UNIVERSITY OFCALIFORNSA AT LOS ANGELES The volumes of the University of Michigan Studies are published by authority of the Executive Board of the Graduate School. A list of the volumes thus far published is given at the end of this volume. Winixftxsitvi of pCicMgan ^tWidijcs SCIENTIFIC SERIES VOLUME II STUDIES ON DIVERGENT SERIES AND SUMMABILITY ^T^^ THE MACMILLAN COMPANY NEW YORK BOSTON CHICAGO DALLAS SAN FRANCISCO MACMILLAN & CO.. Limited LONDON BOMBAY CALCUTTA MELBOURNE THE MACMILLAN CO. OF CANADA. Ltd. TORONTO STUDIES ON DIVERGENT SERIES AND SUMMABILITY BY WALTER BURTON FORD, Ph.D. MICHIGAN SCIENCE SERIES-VOL. II Btto Pork THE MACMILLAN COMPANY 1916 All rights reserved Copyright, 1916 By The University of Michigan PRESS OF THE NEW ERA PRfNTING COMPANY LANCAiTES, PA. LJOrary To My Fathek SYLVESTER FORD This Book is Gratefully Dedicated. Ill CONTENTS Page Chapter I. The Maclaurin Sum-Formula, with Introduction to the Study of Asymptotic Series 1 Chapter IL The Determination of the Asymptotic Developments of a Given Function -, '^-■• Chapter III. The Asymptotic Solutions of Linear Differential Equations. 64 Chapter IV. Elementary Studies on the Summability of Series 75 Chapter V. The Summability and Convergence of Fourier Series and Allied Developments 102 Appendix 1' ^ Bibliography 184 PREFACE During the academic year 1908-9 the author was privileged to give as a part of his work at the University of Michigan a course of lectures on infinite series, with especial reference in the second semester to divergent series — a subject which, despite the uncertain value so long attached to it, seemed clearly to be coming into increasing prominence and importance in mathematical analysis. Little was accomplished, however, as regards divergent series beyond the merest beginning; yet this was sufiicient to awaken a desire to continue farther and this in turn resulted in a course being given throughout the whole of the following year devoted entirely to divergent series and the related topic of summability. But this year also closed with much less ground satisfactorily covered than had been expected, unforeseen difficulties having arisen from time to time, some due to the inherent complexities of the subject in hand and others to the somewhat hastily conceived and hence unsatisfactory state in which much of the related literature was found to be. Thus the course still seemed altogether incomplete. It was therefore decided to continue it once more throughout the following year, 1910-11, and indeed for a like reason it was finally continued throughout 1911-12. As the lectures and class-room dis- cussions progressed, permanent notes were kept in the hope that the whole might possibly pass through the press at some future time and appear in book form — a hope which, after various delays during which the original notes have been considerably supplemented, now reaches its realization in the appearance of the present volume. In its final form it certainly presents a large mass of detail and is doubtless open to criticism in many respects, but it does not seem advisable to attempt any further defence for it than is contained in the remaining sections of this preface wherein, after certain generalities, the content and motive of the various chapters are discussed in some detail. Speaking roughly, the study of divergent series, at least as the author has come to conceive of it, may be divided into two parts, the one concerning the- so-called asymptotic series and the other the theory of summability. Of these the first, representing the older aspect, originated in an isolated note by Cauchy in 1843^ relating to the well-known series of Stirling for log T(x), viz.: (1) \ognx) = ilog2r+(x-i}logx-x+^'^l- §^^^, + ^l^,- .... {Bm = mih Bernoulli number.) Cauchy pointed out that this series, though divergent for all values of x, may be 1 "Sur I'emploi legitime des series divergentes," Coinpt. Rend, de I'Acad. des Sciences, Vol. 17, pp. 370-376. vii viii Preface used in computing log r(.r) when .-r is large (and positive) — in fact, it was shown that, having fixed the number n of terms taken, the absolute error committed by stopping the summation at the 7ith term is less than the absolute value of the next succeeding term, and hence becomes arbitrarily small (n > 3) with in- creasing X. Cauchy's work on divergent series was confined, however, to the single series (1) and, owing to the emphasis placed upon convergent processes exclusivel}' by the successors of Cauchy and Abel, no further progress was made in this interesting field until the subject at last reappeared after more than forty years in connection with the researches of Poincare upon the irregular solutions of linear differential equations.