrete. Let 21 be the union of these aggregates. Then 21 is measurable^ and Cont 21 = Cont 2li + ••• + Cont %,. For, we may divide 2t into 2ti, • • • 21^, and these latter aggregates are unmixed, by hypothesis. The aggregates 2li, ••• 21^ are also measurable by 706. 712. 1. Connected with any limited complete aggregate 21 of upper content 2t > is an aggregate ^, obtained from 21 by a pro- cess of sifting as follows : Let i)j, i>2' "■ b® ^ S6t of rectangular divisions of space, each formed from the preceding, by superimposing a rectangular divi- sion on it. Let the norms of these divisions converge to 0. The division B^ effects a division of 21 into unmixed partial aggregates. Let %^ denote those partial aggregates whose upper content is > 0. Then, by 711, 2, % = %. Similarly, the division D^ defines a partial aggregate of 21^ and hence of 21, such that %^ = 21, etc. Let us consider the cells of i)„ which contain points of 2l„. As ti = oo, these cells diminish in size, and in the limit define a set of points ^. The upper content of the points of 21 in the domain of any point of ^ is > 0. Thus each point of ^ is a limiting point of 21, and hence a point of %. We shall prove, moreover, that ^ is perfect. I PROPERTIES OF CONTENT 521 For, suppose h were an isolated point of ^. Let (7 be a cube whose center is h and whose volume is small at pleasure. Let a be the points of 21 in C. Let us divide Q into smaller cubes, say of volume -a. The points of 31 in at least n of these new cells n must have an upper content > 0. Thus there are other points of ^ in O besides h. Hence ^ has no isolated points. To show that ^ is complete, let ^ be a limiting point of :^ ; it is therefore a point of 31. The upper content of the points of % in any domain of /3 is > 0. /3 will therefore lie in one of the cells of 2>„, w = 1, 2, •••. Hence it is a point of ^. Finally, _ For, any cell of i)„ which contains a point of ^ contains a point of 2l„, and conversely -dnj cell which contains a point of 2l„ con- tains a point of ^, or is at least adjacent to such a cell. 2. The aggregate ^ may be called the sifted aggregate of %. 713. 1. We shall find it useful to extend the terms cells, division of space hito cells, etc., as follows : Let us suppose the points of any aggregate 2t, which may be 9?^ itself, arranged in partial aggregates which we shall call cells, and which have the following properties : 1°. There are only a finite number of cells in a limited portion of space. 2°. The frontier of each cell is discrete. 3°. Each cell lies in a cube of side ^ S. 4°. Points common to two or more cells must lie on the frontier of these cells. We shall call this a division of%of norm S. 2. Let A be such a division of space. Let 21 be a limited aggre- gate. As in 702, %^ may denote the content of all the cells of A which contain at least one point of 21 ; while 2t^ may denote the content of those cells all of whose points lie in 21. 522 MULTIPLE PROPER INTEGRALS 3. Let 21 he an aggregate formed of certain of these cells., 2lj, ••• 21^. Then 21 is measurable ; and Cont 21 = Cont 2li + ••• + Cont 21,. This is a corollary of 711, 3. 714. Let %he a limited point aggregate., and A a division of space of norm S, not necessarily a rectangular division. Then limi^ = I, lim2l^ = 2l. (1 6=0 6=0 Let us prove the first half of 1); the other half is similarly- established. For each e > there exists a cubical division D of norm c?, such *^^* • i there exists a pair of numbers Uq, d^, such that %-'^u,D of di^ of norm d. This effects also a division of norm d of the a;,-axis. Let d^, d^ ••■ denote those intervals on this axis, embracing at least one point for which the frontier points of the corresponding plane section have upper 626 MULTIPLE PROPER INTEGRALS content >a-. If ^^ denote the volume of those cells containing frontier points ^ of 51, we have ^2) > o'^d,, for any D. Let c? = 0. As SI is measurable, a Cont ^^ = 0. As o- > 0, Cont j^ = 0. 