REGULARI POINTS 01 LINEAR DIFFERENTIAL EQUATIONS SECtOND ORDER BY MAXIMIE IBCHEIR, Pih.D., ASST. PROFESSOR OF iMATHE-IATICS IN IHARVARID UNIVERSITY (;AMiBRIDiGE lbarvarb UflniveritV 1896 The following pages are intended quite as much for students of mathematical physics who may not be able to carry the subject farther than is here done as for those intending to make a more extended study of the modern theory of linear differential equations. Students who have mastered Byerly's Inteyral Ccccuclus (revised edition), or its equivalent, are in a position to read what is here given. Treatises on the subject of Linear Differential Equations have recently been published by Heffter (elementary), Schlesinger and Craig. Picard's Traite dt'Analyse and Jordan's Course d'Analyse, Tome III. may also be consulted with advantage. REGULAR POINTS OF LINEAR DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. INTRODUCTION. Ally homogeneous linear differential equation of the second order may be written in the form: d2y d y ( 1 ) ' pd X2 dx (x) q )y 0. Two solutions of this equation are said to be linearly independent if neither is a constant multiple of the other. If yi and Y2 are solutions of (1) it is clear that (2) y = Ciyi + 2Y2, where c1 and c2 are constants, will also be a solution. The converse of this is also true, namely, that if y, and Y2 are linearly independent every solution of (1) can be thrown into the form (2) which is therefore known as the general solution of (1). It is with equations of the form (1) that we shall have to deal in the following pages, and we will assume for the sake of simplicity that the coefficients p and q are rational functions of the independent variable x.t This variable x will be allowed to take on complex as well as real values so that we shall have to represent it geometrically by a point in the complex plane. The points, necessarily finite in number, at which at least one * For a strict proof of this theorem cf. Heffter, pp. 45-48, 233-235. t The reader should notice, however, that the following p.ges apply almost without change to the more general case in which p and q, though not rational, have no singular points except a finite number of poles in the portion of the complex plane considered. 2 INTRODUCTION. of the functions p and q becomes infinite are called the singular points of the differential equation, while all other points are called non-singular or ordinary points. A justification of these terms lies in the following, -- Fundamental Theorem: If a is an ordinary point of the differential equation (1) the genercal solution can be expressed in the neirghborhooc of this point in the form y = go + gl (x-a) g+ g (- a)2 + where go and gy are arbitrary constants. We shall find it convenient to postpone the proof of this extremely important theorem until later, and to go on at once to the following, Definition: A regular point of the equation (1) is a point at which p does not become infinite to an order higher than the first or q to an order higher than the second. It will be seen that all ordinary points and some singular points are included among the regular points. The other singular points at which p becomes infinite to an order higher than the first or q to an order higher than the second or at which both of these things happen we shall naturally speak of as irregular points. In the following sections we shall obtain in the neighborhood of regular points solutions of (1) in the form of series and in the course of this work we shall obtain as a special case the fundamental theorem above given concerning non-singular points. In discussing the behavior of the solutions of (1) in the neighborhood of a point a we shall find it convenient to throw the differential equation into the form: d2y dy (3) (x-a)2po () d + (x- a) p (x) +p2 (x) Y - where po, Pi, P2 are rational functions of x none of which become infinite when x -- a and which do not all vanish when x - a. It is clear that (1) can be reduced to the form (3) in an infinite number of ways, namely by multiplying (1) through by a suitable power of x —a and by any rational fraction neither the numerator nor the denominator of which contains x- a as a factor. This form (3) is peculiarly convenient in FORMAL TREATMENT OF REGULAR POINTS. 