- Poincare considered those divergent series (normal series) of the form „ , , / . , „ . s fix) = volynomial in x, which for some time had been known to satisfy formally linear differential equa- tions of certain types having the point a: = oo as an " irregular " point, and he showed essentially that in general to every such formal solution there corre- sponds an actual solution which can be represented by (2) in much the same sense as (1) was described above as representing log T{x)} In view of the important significance of such results both from the standpoint of the possible use of di- vergent series as well as from that of the theory of differential equations, Poin- care set apart and discussed in some detail a broad class of divergent series of the special form (2), applying to them the name of " asymptotic series." PoiN- care's results, however, in so far as they concerned differential equations, were noticeably incomplete, being limited by certain unfortunate restrictions, and thus his original studies have given rise in later years to numerous researches, notably by Horn, in which noteworthy advances have been made, though open questions in this connection still remain. Corresponding investigations (likewise begun by Poincare) pertaining to linear difference equations have been undertaken in recent years and carried to an advanced stage by Horn, Norlund, and others. Meanwhile an important aspect of the theory of asymptotic series has come into \'iew, especially in England under the leadership of Barnes and Hardy; namely, that of actually determining the asymptotic developments of a given function — a problem of decided interest for the study and classification of functions in gen- eral. This latter aspect of the subject presents a high degree of complexity and doubtless has made hardly more than a beginning at the present time. In fact, it has thus far been approached only by confining the attention to a very limited number of special functional tj^T)es.'* * "Sur les intdgrales irregulieres des Equations lin(5aires," Acta Math., Vol. 8 (1886), pp. 259- 344. Mention should be made also of Stieltjes who simultaneously with Poincare resumed the study of divergent series, confining his attention, however, to the computational aspects of certain special series. (Thesis, Ann. de I'Ec. Nor. (3), Vol. 3 (1886), p. 201.) ' For the more accurate statements, see Chap. III. * For details, see Chap. II. Preface ix The theory of summability, or second general aspect of divergent series mentioned above, is essentially concerned with the question as to whether in any proper sense a " sum " may be assigned to the series, assumed divergent, (3) Han. n=Q This question has been scientifically attacked only within comparatively recent years, the most common avenue of approach being through the so-called boun- dary-value (Grenzwert) problem in the theory of analytic functions.^ Thus Frobenius, without having in view the study of divergent series, showed in the first place that if one has a power series whose radius of convergence is equal to 1 : (4) XI (inX'^ ', ^ = radius of convergence = 1 n=0 and writes Sn = ao -{- ai -\- •••+««, the 00 (5) lim HanX^ = lim and writes Sn = ao -{- ai -\- •••+««, then -^0 + ^1 + • • • + ^n a;=l— n=0 m=oo 71 -p i whenever the indicated limit on the right exists.^ Now, the first member of (5) is naturally associated with the corresponding series (3) (in general divergent) obtained by placing x = 1 in (4). Thus, at least if one confines the attention to divergent series (3) of the particular type just mentioned, it becomes natural to assign sums in accordance with the formula ,. 5o + *1 + • • • + Sn (6) * = !™ — ^+1 • whenever the indicated limit exists. Moreover, this formula finds additional justification in the demonstrable fact that for any convergent series (3) the sum, regarded in the ordinary sense, viz., s = lim Sn, agrees with that given by (6) — w=oo i. e., formula (6) is consistent Aside from this one formula (6) many others are now known which serve with more or less appropriateness to define the sum of a divergent series, both when the series is of the special type above mentioned and when otherwise. To what ultimate extent these formulas are appropriate, how far the theories of summability erected upon them serve any justifiable purpose in analysis, whether the different sums thus assigned involve mutual incon- sistencies — these and other questions may well be asked and more will be said on this point presently.'' Suffice it to say here that formula (6) has been found in 6 For an elementary description of the problem, see Jahuaus, "Das Vcrhalten der Potenz- reihen auf dem Konvergenzkreise historisch-kritisch dargestellt." Program des Gymnasiums Ludwigshafen (1901), pp. 1-56. See also Knopf, "Grenzwerte von Reihen bei der Annaherung an die Konvergenzgrenze." Dissertation, Berlin, 1907. 