2. In a similar manner we prove : Let 7i^ denote those points of the projection of the measurable aggre- gate % on the plane x^ = ^^for ivhich the content of the frontier points on the corresponding rectilinear sections is ^ cr. Then Ti^ is discrete. 3. Let TC^ be the projection of the measurable aggregate % 07i the x^-axis. Let D be a division ofdl,n of norm d. Letf^^ fi'" denote those intervals on the x-axis contahmig frontier points of ^^. Let 7>0, o->0 be taken small at pleasure. If f'li f'i ■•• denote those f -intervals contaiyiing points of ^*^, for ivhich the upper content of the corresponding plane sections "^ is ^ 7, we can take d^ so small that ^f[<(r, dKd^. For, in the contrary case, the upper content ^ of the frontier points of 21 is =^^_ ^^ But 21 being measurable, 0^ = 0, which contradicts 1). Classes of Integrdble Functions 719. 1. Let f(x-^ •■■ x„i) be conti7iuous at the limiting points of the limited complete field 21. Then f is integrable in 21. For, reasoning similar to that of 352 shows that we can effect a cubical division 2>, such that the oscillation of / in each cell of D containing points of 21 is ,(^.<^-5;ft,,, 0. Let ^ be the partial aggregate formed from 21 by the process of sifting, considered in 712. Let _Z> be a rectangular division, and d one of its cells containing points of ^ ; we can choose _Z> so that no cell has points of ^ only on its sides. Let o be the points of 21 in d. Since a is a partial aggregate of 21, /(a;^ ••• a;^) is integrable in o. The reasoning of 508 shows now that / must be continuous at one point, at least, of a and hence at an infinity of points of a. Among these points, lie points of iB. Thus every cell of the division 2>, which contains a point of ^, contains a point of ^. Hence I = f . Generalized Definition of Multiple Integrals 722. Letf(x^ •■■ x^') be liynited in the limited field %. Let A be any division of space of norm S into cells 8^ 8^., •••, not necessarily rectangular. Let Wl^, VX^ be respectively the maximum and Tninimum off in S^. Then __ lim E^ = lim ^m8^ = Cfd% (1 6=0 5=0 »/2l lim S^ = lim 2m^a^ = ffd^. (2 S=o &=o ^21 Let D he a, cubical division of norm d. Let c?^, d^., ••• be the cells of D containing points of 21. We may denote their volum.es by the same letters. Let M^ = Maxf in d^; also #^Maxj/| in 21, and ^ 1. Then for each e> 0, there exists a d such that ^%\ 2 where, as usual, ^ ^^.^ ^ GENERALIZED DEFINITION OF MULTIPLE INTEGRALS 529 Furthermore we may choose d so small that Sz)-I containing points of :S9„, and d'l, d'2, ••• the cells containing onl}^ points of 21. Then S^^=^M^d, + XM[d[, S^^^^=2N^d,, where iV^ = Max / for points of ^ in d^. Hence \^%,-S^^,J<^CM^-nd. + F2d[ M-a, however small cr > is chosen. Hence ASj,-atd.. (2 Let g CMt2l 534 MULTIPLE PROPER INTEGRALS Then passing to the limit d — in 2), we get A-e< i < I X in 21, _ r/c^2l^Xl. ^% 6. Let f{x-^ ••• x^ he limited in the limited field 21, and ^0. If ^ is a partial field of 21, JfM^Jfd^. For, >^2t^^A^53^. 7. Letfy, ••■fn he limited in the limited field 21. Then jSfl + - +fn)d% <( f^d%+..- +£fM. iiSl ^l2l il2l For, in any cell d^ of the division D, Max(/i+ ..