3 distinguishing between regular and irregular points, the necessary and sufficient condition that a should be a regular point of (3) being that Po (a) shall not be zero, while the further condition that a should be a non-singular point is that p1 should vanish at least to the first order when x - a and P2 at least to the second order. ~ 1. FORMAL TREATMENT OF REGULAR POINTS. GENERAL CASE. Without at first making any assumption concerning the nature of the point a we will try to find a solution of (3) in the form: (4) y = (x - a) K V g (x - a). V=0 We shall here assume, as we may do without loss of generality, that y0 == 0. The above expression (4) involves an undetermined exponent K and an infinite number of undetermined coefficients g/. all of which quantities must be determined by substituting (4) in the equation (3). Before performing this substitution let us note that the result of substituting (x- a)P in the first member of (3) is (x - a)P[p (p - l)po(X) + ppl(X) + p2(X)]. Let the expression in square parentheses be denoted by f(x, p). This function does not become infinite when x- a and, being a rational function of x, can therefore be developed by Taylor's theorem according to powers of x - a: (5) f(x, p) p (p - l)o(x) + ppi(x) +2p2(x) - fj(p)(x-a)< M=0 The coefficients fi involve p integrally and rationally to a degree not higher than the second. If now we substitute (4) in the equation (3) we get: v = oo sj gf(X, K + V) ( - a)+V o v=0 4 FORIMAL TREATMENT OF REGULAR POINTS. or replacing f(x, K + v) by its development (5): 0 -— a oo --: gV: f (K + V) ( - )KV '++ = 0. v= —O /=-O We must here collect the terms involving like powers of x - a into a single term and equate the coefficients of the terms thus formed to zero. This gives us the following infinite set of equations: goo (K) 0, g(6) o(K + ) + gof((K) =0,.g2f(K + 2) + lfl (K + 1) + gof2 (K)= 0.3fo((K + 3)+ g2 (K. + 2) + glf2(K + 1) + g03(K) = 0, From these equations we must compute the exponent K and the coefficients g^. Since as we have seen we may assume qo =-= 0 the first equation reduces to fo(K)= 0 an equation to determine the exponent K and known as the indicial equation of the point a. Since f (K) is the constant term in the development of f(x, K) we have: fo(K)- f (a, K) = K(K - 1) po(a) + Kp(a) + p2(a). The indicial equation will therefore be of the second degree unless po(a) - 0 in which case it will be of lower degree. The necessary and sufficient condition that the indicial equation of a given point should be of the second degree is that the point should be regular. In what follows we will assume that a is a regular point and we will denote the roots of the indicial equation, which in this case are known as the exponents of the point a, by K' and K". Having thus determined K by means of the first of the equations (6) the remaining equations give us in succession the values of gy, g2, g3,.. in terms of go. The equations being linear the first v of them can be solved at once by determinants and give: FORMAL TREATMENT OF REGULAR POINTS. I (7) (-yV II)',,h)g o where: fo(K + 1)fo(K+ 2).... (K + v) fi(K+V — ) f2(K+v-2)... f-(K+ 1) f,(K) fO(K+ V —1) l(K + -- 2).. f_2(K+ ) f,_l(K) O fo(K+v-2)... f_3(K-+ 1) fv-2(K) hv(K)- 0 0 *...f-4(K+1) f/-_3(K)... fo(K+ 1) fi(K) We thus see that g/, which occurs merely as a factor of the whole series, may be chosen at pleasure and that then g1, g2,.. are completely deterlinecl the determination of course being different according as we have given to K the value K or K. When it is necessary to indicate this last mentioned dependence of g, upon K we may write g/,(K). We have now obtained the following two series of the form (4): y1 - (x- t)a g (K') ( - ), v=O v = -0 2 - (X - Ca)K" gV(K") (x - a) It will be proved in 2 that these series converge in the It will be proved in ~ 2 that these series converge in the neighborhood of a. Assuming this it is clear that yi and Y2 are solutions of the differential equation (3), and, since they are in general linearly independent, that the general solution of (3) is c1y1 + c2y2. Two difficulties may, however, arise: 1st, the indicial equation may have equal roots K' = K" in which case we get only one solution in the above form. 2d, some of the factors fo(K + 1), of(K- 2),... which occur in the denominator of g,(K) may be zero, so that the determination of gY(K) is impossible. This can happen only when K +- n (n being a positive integer) is a root of the indicial equation, i. e. it can happen only when the difference of the exponents is an integer and even then only in the case of the series corresponding to the exponent with the smallest real part. We have then, except for the proof of convergency to be given later, established the following proposition: 6 FORMAL TREATMENT OF REGULAR POINTS. In the neighborhood of a regular point two and only two linearly independent solutions of the differential equation can in general be found in the form (4) corresponding to the two exponents K', K" of the point. Exceptions occur only when K- K" is an integer in which case only one solution of the form (4) can in general be found. Owing to the fact that it is impossible to find more than two series of the form (4) which even formally (i. e. without regard to convergence) satisfy the differential equation it is obviously immaterial whether we use the above method for determining the exponents and the coefficients or any other method which will make the series satisfy the differential equation formally, the result of the determination must be the same in either case. In particular if we wish to use the above method it will make no difference how we reduce our differential equation to the standard form (3) i. e. instead of the functions po, pi, P2 we may take any other rational functions proportional to them none of which become infinite and which do not all vanish when x - a. We may for instance let po0 1 as we shall find it convenient to do in ~ 2; or we may make po, pm, P2 all integral rational functions as we should usually find it convenient to do in applying the general theory to special problems since then all the coefficients f,(p) in the development of f(x, p) after a certain point will be zero. Finally let us glance at the case of an irregular point. Here the indicial equation is either of the zeroth or of the first degree. In the first case no solution of the form (4) exists. In the second case one and only one series of the form (4) exists which formally satisfies the differential equation, but even this series is of comparatively slight importance as, unlike the series for a regular point, it will not in general converge.* Exercises. 1. Find the singular points of the following differential equations; distinguish between regular and irregular points and find the exponents of the regular singular points: * For examples illustrative of the matter here touched upon see Heffter, pp. 43-44. CONVERGENCE. 7 2c12y dy (a) x 2 dX d2y dy (b) (1-x2) -2x +m(m+l)y O (Legendre's Equation). 21)d2)dy n2 (c) (1 —2) — 2x c+ [ (m+ l) 2 ]y-O. an y ]4 1 1+ 1 1 y (cl) 2 e3 c x 2 x-e, x —e x-e- cdx Ax-+-B y 0 (Lame's Equation). 4(x- e1) (x - e2) (x - e). 2. Determine the coefficients gy in the developments about the regular point 0: (a) for equation (a) Exercise 1; (b) for Bessel's Equation: d2y 1 dy 2( n) 2T + x1 dx+( ( Y=O; do not attempt to consider at present the cases in which the difference of the exponents is an integer. ~ 2. CONVERGENCE. GENERAL CASE. In the preceeding section we saw that in the neighborhood of a regular point a two series of the form (4) will in general exist which formally satisfy the differential equation, while in all cases where we have a regular point one such series exists. It is the object of the present section to prove that these series converge within a circle described about the point a as centre and passing through the nearest singular point of the differential equation. In the proof of this theorem we shall find it convenient to assume that Po = 1, an assumption which as we have seen is allowable. 8 CONVERGENCE. Call the distance (real and positive) from the point a to the nearest singular point of the differential equation R. We wish to prove that the series (8) Y g~(K) (x - a)" P=0 in which the coefficients have been determined by equations (6) is convergent at any point x within a circle described about a as centre with radius R. Draw a second circle of radius K and centre at a large enough to include the point x but smaller than the first circle: R>Kr> I x-a. We will speak of this second circle as the " circle K." Our method of proof will be to find a set of positive real quantities a > gI I and to prove that the series E aC x - a I is convergent. From this the absolute convergency of (8) follows. To find these quantities a. we proceed as follows. From equations (6) we get: 1 Yv+l = I ( K+- +- 1EsA(K + v) + g-lf2 (K + - 1) fo(K + V + I) +- ~ ~ gof+l (K)] whence it follows that: |g"+iiS i <i) i A ( ^c-+ V) I fo(K + V + 1) + ~ + 0o. ( +I(.K). Moreover we have: af(X, p) ~:~ dax — ' (V + l)fv+l(p) (x- a) ", the radius of convergence of this series being R. If then we denote the greatest value of I af(x, p) /ax on the circle K by M(p) we have by a well known theorem concerning power series:t M(p)~(v+l ) lf5+i(p) I K, * By | z is meant the absolute value or modulus of the complex quantity z. t The following theorem is refered to: If F (z) = c, + c z + c2 z2 + * * is a power series convergent in a circle described about the origin with radius R and if K < R then the CONVERGENCE. 