6 "Ueber die Leibnitzschc Reihe," Jour, fur Math., Vol. 89 (1880), pp. 2G2-264. ^ Interesting comments by Pringsheim relative to such questions are to be found in Vol. I of the "Encyklopiidie der math. Wissenschaften," §§ 39-40. X Pkeface particular to yield interesting and valuable results when applied to Fourier series and the other important allied developments in mathematical physics — develop- ments in terms of Bessel functions, Legendre functions, etc. Such applications alone go far toward assuring a permanent place in analysis to the theory of summability as now commonly understood. Turning now more specifically to the contents of the present volume. Chapter I considers certain aspects of the so-called Maclaurin Sum-Formula, the especial aim being to develop and summarize into actual theorems those results which are of importance in this connection to the study of divergent series. These when once obtained are of particular service in the problem of determining the asymptotic developments of a given function, and it is to this that Chapter II is then devoted. Beginning with very easy illustrative studies, the Chapter proceeds to problems of greater and greater difficulty and eventually treats the general problem already considered by various investigators of determining the asymptotic developments of the general integral (entire) function of rank p (order > 0), following which, at the close of the chapter, the problem of deter- mining the asymptotic developments of functions defined by power series is briefly considered. Chapter III concerns the asymptotic solutions of linear differential equations and is an attempt to summarize briefly and without proof what are deemed to be the most essential results thus far known in this field, with mention also of the corresponding results obtainable in the study of linear difference equations, and with indications as to certain open questions still remaining in both connections. Chapter IV considers the theory of summability with the especial attempt, as in previous chapters, to single out what seems most essential. More specifically, it makes an examination of a few of the standard definitions of " sum " with the idea of subjecting each to a number of tests which, as the author has come to view the subject, every such definition should satisfy. For example, it is well known that if a really logical general theory of summa- bility is ever to be constructed it cannot include all definitions of sum that satisfy merely the condition of consistency (§ 37) since this alone does not insure unique- ness of sum. Therefore, observing the genesis of the whole subject from the boundary value problem as described above, it is proposed to arbitrarily limit the general theory to those series (3) for which the corresponding power series (4) has a radius of convergence equal to 1 and then retain only such definitions of sum as give the unique value s = lim 2_/ fln''^'". « = ]— 71=0 Definitions which do this are said to satisfy the boundary value condition (§ 39). Such definitions not only all give the same sum to a given series (convergent or divergent) (3), but they at once serve a useful purpose in analysis from the fact that they frequently come to furnish the analytic continuation of the series (4) Preface xi over some portion of its circle of convergence, or indeed in some cases, as in the definitions of Borel, throughout regions lying entirely outside that circle. How- ever limited the scope of a general theory of summability as thus conceived, it at least has perfect definiteness and logical coherence and finds immediate use- fulness in the theory of functions of a complex variable, and we venture the opinion that some such characteristics as these must be preserved in any general theory of summability that is to retain a permanent place in analysis.