+/„)o-. If %^ is discrete for any ifd'^= ifdu = o. For, let _ , _ . (7o-, while d^l denotes the cells in which |/|< cr. Then ^^°^^ \Js\ a, an arhitrarily small positive number^ form a discrete aggregate. Let 3 denote the points at which f= 0. If 21 is complete^ __ _ 3 = 51. For, 0= r> f, by 729, 6. MULTIPLE INTEGRALS TO ITERATED INTEGRALS 537 But r ,/^^3I., by 729, 5. Ifence __ ^21, = 0. Since o- > 0, %^ must = 0. To prove the second part of the theorem, let 5 be a point of ^, the sifted aggregate of 21, at which/ is continuous. Cf. 712. Then if /> 0, we can choose 3>0 so small that />A,>0 in V^(h'). But the upper content of the points a of 2t in Fg is a>0. Hence Hence /= at every point of continuity of 21 in ®. Let now 7) be a rectangular division of space. The reasoning of 721 shows that every cell which contains a point of ^ also contains a point of continuity lying in ^. Hence, which gives, 3^a or 3 = ^. Reduction of Midtiple Integrals to Iterated Integrals 732. 1. Let f(xi ■■■ x,„^ be limited in the limited field 21. Let j^ be the projection of 2t on the a;^-axis. Let ^^ be a plane section of 21 perpendicular to the a;^-axis. Then the (m — l)tuple upper and lower integrals _ ffd^^, ffd'^, (1 are one-valued limited functions of x^, defined over J^. For, let 21 lie in a cube of side C. Let If^x^^ ••• a;^)|<^. Then both inte- grals are numerically for any x, in f ^. 638 MULTIPLE PROPER INTEGRALS 2. Each of the integrals 1), considered as functions of x^ defined over j^, have therefore upper and lower integrals in ^^, viz. : fdxjfd^ fdxffd% For brevity these may also be written ff If SI If- tLvr-^ ^v.^"^. ^vy%\ ^vy^c 733. 1. Let f(x-^ ■•■ x„i') he limited in the measurable field 21. Let ^\ be the projection of 2( on the x^-axis. Let '^^ be the plane sections of % corresponding to the points of j^. Then ffd^< fdx^ ffd'^,< fdx, ffd^ < ffdn-, (1 ffdn< fdx Tfd'^^K fdx^ f.fdf,< ffd^. (2 Let us establish the relation 1) ; the demonstration of 2) is similar. Let 21 lie in an outer cube ^, whose projection on the a;^-axis is b, and whose plane sections perpendicular to this axis, we denote by O. We introduce an auxiliary function g(x^--- x^) =f(x^ • • • a:^), at points of 21 ; = 0, at other points of iB. Let I) he a. cubical division of 9?^ of norm d. This divides ^ into cells which we denote by h. It also divides the planes O into cells which M^e denote by 8' ; and the segment b into intervals which we denote by h" . Let M, M' denote the maxima; and w, m' the minima of g(x^ ••• x„^ in the cells 8, h'. Let G-^ G^ be the upper and lower £1 ^Q a Or, since MULTIPLE INTEGRALS TO ITERATED INTEGRALS 639 integrals of g in the field ^. Let \f(x-^ ■■■ x„,)\ 0, there exists u d^ such that a-et-e, d0 he arbitrarily small. Let j^. denote the points ofljorwhich - 542 MULTIPLE PROPER INTEGRALS Then ^'^ is discrete. If 21 is complete., the upper content of the points where this difference vanishes is ^^. A similar theorem holds for the differences -^ ^ ) fdx,- \ fdx^. For, by 734, 1, ^ r r r r • ' i fd^=i i =i i . Hence f J -//=/!/-/! = 0. The theorem follows now, by 731. 737. 1. Let "^ he a complete 7neasurable field. Let ^^ he its pro- jection on the x^-axis. Let t)^ he the points of ^^ for which the corresponding plane sections are measurahle. Theyi X\^ = ^^. A similar relatioyi holds for the projection 2i^. ' /(a;j ••• x„^)= 1, at frontier points ^ of 21; = 0, at other points of 21. Then, by 702, -^ Since 21 is measurable, g" i^^ discrete, and hence measurable. Hence, by 734, 1, r r O^jdxl fd^. Hence, by 731, the points t}^ at which X/^g = have the same upper content as ^,. 