9 or fv~i(p) < < 1 (p) v + therefore, - + f < -1 1K+ V) +Ii gv —lI I IM(K+V -1) ~ K-1-+.+~~+Igo I.IY(K) ~K-P].y] The second member of this inequality we will denote by a,+, so that: a. - g_ IV~j7(K+ V —l)+ I fo(K+V) I I go 3(K) This is the quantity a 2which we wvished to delie. From these expressions for a,+, and a, we obtain at once the equation: Kjjl li'/fo(K +~ fo(K ++v+1) I In order to prove the convergency of the series a,2 I x-a j "we need merely to show that lim a,+,C~ Now since, <a a~1< Ifo(K +v) + M((K+V) a. rK fo(K + V ) + Ifo(K+ VH-) and we have to find the limits of these two fractions which we will call P and Q. We evidently have: fo(P) - p2 + Ap + B, where A and B are independent of p. We have then: lim P- 1 lim (K +v)2 +A(K+v)+B 1 V = 00 KV"::o (K+V-+ 1)2+ A(K+ V+1) +B K To compute the limit of Q let us denote by xo that point (or one of the points) on the circle K at which If(x, p) / x has greatest value whichI F (z) I takes on on the circle of radius K described about the origin is at least as great as I el I K". For two different proofs of this theorem cf. Forsyth Theory of Fu'nctions, p. 33 (where however the theorem is not stated), and Harkness and Morley: Theory of Functions, p. 107. 10 NON-SINGULAR AND SEMII-SINGULAR POINTS. the greatest value M. Then since we have assumed that po = 1 (and it is here that this assumption is essential to our proof): M (p) - Pp1' (Xo) + P22' (.X) I, where the accents denote differentiation. Accordingly: (K + V)pI' (xo) +P2' (Xo) (K + V 1)2 + A (K + v+ 1) +B < K+ I P I'(X) )+ iP2'(X0) - (K+v+ 1)2 + A(K + V +1)+B Let us denote the greatest values of [ p1' (x) and 1 2' (x) on the circle K by M, and M2, then: Q < I K+V IMM1+ M2 I(K+ V+ 1)2+A (K+V+1) +B I From which it follows at once that the limit as v increases indefinitely of Q is zero. Thus we get finally: lim _,+,1 <1)< I a < V = n \av Ic x )~ Kx l and the convergency of our series is established. We have here merely proved that the radius of convergence of the series (8) is at least as great as R, we have not proved that it is exactly equal to R and in fact in some cases it may be greater. It would not, however, be hard to show that in general the radius of convergence cannot exceed R. ~ 3. NON-SINGULAR AND SEMI-SINGULAR POINTS. Among the regular points of a differential equation are included as we have seen all non-singular points. We will begin this section by attempting to apply the results so far established for regular points in general to non-singular points. If a is a non-singular point of equation (3) the coefficients of the equation must be developable as follows, a0 being a constant different from zero: o (x) = -O + a, (x - a) + a2 (x - a)2 + pl (X) b, (x- Ca) + b (x - a) 2 + p2 (:) = 2 (x - )... NON-SINGULAR AND SEMII-SINGULAR POINTS. 11 We have then: fo(K) K (K - ) a0 fl(K) - K [(K - 1) a6 + bl] From the first of these equations we get at once the theorem: The exponents of every non-singular point are 0 and 1. The difference of these exponents being an integer we have here one of the exceptional cases of ~ 1. All that we know from the preceeding sections is that corresponding to the larger of the exponents 1 a solution of the form: V = oo y -(x —a) y g (x -a) v=O can be found. From the general theory given in ~ 1 we should expect that no series corresponding to the exponent zero would exist. If, however, in determining the coefficients gy of the series: y g(= x a v= we use not formula (7), which as we have seen contains a zero factor in the denominator, but the equations (6) from which this formula was deduced, we see that the difficulty which usually presents itself disappears in the case we are now considering. In using equations (6) we must of course let K - 0 so that the second of the equations (6) now becomes: fo(1) + gofi (0) - 0. This equation is to be used to determine yg, but since fo(1) = 0 it appears at first sight that, since go =|= 0, no value of gy will satisfy the equation. This is the difficulty above refered to which might be expected to present itself here. That it does not present itself is due to the fact that f,(0) 0 (see the formula for f, just given). It follows that no matter what value is given to gy the second of the equations (6) is satisfied, i. e. Not only go but also y, are entirely arbitrary in this case. From this point on there is no difficulty in using the equations (6) to determine in succession the coefficients qg, g3, 12 NON-SINGULAR AND SEMI-SINGULAR POINTS. in terms of go and g,, and since the proof of convergence given in ~ 2 depends not on formula (7) but directly on equations (6) it will apply here without change. Without actually computing gv in the terms of go and gy it is clear from the form of equations (6) that we shall have: gV - Yogy' + glgv" where gy' and g," are ildependent of the arbitrary constants go and gl. If then we assign in succession to the arbitrary constants go and yg the values go 1, g = 0 and g0o 0, gy =1 we get the following two particular solutions: Y1 - 1 +: + Y' ( — a CL)2+ (- C)3+... Y = ( - a) + 2" (x -a) + g31 (x)- a)3 +.