^ No attempt will be made here to describe the other tests which Chapter IV sets up, but it should be remarked that only a few of the standard definitions of sum are tested out since they suffice to illustrate the spirit of the undertaking. The chapter closes with a brief account of absolutely summable series and a state- ment of certain supplementary theorems and corollaries upon summability in general. A most important aspect of the theory of summability, as the author regards it, lies in its applications mentioned above to Fourier series and other allied developments in mathematical physics, and this forms the subject of Chapter V. For the sake of completeness the treatment is made to include both convergence and summability. It is based upon a general method for the study of all such developments due to Dini and appearing, though in somewhat diffuse and inaccessible form, in his great work entitled " Serie di Fourier e altre rappre- sentazioni analitiche delle funzioni di una variabile reale " (Pisa, 1880). Dini naturally considered at the time of his investigations only the question of con- vergence (not including uniform convergence), but his methods are here shown to be readily extended so as to be applicable to studies in summability. Especial effort has been made here as in the other chapters to summarize all essential conclusions from time to time into actual theorems. To Professor Alexander Ziwet the author would here express his deep grati- tude. Not only has the book enjoyed the benefits of his critical judgment in many ways, but his sympathy and kindly interest have served as a constant en- couragement, and indeed they are responsible in no small measure for the ap- pearance of the whole in its present form. The author is much indebted also to his colleagues Professors C. E. Love and Tomlinson Fort, the former for various suggestions and criticisms, and the latter for the valuable aid he has rendered in reading the proofs. Ann Arbor, AprU, 1915 8 The adoption of any one definition for "summable series" evidently involves the exoludinp; of many scries previously classed as summable; yet we believe the time has arrived when a single universal definition should if possible be agreed upon, however disastrous its immediate efi'ects upon one or more of the special forms of definition now current. The present situation in this matter is strikingly analogous to the state of confusion which led Cauchy and Abel to the formu- lation of their universal definition for "convergent series," notwithstanding the exclusions brought about and the consequent objections urged by contemporary mathematicians. CHAPTER I THE MACLAURIN SUM-FORMULA, WITH INTRODUCTION TO THE STUDY OF ASYMPTOTIC SERIES 1. The following formula (Maclaurin Sum-Formula^) Efix) = i rf{x)dx - i [/(6) - /(a)] + ^.f [fib) - fia)] + • • • ; Bm = inih Bernoulli number plays an important part in the modern theory of divergent series and we shall therefore begin by pointing out certain facts (cf. Theorems I, II, III and IV) connected with its legitimate use. These will form the basis of the studies undertaken in Chapter II. Following the discussion of (1), we shall also give in the present Chapter (cf. §§ 13-17) an outline of the general theory of asymptotic series as originally developed by Poincare in his classical memoir in the Acta Mathematica (1886), the elements of this theory being likewise needed for the proper development of Chapter II. 2. In order to carry out the desired studies relative to the formula (1), let us begin by supposing that there is given any function Ux (real or complex) of the real variable x which, together with its first 2m-j-l derivatives, is continuous within a certain interval (a, b) . For any value of x such that a ^ x ^ x -\- h < h (h = constant) we may then write 2! " ^ ^(2m)! Aux = Ux+h — Ux = hux + w, Ux" + • • • + yc,_Vi Ux^^""^ ^^^ ^" (h - z) ^ i (2m) 1 '''^' ^'^' as appears directly upon applying an integration by parts 2m times to the last term in the second member. More generally, it appears in like manner that when ^ ^• ^ 2m — 1 we may write ^ Known also as the Euler sum-formula. For comments upon the historical aspect of the subject, see Barnes, Proceedings London Math. Soc. (2), Vol. 3 (1905), p. 253. 2 1 2 The Maclatirin Sum-Formula 12 I2m-fc while the corresponding formula for the case k = 27?i is (4) Awx^""^ = J wf™+^Wz. Whence if //o, i^i, H2, • • • ^2m be the 2m + 1 constants determined by the equations Ho = 1, H2m = 0, (5) we shall have or 2m (6) hi,' = E Hj^h^i^u,^^^ + r^(.r, A) where (7) rirK) = - f ,(--) V ^^^^^^^^^^^' ^. (7) r.(.r, /»)- j^ ^^.+. 2^^ {2m - k)\ '^^' Formula (6) bears a close relation, as we shall see, to the Maclaurin Sum- Formula (1). We first proceed to determine the values of the constants Hu, noting certain changes which thereby become possible in the form of (7). If we place '^""^"'^ ~ (2m)!"*" (2m- 1)^ (2m - 2)r (2m - 4)!"^ " * (8) 77„ „^2m-2~2 and (9) 4^uz) = (^^,^^3)-,+ (o^^T^T^!"^ • • • "^ — rr~ we have .