2. Let f(x-^ ••• x^ he integrahle in the measurahle complete field 21. Let t)^ denote those points of ^^^ for ivhich the integrals over the corre- sponding plane sections ^, exist. Let 2), denote the points of H, for ivh'ch the integrals over the corresponding rectilinear sections a^ exist. -L lien ^ /* /* f* f* AFFLiCATlOlN TO rNTE-RSIOlM 643 Let us prove the first half of 1) ; the other half follows similarly. By 734, 1 = Cdx^ f fd%, by 726, 737, Since at the points tf^ ^?c ^^. Application to Inversion 738. 1. In 570 we saw that admits inversion, if /(a;, «/) is limited in the rectangle R= (ahafi^^ and continuous except on a finite number of lines parallel to the X and ?/-axes. We can generalize this result as follows : Let f(x^ if) he limited in the rectangle R = (ahajS^. Let the points of disconti^iuity A in R he discrete. Let the points of A on any line parallel to either axis form a discrete aggregate on that line. Then j dy I fdx, I dx I fdy exist and are equal. For, by 719, 2, the double integral exists. The theorem now follows from 734, 2. 2. If the points of discontinuity oifixy') on any line parallel to the x or ?/-axes do not form a discrete aggregate, we may apply the following theorem : 544 MULTIPLE PROPER INTEGRALS Let fixy) he limited in the rectangle R= (^aba^y If the points of discontinuity of f in R form a discrete aggregate^ we have \ dy ) fdx = i dy I fdx = i dx i fdy = \ dx \ fdy. 739. 1. Let f(x^ y, z) he limited in the rectangular parallelopiped jR, hounded hy the planes x= a, x= h; y = a, y = ^\ z = A, z = B. Let it he continuous in -B, except at the points of a discrete aggre- gate A. Let the jjoints of A on any line or plane parallel to the axes be discrete with respect to that line or plane. Theti the triple iterated integral Jdz \ dy \ f{xyz')dx exists, and admits unrestricted inversion. For, }/{xyz-)dB exists by 719, 2. Let P denote a plane section of R parallel to the a;, ^/-plane. Then the double integrals { f(xyz')dP, A 0. 546 MULTIPLE PROPER INTEGRALS 2. Let E be the rectangle (0101). Let us suppose the coordinates of its pointa X, y expressed in the dyadic system. Cf. 144. We define now a jmrtial aggregate % of Ji as follows. All its points lie on certain lines parallel to the (/-axis, viz. x = a, where < a < 1 is any number having a finite representation. Let a particular value of a embrace p digits in its representation. Those points of the line x = a belong to %, whose ordinate is expressed inp digits. Obviously this set of points is symmetrical with respect to x and y. If the representation of a is not finite, there is no point of 3t on the line x = a, or on y = a. In any case, there are but a finite number of points of SI on any line parallel to the x or y-axis. Not so, for lines passing through a point of % making an angle of 45° with the a;-axis. Obviously, any little segment of such a line has an infinity of points of % in it. Thus % is dense. Let us define now f(xy) as follows : f(xy) = 0, for any point of % ; = 1, for a point of B not in % Since the oscillation of / in any cell of i? is 1, the double integral I fd% does not exist. However, Hence both iterated integrals exist and are equal, (Pringsheim.) 3. In the rectangle B = (0101) let us define another aggregate 33 as follows. As before, the x of every point of i8 must have a finite representation. If the repre- sentation of a embraces p digits, all the points of the liyie x = a belong to 33 whose ordinates are expressed by p or less digits. Thus on any given line x = a, are only a finite number of points of S3. On the contrary, on any little segment of the line y — a, lie an infinity of points of 33. If the representation of a or & is not finite, there is no point of 33 on the lines x= a, OT y = b, as in Ex. 2. Let us define f(xy) as in Ex. 2. f(xy) = 0, for any point in 35 ; = 1, for a point of i? not in 55. Then as before, the double integral does not exist. k^'"' TRANSFORMATION OF THE VARIABLES 647 The integral ^i does not exist for any y whose representation i^ finite, since On the other hand, -^ J. fdy = \, for any X. Hence /.i -.