* Since these two solutions are evidently linearly independent the general solution will be y - go Y + g1 Y2 which is precisely the most general series which corresponds to the exponent zero. We have then established the fundamental theorem stated in the introduction. Exercises. 1. Evaluate the coefficients g- = gog' -+ ygg" in terms of the functions f0, fl, f2,.; and show that when g - 0 these coefficients are precisely those given by formula (7) for the series corresponding to the exponent 1. 2. Develope the general solution of Legendre's equation (~ 1, Exercise 1 (b)) about the non-singular point x - 0. Besides the non-singular points we have just considered there are also other regular points the difference of whose exponents is an integer and yet where the difficulty which as we saw in ~ 1 will in this case in general present itself does not occur. Such points we will call semi-singular points according to the following definition: A semi-singular point of the differential equation is a regular point the difference of whose exponents is an integer, but in the neighborhood of which, nevertheless, two linearly independent solutions of the form (4) still exist. NON-SINGULAR AND SEMI-SINGULAR POINTS. 13 Let us find the condition that a regular point with exponents K' and K", where K'-K"- n is a positive integer, should be a semi-singular point. It is in the determination of the coefficients of the series which corresponds to the exponent K" that the difficulty here presents itself, or, to be exact, in the determination of the coefficient gy and of the subsequent coefficients. The determination of the coefficients g1,.. gn- in terms of g0 presents no peculiarity. After these first coefficients have been determined the equation for determining gY is: Y.nfo (K+ n) + gnlfl (K"+ n-1)+ ~ *. +l/ofn(K")=0. Since of(K"- + n) =of(K') = 0 the necessary and sufficient condition that there should be any value of gj which satisfies this equation is that: g-_lAf (KI"+ r-l + gn-2f2 (K" +n - 2) +. + of (K") 0, or, as is at once seen by eliminating g0,.. gn-1 between this equation and the preceding equations (6) which have been used to determine g1,... yg, that: h,(K") - 0 where h has the same meaning as on p. 5. If this relation is fulfilled the value of g, is entirely arbitrary. From this point on there is no difficulty in computing the subsequent coefficients g i+l, Un-2,.. in terms of go and gY. The necessary and sufficient condition that a point, the ditference of ihose exponents K' - K is ct positive integer n, should be a semi-singular point is that the determinant hn(K") should vanish. It should be noticed that non-singular points are merely special cases of semi-singular points. Exercises. 1. Obtain a formula similar to (7) for the coefficients gn+i, gn+2,... in the series just discussed. 2. P ove that every semi-singular point with exponents 0 and 1 is a non-singular point. 3. Can a regular point whose exponents are equal be a semisingular point? 14 LOGARITHMIC CASE. 4. For what values of n will the point x - 0 be a semisingular point of Bessel's equation (~ 1, Exercise 2 (b))? Determine the coefficients il the development in this case. ~ 4. REGULAR POINTS. LOGARITHMIC CASE. We come now to the case of a regular point which is not a semi-singular point, for which the difference of the exponents Kr- K" = n is zero or a positive integer. Here one solution yi of the form (4) exists, namely the series corresponding to the exponent K', but no solution of the form (4) linearly independent of y,. In order to get a second solution we may proceed as follows. Let y2 be any solution of (3) linearly independent of y1 and introduce the auxiliary variable: z d (Y2 /dx or 2 yf zdx. (9) Substituting this last expression for Y2 in the differential equation (3) gives us when we remember that y1 is also a solution of (3): (10) dz+(2 Y1 p 1() ) 0. dx k Y1 p0)o(x) (x - a) Not only does z satisfy the differential equation (10) but conversely it is easy to see that every solution of (10) gives when substituted in (9) a solution Y2 of (3) which except when z is identically zero is linearly independent of y,. We can, however, solve (10) getting: z-e/(2i + P(- ) Z -- e ~ Po (x) (X —a). Remembering now the form of yi we see at once that yl'/y1 can be expressed as a series proceeding according to ascending integral powers of (x- a) and begining with the term Kt (x -a) —1. Since p0(a) == 0 the second term under the sign of integration above can also be expressed as a series proceeding according to ascending integral powers of (x - a), LOGARITHMIC CASE. 15 and the first term here is p1 (a) (x -ac)-1/p0(a). Since K' and K" are the roots of the equation K (K - 1)p0(a) + Kp1)(a) + P2 (a) = 0 we have p1 (a) /po (a) =1 - K'- K". Accordingly if we expand the whole expression which stands under the sign of integration above according to ascending powers of (x - a) we get as the first term: (1 + - K") (x - a)-1. Therefore (2 + pl ( -x) z (+ K'- K)log (x-a) + a, a(x — a),, v== Hence it follows that: y = oo z (x-a)K"-K'-le-ZaV(x-a)v = (-ay)K"-K-1 bV(x-a)V v=0 where bo I O. Since K'- K" is zero or a positive integer there is a term in the development of z involving (x- a)-1. We have therefore by (9): Y2-Y1 [(xa) -a cV(x- a)V +COlog(x-a)] v=o where Co =-= 0 except perhaps when K' K". Multiplying out and replacing yi in the first of the two terms by its development we can finally throw the above expression into the form: v= 00 Y2- (x — a)K" Y sg' (x — a)y+ Cyllog (x- ca), v=o where g0' i= 0 except perhaps when K' = K" The constant C can not be zero for if it were we should have the case of a semi-singular point with which we are not at present dealing.* We may therefore simplify Y2 by dividing it through by C. Having done this we can simplify our solution still further by adding to it such a constant multiple of y1 that the term in (x - a)K' drops out. * When K' = K" the reason here given no longer holds good, but in this case C is equal to bo which as we saw is not zero. 16 LOGARITHMIC CASE. In the case we are considering in this section besides the solution Y1 corresponding to the exponent K' there is also a solution Y2 of the form: V= oo (11) Y2- log (x-a) Y~ y- (x —a')K': g'(x —a) v=0 vwhere g' = 0 but (except when n =- 0) go' =-= 0. We proceed now to the theorem: The series in (11) converges within a circle described about a as centre and passing through th e nearest singular point of the differential equation. Although the truth of this theorem might be established by a careful consideration of the steps by which we deduced formula (11) it will perhaps be easier to proceed as follows: We know from the general theory of functions that the circle of convergence of any power series passes through the nearest singular point of the function developed. Suppose now that the circle within which the series of formula (11) converges (we know from the way in which the formula was deduced that it converges within some circle) did not reach out as far as the next singular point of the differential equation. Then it is clear that Y2 would necessarily have a singular point at some point on the circumference of the circle of convergence. This, however, would be impossible as the fundamental theorem concerning non-singular points which was stated in the Introduction shows us that Y2 being a solution of the differential equation cannot have a singular point at a non-singular point of the differential equation. The above method although it gives us the form of y, does not give us any convenient means of computing the coefficients g,'. For this purpose we will again make use of the method of undetermined coefficients. Let g_-= g_2 — ~ ~ ~ - n - 0. WVe wish to satisfy equation (3) by a series of the form: = oo =2 (v-nlog (x -a) +vl) (x -a)V+K' V=O We saw in ~ 1 that the result of substituting (x —a)P in the first member of (3) is (x - a)Pf(x, p). In the LOGARITHMIC CASE. 17 same way it is easily seen that the result of substituting log ( x- a) (x - a)P in the first member of (3) is ( -a)P [log (x- a) f(, p) +f'(x, p)] where f' (x, p) - f (x, p) / p. If then we substitute Y2 in (3) we get: [s -_,, (log (x —)f(x, -r K") +f'(x, V+ K")) + g'lf (, V + K) ' (x ---,) V+' 0. Now we have: /t= 0A — coo f(x, p)= fA(p)(x —ca) and f'(x,p) — fJ,'(p)(x-a)) t~ —O A-=O where the accents denote differentiation with regard to p.