(.r, h) = - f u^:!:t'\^2m(h -z) + hUh - Jo z)]dz. Let us now develop (p2m{h — 2) + \l/2m{h — z) in ascending powers of z. We obtain General Theorems 3 (p2m(.h — 2) + \l/2m(h — z) = J2 Hk —7^ -TTj- But from (5) we have 2m-j TT 1 JJ E {2m- k-j)r'^ +"'--' ="'--' 0- + 2™-l); E^j^^ =-//,. Whence, |/2-(a:) | ; a^x^b, so that we reach in summary the following result: " If f{x) be any (real) function of the real variable x which together with its first 2m derivatives is continuous within the interval (a, b) we may write formula (16) in which, if M represents a value as great as the maximum value of |/(-'"^(a;)| within the same interval, the expression Rm satisfies relations (19) and (20)." 6. Other important forms for the remainder in the Maclaurin sum formula may be obtained when further hypotheses are placed upon f{x). Thus, let us suppose in the first place that /^-'"^ (.r) does not change sign between x = a and X = b. By applying the first law of the mean for integrals we may then write Rm = ^-h (26) HP'^Hx + z), Zp-'+'Kx + z) x=a ■r=a changes sign between z = and z = h. Replacing m by m + 1 in (16), using 8 See Malmsten {I. c), P- 70. 8 The INlACLAimm Sum-Formula therein the form for i?mfi determined by (19), and comparing the result with that of § 5 (in which m is left unaltered), we obtain D 7,2m+2 h-h I o2m _ 1 ID Z,2m-1 But /th b—h /(2'»-l)(fe) _/(2m-l)(a) = I J2P"'KX+Z)dz. Whence, upon recalling that Bn and Bm+i are both positive, we see that the expression 22m 1 " 22m— 1 ■'■ will be negative and numerically less than 1 in case expressions (26) are of the same sign between z = and z = h, while it will be positive and no greater than 22"» _ 1 22'"-i — 1 22m— 1 -'■ ~~ 22™~1 in case expressions (26) are of opposite sign throughout the same domain. Thus we reach the following result : " Let f(x) be a (real) function of the real variable x which together with its first 2m derivatives is continuous within the interval (a, b) and is such that neither of the expressions ZP'^Kx + z), ZF-'+'Kx + z) x=a z=a changes sign between 2=0 and z = h. Then, according as these expressions preserve the same or opposite signs for the indicated values of z, we may write E/Or) = \ fmdx - \ \m - /(a)] + ^ [/'(&) - /'(a)] (27) --^ [/'"(&) -/'"(«)]+ ••• + ^" ^^'" ( 2m -2)! [/'"""''(^) - F-'^-'^i^)] + Rn.> where B ^2m-l R^= i- ir+^e -g^ [/(2-i)(6) _ /(2-i)(a)]; < e < 1 and EV) = \ fmdx - i [/(6) - /(a)] + 1^ [j\h) - f (a)] (28) -^' [/'"(&) -/'"(«)]+ ••• General Theorems 9 where 02m—l ID Jj2m—1 R,n = (- i)-+^e ^,^_, ^^ [f'-Hh) - r---Ha)]; < e < 1." Formula (27) was first established by Jacobi'^ in 1834. Whenever the con- ditions for its use are satisfied it is seen that the sum of any number of terms in the series (1) (convergent or divergent) gives the value of (25) with an error having the same sign as that of the first term- neglected and less numerically than the absolute value of that term. Formula (28) is due to Malmsten.^ Whenever it may be used the sum of any number of terms in the series (1) gives the value of (25) with an error having the same sign as that of the last term taken and less numerically than the absolute value of that term. 8. Another important and well-known form for the remainder in the Mac- laurin Sum-Formula may be obtained when the function f{x) may be regarded as an analytic function of a complex variable. To see this we recall in the first place that if /(w) and ^(w) are any two func- tions of the complex variable w {w = x -\- iy) both analytic and single-valued in the neighborhood of the point w = a and of which the second has a zero of the first order at the same point, then we have the formula J a). For the contour C„ let us take that formed by the line w = a -\- iy (the point lo = a excluded), by the line w = h -\- iy {w = h excluded) and by the lines id = x ± ij (j = constant > 0) together with small semicircles of radius e > h about the points IV = a, 20 = h, the former extending to the right and the latter to the left. Since ^(w) has zeros of the first order at the points iv = a -\- jih; y = 0, 1, 2, • • •, while at the same points (p'{w) = 2Tri/h, we obtain as a result of (30) (32) hZfix) = hf{a)+ f fiw) div. We proceed to study in further detail the complex integral here appearing. Fig. 1 First, the contribution coming from the side J A (see Fig. 1) is '7(^ •-r ij) , T7 ax, ip{x - ij) and since (p{x — ij) becomes infinite when j = + oo like e-"-''"', we have but to suppose that/(w) satisfies the following supplementary condition: (33) lim fix - ij)e--''^l'' = 0; x^b in order to have 7i = provided we take j = oo. In particular, condition (33) will be satisfied whenever \f{w) \ remains less than a constant for all values of w within the strip already mentioned. ' See Petersen's " Vorlesungen uber Funktionstheorie " (Copenhagen, 1898), pp. 161-169. '° It is to be understood that the constants a, b, h have the meanings already introduced; viz., h = a + 7ih; h > 0, n = positive integer. General