^ I dx I fdy — 1. {Pringsheim.) 4. Let/(x2/) be limited in the rectangle B = (0101). Let ^^'dyf^]f(xy)clx exist. The reader might be tempted to conclude that therefore C^dyCfdx, 0<«. dm dt. dt^ 501 TRANSFORMATION OF THE VARIABLES 549 For, suppose, for example, that to the point u of Sly correspond two points t, t' of St. If now x corresponds to u; to the point x will correspond at least the two points t, t'. The correspondence is thus not uniform between SI, '^ru- 7. If the functions ^ have first partial derivatives in SI, we call J ^ 5(<^l--- m) ^ the determinant of the transformation. If the first partial derivatives of the ^'s are continuous in a region M, while the first partial derivatives of the ^|r's are con- tinuous in a region containing _By, we have, by direct multiplica- tion of the determinants t/y, J^, and using 430, 6), which we may state roughly thus : The determinant of the product of two transformations is the prod- uct of their determinants. 742. 1. Let have continuous first partial derivatives in the region M. Let the correspondence between M and i^y be uniform. Let the determi- nant of the transformation Jyi^tO in M. In this case we shall say the transformation T defined by the equations 1) is regular in R. 2. Let T he a regular transformation in R. Let t he a point of R, to which corresponds the point u. Let E he the image of D^(f). There exists an ?; > 0, such that B (u) lies in E. Furthermore^ if t runs over an inner aggregate % of R, the rfs do not sink helow some positive number tjq. For, suppose there exists no t; > 0, such that D^(u) lies in E. Then there exists a sequence of points w^, u^^ ••• which = m, and 550 MULTIPLE PEOPER INTEGRALS which do not lie in E. The inverse functions of 1) being one- valaed and continuous about m, by 443, the image of the above points form a sequence ^j, t^^ ••• which =t. Hence all the t^ for 71 > some m lie in Dg(t), and thus u^, ^m+i-, ••• must lie in S, which is a contradiction. This establishes the first part of the theorem. Turning to the second part, suppose ?; = as ^ runs over 31. Then reasoning similar to that of 352 leads at once to a contradiction. 3. Let T be a regular transformation in the region R. Then Rj is a region. Let % he an inner aggregate of R. To inner and frontier points of St, correspond respectively inner and frontier points of ^ = Stlrp^ and conversely. This is a direct consequence of 2. 4. If T is a regular transformation in the region R, T~^ is a regu- lar transformation in R^. The determinant of the inverse transfor- mation is ^ Jrp-\ = -r- This follows at once from 443 and 741, 4). 5. Let T he a regular transformation in the region R. Let % he an inner aggregate of i?, and let ^ he its image. If either % or ^ is measurable., the other is. If one is discrete^ the other is. This follows from 4 and 708, 3. 743. 1. Let T he Si regular transformation T; x^ = (f)^(t^--- t,J, • . . x,„ = (f>,^(t^ — t^') (1 in the region R. Since Jy^ 0, not all the derivatives vanish at any point of R. To fix the ideas, let ^^0 (2 at a point t, and hence, as it is continuous, in a certain domain i>a(0 of t. TRANSFORMATION OF THE VARIABLES 551 We show now how T can be expressed as the product of two special regular transformations. The first transformation we define thus : By virtue of 2) this system may be inverted, giving Here 6 is one- valued, and has continuous first partial deriva^ tives in a certain domain I)^(u). If h' , -yjr have continuous first derivatives in the respective domains, the two transformations T^, T^ are regular. 