* Substitute these expansions of f and f' above and cancel out the factor (x -a) " and we get: r —o, /xLa) log (x —a) Y. g- y,,f/(v+ K") (x —a)Vr+ v=O -=0 v= U = -+ - ( Ig_,fl'(V - K") + f g)'f(V + K")) (x —a) + A- 0. V=O -1=O It is easily seen that the series which is here multiplied into log (x - a) is identically zero; for if we collect the terms involving like powers of (x - a) and remember that the g's with negative subscript are zero it is clear that the coefficients of the successive powers of (x - a) are the first members of equations (6) in which K is replaced by K'. We have left then only the second term of the above equation which must vanish identically. If in this series we collect the terms and equate the coefficients of the individual powers * The development of f' need not to be obtained by differentiating the series for f term by term. We need merely to notice that: f (x, p) = p (p- l)po(x) + pp,(x) p2(x), f' (, p) = (2p- 1)po(x) + p,(x). If we here expand po(X), p,(x), p2(x) and collect terms we see at once that the coefficients in the development of f' are the derivatives of those in the development of f. 18 LOGARITHMIC CASE. of (x - a) to zero we get, remembering that not only the g's with negative subscripts but also g,' vanish, the following infinite set of equations to determine the coefficients g ': gofo(K")= gl'fo(K"+ 1) + go'fi(K") _ O (12).. Sn- fo(K'- 1) + g'n-2f (KI- 2) +. *.+gofn-(K") 0 gofo'(K') +, + - n-_lfl(K'- 1) +... + glfn-l(KI+ + 1) +gn(K) 0= glfo'(K'+ 1) - go0f'(K') + gn+lfo(K' + 1) + + * + 1(K"+ 1) + 'gofn 1 (K)) 0 Since of(K") -0 the first equation is fulfilled by any value of yo' and may therefore be left out of account. By means of the 2d, 3d,... nth equations we can determine g1',2 2... g 'n- in terms of go'. The (n + 1)t of the above equations introduces no new unknown quantity, but is to be regarded as an equation for determining in conjunction with the preceding equations the as yet undetermined quantity yg'. Solving these n linear equations we find the value of go' as the ratio of two determinants of which the denominator is the expression hn(K") introduced in ~ 1. This expression, however, is not zero since a is not a semi-singular point (see the theorem at the end of ~ 3).* Having thus determined go', g'.. g'n- the subsequent equations allow us to determine g'n+l, ' n+2,... in succession without difficulty. Equations (12) allow us in all cases to determine the coefficients gv. Having thus obtained a series of the form (11) which formally satisfies our differential equation it might seem at first sight that we ought to go on and prove that the series we have * It is true that the above does not apply to the case in which K1 = Kf, but then the first n + 1 equations which we have been considering reduce to the single equation gofo,'(") =0 which is identically fulfilled since fo(K) has K = K"l as a double root. THE POINT AT INFINITY. 19 found converges. An explicit proof of convergency is not, however, necessary; for the considerations of the first part of this section establish the existence of a solution of the form (11) and the considerations we have just completed show that only one such series can even formally satisfy the differential equation. The formal solution we have just found must then be the true solution. Exercises. 1. Prove that if we do not assign to gn' the value zero the equations similar to (12) which we should obtain would leave gn' entirely undetermined; and that if we assign to it a value different from zero we thereby add to our solution Y2 a constant multiple of yi. 2. Prove that it is only when a is a regular (not semisingular) point the difference of whose exponents is an integer that we can have a solution of the form: / y=oo V=oo \ (x —a)K log(x-ac) E ca(x —)v+ - by(x-a)v v=O- v=O in which all the coefficients ac are not zero. 3. Develope about the point x - 0 two linearly independent solutions of Bessel's equation (~ 1, Ex. 2 (b)) when n is an integer. 4. Develope about the point x - 1 two linearly independent solutions of Legendre's equation (~ 1, Ex. 1 (b)). ~ 5. THE POINT AT INFINITY. In the preceeding sections we have obtained solutions of our differential equation in the form of series which converge in the neighborhood of some point a of the x-plane. It is often desirable to have solutions of our equation expressed in the form of series which converge for all distant portions of the plane. Such series we shall speak of as converging in the neighborhood of the point at infinity. In order to obtain solutions in this form we will introduce the auxiliary variable 20 20 ~~~THE POINT AT INFINITY. -I 1 / x in the dliff erential equation (1) whieh then takes on the f orm: (1) cl2y 2 x'-p (1/x') cdy + c(1 x') y 0 dX12+ x12 cXI x14 anl equation which as we see is of precisely the formn (1). Definition: The equation (1) is said to have the point (non-singular - x - 0 as a semi-singular ~ point when the equation (13) has (reular the point x' 0 as such a point. The following