Theorems 11 Secondly, let us consider the contribution coming from the portion DEFG. By writing (p{w) -^''"'^' = 0; a^x^h, where t] is an assignable positive quantity. If we then take account of the two remaining contributions, viz., those arising from the sides AB and IJ , we obtain in summary h 1^ fix) = fiw)dw - r [fib) - fia)] + i , . dij x=a Jghcjd ^ Ji a. Also, lei it be supposed that the following series is convergent-}" (37) E r f^'^'Ky + t)(a)] + R„ /»1 x-l IL= - Z/^""^^/ + t)^im{t)dt = Qmix) - Qmia). Jo y=a But this result is coextensive with that indicated in the theorem. As to the second form there given for fim(a;), we observe that by virtue of statement (a) of § 3 we may apply the first law of the mean for integrals to each term of the series representing fi,„(a-), thus writing ^ix) = Zf^'^'Hy + Oy) f a. Also, let it be supposed that f^-^^{x) does not change sign within the same interval and that lim /^-"""^^ (.r) = 0. We may then ivrite Zm = Cm + ffi'-^-Hv - hfi'V) + |jV'(aO - ^f'ix) + (2m) where ^'^ ,e,„(.T)r-^>(.r); ^•''^'' (2m) r'"^-"^-' ^-^^^ [- l^e^(a:)^l, and where Cm is a constant as regards x, defined by the equation Cm = hKa) - -^^f'ia) + -^r(a) + (2^)1 "/^^"""(«) " "-(«)• To prove this theorem we first observe that, as a result of our hypotheses upon /^-'"^ (a:), the terms of the expression Fm{b, X) = E f f^'^'Ky + t)ip2m{t)dt; b > X y=x I/O will all have the same sign, so that Fm{b, x) is either an ever increasing or an ever decreasing function when b increases; also by treating Fm{b, x) as we did the Rm of (17) by means of (21), (22) and (23) we obtain ^-^^' •''^ = (2m) l'" -^rrT-r Vm{b, x) {/<^'"-^>(6) " f^'-'\x) } J < vmib, x) < 1.1^ " The expression G appearing in § 5 is in general a function of m, a, b and h. Since, in the present instance we have h = 1, a = x we represent 9 by ??m(6, x). General Theorems 15 Whence, the expression Fm{x) = Hm Fm{h, x) exists, and since by hypothesis limj(2m-l)(5) = 0, 6=00 we shall have ^ Vmix) ^ 1 or (40) (_ n^+iD f o-"* — 1 1 Thus, fim(-^) exists and has the form indicated in the theorem. Now, by equation (16) we shall have also E /(.!•) = fj{x)dx - i [j{:x) - f{a)] + ^ [f (.t) - f (a)] + (2m)! " [Z^'""'^^-^) -P'^-'Ka)] + 0.(:r) - i2.(a). Thus, we reach the desired result. Again we note that Cm will be independent of m as well as of x whenever lini/(2p-i)(^) = 0; 2^ = 1,2,3, •••. «=00 Theorem III. Let f{x) he any {real) function of the real variable x which together with its first 2m + 2 derivatives is continuous throughout the infinite interval X > a. Also, let it be supposed thatf^^'^\x) and f^^"^'^^\x) do not change sign within the same interval, ivhile \\mf^'^-'\x) = 0; p= 1, 2, 3, •... a;=oo Then, iff^^'^^x) and f^^"''^^'' (x) preserve the same sign (x > a) ice may write Zm = C + rfix)dx - y{x) + §lf'{x) - ^f"'{x) + • . . ivhere ^mix) = E rf''-\y+t) a) {other conditions remaining as before) we may write %m = C + £f{x)dx - I fix) + fff'ix) - ~-f'"(x) + . • . + ---(^2,n)r^P'^-H^-) + 9.^^), u'here ^r.{x) = ^-^y,-/^^-"Cr) + E j^ f''-Ky+t) fl <. iJ «..U; 22--1 (2w)!-^ ^•'^' 1 ^ e„.(.r) ^ 1, and ivhere C is a constant as regards both m and x, defined by the equation C = hfia) - ^f'{a) + -^f"\a) + --(2m)r'-^'''"""^^^ " "-^"^- For the proof of this theorem we first observe that the conditions for theorem II, and hence also those for theorem I, are here fulfilled both for m = m and m = m + 1 ; also the conditions that Cm shall be independent of m. Upon applying theorem II with V.m{x) as given by (40) and comparing the result with that obtained by placing m =7/1+1 in theorem I, we obtain J) [ 92m 1 I n 00 m (2;^ { .-W -^s^ - 1 |/«->(x) = (^„^ £/-«'(. + ex Let us now write /(^m-i) (^^-^ jj^ ^j^^ form -lim ( 'llf^"^Ky+t)dt b=co I/O y=^ and let Qm'ix) represent the expression Qm{x) of theorem II in the present dis- cussion. Then, in case f'--'"\x) and /^-'"+-^(a:) preserve the same sign it follows from (41) that (42) njix) = ~^^^^^ ;-^ 4^m{x)f^'"^-'^ (x); - 1 ^ ^mix) ^ and hence, for the first expression Qmix) of the present theorem, we shall have ^ e.(.r) ^ 1 with which the first part of the theorem becomes established. 1" Cf. Makkoff (L c), pp. 131-133. General Theorems 17 If, on the other hand, /^-"'H^') and /'^-'""'"'^ (a;) preserve opposite signs {x > a) we shall have equation (42) in which 22m 1 92to— 1 1 ^= VmUV = 92m— 1 ■'■ ~ 22"i— 1 and thus the second part of the theorem becomes established, upon observing finally that we here have 9.