2. Let %he a limited inner aggregate of the region R. We can effect a cubical division of the t- space of norm d such that for the points of 51 in each cell d^, there exist two transformations 2\^*\ T^"^ of the type just considered^ such that For, we can take d so small that not all the first partial deriva- tives vanish in any cell. For if they did, reasoning similar to that of 264 shows that they must then vanish at some point of M, which would require t7= at that point. Thus these cubes may be taken as the domains J)sCO i^^ !• ^J reasoning similar to that of 352, we show that the norms 7} of the domains D (u) considered 552 MULTIPLE PROPER INTEGRALS in 1 do not sink below some positive number. The same reasoning applied to the norms B' of the J)s'(t} above, shows that the norms S' of the above Z)j.(i), are all greater than some positive number. Thus if d is taken small enough, the relation 4) will hold in each cell containing points of 21. 744. Let define a regular transformation of determinant J in the region R. Let % he any inner measurable perfect aggregate of M, and let 3: be its image. Letf(x-^ ••• x^') be continuous in H. Then Jj/(^i ••• ^m)dx^ "• dx^=X\J'\fdh ••' dt^' (1 For m=\ the relation 1) is easily seen to be true, taking account of direction in %. Let us therefore assume it is correct for w — 1, and show it is so for m. Let _Z> be a cubical division of the t space, such that in each cell containing points of % the trans- formation ^can be expressed as the product of two transformations -^1 ' **! == ^11 "■ '^m-l =^ '^m-\i "^m == H>m\P\ '" tmji T^\ X^ — <^i(Wi ••• W^_i^), ••• X^_i = , it obviously holds in 3^, by 728. We may therefore assume, without loss of generality, that the same transformations T^, T^ may be employed throughout 9£. Let the image of H in the u space be U. We have now Lf'^^x ••• dxra=j dx„,j fdx^ ••• dx^_i, by 737, 2, = I du„i I IJ^lfdu-^^ ••• du„,_i, by hypothesis, where ^^, S^'^n are the transformed Q,„, *!)3„„ respectively. TRANSFORMATION OF THE VARIABLES 553 Since X is measurable, U is so, by 742, 5. The same is true of ^[. Hence, by 737, 2, j^fdx^ ■ ■ . dx„, = J^ 1 J^ I fdu^ • • . du„ Applying the transformation T^ to the integral on the right, similar considerations show that j^fdx^ ••• dx„,=jjJ^\\J^\fdtj_ ... dt^, which is 1). 745. Let , .. . . , ., ^ - define a regular transformation of determmant J^ in the region R. To a rectangular division D of norm d of the t-space into cells d^, corresponds a division A of norm 8 of the x-space into cells S^. Let % he any inner region of R, and X its image. The cells of ^falling within 36 are unmixed, and their contents are ^K=\'^\d^ + ^^d^, t in d^ (1 where | e^ | < e uniformly, on taking d sufficiently small. For, % being an inner region, to each inner rectangular cell d^ of X, corresponds a measurable cell S^ of H by 742, 5. Hence the cells S^ within H are unmixed, limited, perfect, and finite in number for any A. Since the determinant J is continuojis in St, we can take d^ so small that in any d^ in %, l^l = I^.J + ^<; (2 where t^ is any point d^, and for any division I) of norm < c?q. From 702, 2 and 744, 1), we have for divisions of norm <, <, -,313,314. f , C, 337, 510. Sj), Sj), 337, 506. Md, ^d, 354, 513. 21a) ^A) 521. (A general division.) rn Date Due ' ■ "" W' \ .] H^ JftN 3 '(;7 1 .muz'^sf 1 4^^/3-/7 S^/'?i^^O SHao'&ip ^ A BOSTON COLLEGE U, T ' '^ATH, OEPT, 3 9031 01548913 1 H BOSTON COLLEGE LIBRARY UNIVERSITY HEIGHTS CHESTNUT HILL. MASS. Books may be kept for two weeks and may be For Reference Not to be taken from this room