theorems imay be deduced at once from this definition The necessar-y and sufficient condition that x -- o should be a regular point cf (1) is that wchen x = p (x) and q (x) should vanish respectively to at least the first and second orders. The necessary and suzfglcient condition that x - oo should be a non-singular point of ( 1) is that when x - po1 (x) should vanish lihe 2 / x and q (x) should vanish at least to the fourth order. Finially we will give the following: Definition: If x = (x is a r-egular point of (1) then by the exponents of x -- for the equation (1) ar-e meant the exponents of X'I -Ofor the equation (13). Suppose now that x — o is a regular point of (1) with exponents K' and KII; then we can in general develope two linearly independent solutions of (13) about the point x' 0 in the f orm: yl-XK g (K') X'~ (14)~~~~~~~V= Y2:, gp (K") X' Now replace XI by its value 1 / x and we get as two linearly independent solutions of (1) THE POINT AT INFINITY. 21 The series (14) converge when x' < R if R is the distance from x1- 0 to the nearest singular point of (13). The series (15) will therefore converge when Ix I > 1/R i. e. the series (15) converge outside of a circle described about the origin with radius 1 /R. Now the form of equation (13) shows us that the singular points of (13) may be obtained from those of (1) by the transformation x' - 1/x. Accordingly if R is the distance from x' - 0 to the nearest singular point of (13) 1/R will be the distance from x - 0 to the farthest singular point of (1). Therefore: The series (15) converge outside of the circle described about the origin and passing through the most distant singular point of (1). In precisely the same way we get the following theorem: If x = o- is a regular (not sezmi-singular) point the difference of whose exponents K'- K" is a positive integer or zero tuo linearly independent solutions can be found in the form: I KI VY- ()o V I K" P0 /iv X \X v=0 X/ Y2 Yllog(-) + where the series converge outside of the circle described about the origin and passing through the most distant singular point. In any case then we see that the solutions in the neighborhood of the regular point x _ co differ in form from those in the neighborhood of a finite regular point x = a only in having (x - a) replaced by 1 /x. Of course the exqponents and also the coefficients in the series would be obtained in any given case by actually forming equation (13) and developing its solutions about the point x' - 0. Exercises. 1. For each of the differential equations in the Exercises of ~ 1 determine whether the point x - o is regular or irregular, singular or non-singular; and in those cases in which it is regular determine its exponents. 22 THE POINT AT INFINITY. 2. Develope two linearly independent solutions of Legendre's equation (~ 1, Ex. 1 (b)) about the point x -o. 3. Prove that the most general homogeneous linear differential equation of the second order with rational coefficients whose only singular points are el, e2,... en all of which are regular and have respectively the exponents Kl', K111 K21 K2... KnIK 15i s (a) when all the singular points are finite: ciX2 + Lx-el + + x-e 1 (I X:'. n ~ n KflKn~f (el) +f(X) K fe) +x-e+ x+ cn4 Xn-4 + Cn-5n-5+. + co]Y- y, where f(x)-(x-e)(x -e2)... (x-en) and f'Qr) - dfc(x) /dx. (b) when e = GO: c2y ~[ KI K (1 - K'n __-I rf1 d1Y [1x1-e X — en-1l] +f~x) [Ki'Kl If(ei) f * + K n- 1K' lfn-'(en-l) f (X) x-e, ~~x-en-,_ + Kfl Kl n3+ C Xn4 4 -+CO] y=O, where f () -(x el)... (x - el) Prove that in either case we must have: Kljf + K1 + K2 + K21t I... + K- 4 + Kcn" n - 2. 4. The results of the last exercise apply only when n > 3. Discuss the cases when n < 3. 5. Prove that if we transform the differential equation by introducing the new variable - - (x - a)c y any finite point b will be a regular point of both differential equations or of neither and if regular will have the same exponents in both cases if b 41= a while for the point x - a both exponents will have been increased by c. Show also that if x - b is any finite regular point whose exponents differ by an integer, b will THE POINT AT INFNITYI. 23 be a semi-singular point of both differential equations or of neither. How will the point x =- be affected by the transformation? 6. Prove that if we introduce the new variable x' -= -- + yx+ 8 any point a of the original differential equation and the corresponding point a' a- + of the transformed equation will ya+be either both or neither regular; and if regular will have the same exponents; and if these exponents differ by an integer will be either both or neither semi-singular.