„Xx) = 0„/(a;). Theorem IV. Let f{iv) he any function of the complex variable iv = x + iy ichich is anahjtic throughout all portions of the w plane {ic = co excl.) for ichich x ^ a. Also, let it be supposed that lim f(x ± iy)e^'^-''^^ = 0; .r ^ a, where rj is some assignable positive quantity. We may then ivrite Zfix) = an + rfii-)dx - i/(.r) + ^f'ix) - ^f"'(x) + • • • where Qmix) = i-ir r f^'"H^- + edy) - f^'^^Kx - ejy) {2m)liJo e'^^'y - 1 ^ ^' ^a; = 1 when m = < ^s < 1 ichen m = 1, 2, 3, • • •, and where Cm, is a constant as regards x, defined by the equation Crr. = hf{a) - fj'ia) + ^J"'{a) - • • • + --^2^^ ^'^~"^~'' ^'''^ ~ ^"'^"^' This theorem is, in fact, a direct consequence of the result stated in § 8, being obtained from it by placing h = x and rearranging terms. Generalization of the Preceding Results^^ 10. The results given in § 4-7 and the first three theorems of § 9 require that the function f{x) together with its first 2m derivatives shall be continuous throughout a certain specified interval. When this condition is not satis- fied the same results and theorems no longer exist, at least in general. How- ever, in cases in which fix) satisfies the indicated condition except at a finite '* For a derivation of the Maclaurin sum-formula from the standpoint of Fourier series, see PoissoN [1. c). A still difTcrcnt method may be found in Boole's " Treatise on Finite Differ- ences " (London, 1860), pp. 80-84. The formula has been generalized in various directions by Barnes; see Quart. Journ. of Math., Vol. 35 (1903), pp. 175-188; Trans, of Cambridge Philosophical Sac, Vol. 19 (1904), p. 325; Proceedings of London Math. Soc. (2), Vol. 3 (1905), pp. 253-272. 3 18 The Maclaurin Sum-Formula number of points (at which discontinuity or uncertainty may exist) we may still obtain certain noteworthy results. Tn order to show this we first observe that if u and d be any two functions of the (real) variable x which together with their first derivatives are continuous throughout the interval {a, a -{- h) except at the point x = ^, we may write + ) idv = uv \ - uv \ - ( I + ) vdu € being an arbitrarily small positive quantity. This is, in fact, a direct conse- quence of the ordinary formula for integration by parts.^^ In particular, if 7/.c be a function which together with its first 2m + 1 deriva- tives {u', u", • • • «(2m+i)) jg continuous within the interval {a, a + h) except at the point x = j3 we may obtain by repeated use of (43) the following result (c-/. (3)): p=l V' L P=0 V- Jz=3-a-e "Whence, if ^o, Hi, • • • Ihm be the constants defined by (5) we may write (cf. (6)): 2m-l r2m-l 2m-i-/'L_ Np -|z=3-a+e where Upon introducing the function (pim{^) and making use of the relations (10) and (13) we thus obtain ■L m-1 T> 7,2fc hu: = Az.. - \m: + E (- D^^^^M^'^ (44) where + E //./i^ E ^^ l, ^ y^ir^ + r.(a, A), + J jWx^''"+"^2m(.'C-«)^^. The case of especial interest for our present purpose is that in which i/^ is taken in the following manner »9 As usually stated (cf. Goursat, " Cours d' Analyse," Vol. 1 (1902), § 85) the formula requires that u and v with their first derivatives shall be continuous throughout the interval of integration. General Theorems 19 Ux = I f(x)dx when a ^ x ^ ^ — e, Wi = ( I +1 ) fix)dx ivhen jS + e ^ a^ ^ a + /;, f(x) being any function which together with its first 2m derivatives is continuous within the interval (a, a + h) except at the point x — /S. Such a function w^ together with its first 2m + 1 derivatives will be continuous except at a* = jS. Whence, applying (44), we may write^° '"-1 iR,^2fc (45) + E (- I)'-' ^~ A/«'-»(a) : L A=0 p=0 Pl Jx=-f Let us suppose lastly that the interval (a, a + h) containing the point x = 8 is part of a larger interval (a, b) throughout which (except at a* = B) f{x) satis* fies the indicated conditions; also let us suppose that a is one of the quantities a, a -jr h, a -\- 2h, • • -, b — h. If then we apply formula (16) to/(.T) when con- sidered within the intervals (a, a), (a + h, b) and apply formula (45) to the same function when considered within the interval (a, a + }i) we obtain, after adding the three results and dividing by li, n-l h-h -. / ^3_e r^' \ Z/(a + g/O = Z/(^) =7 + /(.r)f/.r 3=0 x=a ll' \Ja t/g + e/ ~ A U "^ i ) '^^'"^ '^^^'^--^-'^ " "^^''^ E 7/.^'= Z ^- + ^ ^-^ /(^+^i)(^ + :r) (46) .where + (2m -"2) T~ t/^'""''^^) - /^'"-'H^)] + /?. By use of this formula instead of the earlier corresponding one (16) we arrive 20 We note that/(-"(i3 + x) = ?/)3+x and hence /(-"(/3 + x) ]-^=l, = 0. 20 The ^NIaclaurin Sum-Formula at the desired theorems corresponding to the first three of § 9. Since these are long in statement though readily supplied we shall omit them. Analogous results may evidently be obtained whcn/(.r) presents any (finite) number of exceptional points of the type just mentioned. 11. Again, the results stated in §S and the fourth theorem of § 9 require that f{iv) be analytic within a certain domain. If, on the other hand, this function presents singularities at a finite number of points within the domain, but otherwise satisfies the indicated conditions, we may readily make such alterations as are necessary to preserve correctness. For example, let us sup- pose that the function f{tc) of theorem IV satisfies the conditions there stated except at the point w = 13 = p -\- iq; a < j) < x, q < 0. The theorem will then continue to hold true-^ provided that we subtract from tlie second member the residue r^ of the function corresponding to the point iv —■ 3. However, if the exceptional point occurs at y; = fi z= p -\- iq-^ a < p < X, q > 0, then (in view of the manner in which in § 8 the integral of f(rv) over the path DEFG was transformed to one over the path DCIIG) the theorem will continue true provided we subtract from the second member the expression r^ together with the residue r/ of the function 2Trif{w) corresponding to the same point w = jS. Other cases are those in which a singular point occurs on either of the lines ^<; = a -j- iy^ w = X -\- iy or at a real point iv = ^ < x. If, in the last of these cases (which is the only one to which we shall refer later), the singular point is a pole of the first order the theorem is seen to continue as a result of (29) provided that the term mdx be changed to iim e=0 (.r ^^'i[ )-^(->^-^ - ^^ - ^'■^'' where r^, r/ have the meanings already given. Series of Stirling 12. As a preliminary application of tlie preceding general theorems to special functions fix) let us take /(.r) = log x, a = amj real number > 0. We are thereby led to certain well-known results respecting the series of Stirling. The first part of theorem III may here be applied since we have (_ 1)P-1(7; — 1)! " Cf. § 4. Seeies of Stirling 21 Whence, upon observing that log xdx = [x log .r — .r]', = /»'i + x log x — .t; ki = const. £ and that x-l X) log a; = log r(aO — log T(a) = /i-2 + log r(.r); 7^2 = con5^. x=:a we obtain log r(.T) = A + (X _ ^) log .r - .r + j-^ ^ - ^^ ^ + ^ _^ (-ir^.-i 1 "^ (27W - 3)(27?i - 2) a;2"^-3^ ^"^^••^^' where A is a constant as regards m and x and where (- 1)-+^^^ 1 (2m- l)(2m) a;^ (48) Tmix) = e^(.r) ,o^_:■,,,oI^ i^^ ; ^ ©-(-^O = !•' Moreover, by comparing the above results with the well-known formula^^ log T{x) = I log 27r + {x — ^) log x — x (49) _ r/ 1 1 i\ , dt + Jo Vl-e-^ t 2) ^ t it follows (upon placing .x = 00 ) that A = ^ log 27r. Thus, we arrive at the series of Stirling (see Preface) and it appears from (48) that, though divergent, the series may be used to compute log T{x) with but slight error when x (real and positive) is large. In fact, the first term neglected is seen to constitute an upper limit to the error committed by breaking off the series at any one point. This fact was pointed out by CArciiY-^ in 1843 through an independent investigation based upon formula (49), he also noting in this connection the possible value of divergent series in computation. Cauciiy's work was, however, confined to this one series and in this it appears that his results might have been obtained much more directly, as indicated in § 12, from the earlier general investigations of PoissoN and Jacobi relative to the Maclaurin Sum-Formula. We add that the value of the constant K may be obtained independently of formula (49) by use of the well-known formula of Wallis expressing the value of x/2.25 22 In the present case it may be shown that < 0„,(x) < 1. See Malmsten {I. c), p. 75. 2' Usually attributed to Binet. 2'' See Comples Rendus de I' Acad, dcs Sciciices, Vol. 17 (1S43), pp. 370-376. 2* See Markoff [l. c), p. 134. 22 The IMaclauein Sum-Formula Preliminary Discussion of Asymptotic Series 13. The formula of Stirling, by means of which the function log r(.r) - (x — ^) log x+ X may be identified with a certain divergent power series in l/x, affords an illustra- tion of an important class of developments known as asymptotic series. We proceed to give at this point a brief exposition of the general features of this subject, leaving its further development and applications for later chapters, especially chapters II and III. Following PoiNCARE, we adopt the following definition:-^ " A power series of the form (50) oo + Qi (".)+ fl2 (") + ••• ; flo, «!, ^2, • ■ ' constants is said to represent asymptotically the function f{x) for large positive values of x whenever lim X- [fix) - («o + a,fx + a^fx' + • • • + aj.r")] = 0; (51) ^=+'» n= 0, 1, 2, 3, •••."" Thus, for a given value of n the difference between the function and the sum of the first n + 1 terms of its corresponding asymptotic series (in case one exists) vanishes to a higher order than the nth when x = + oo , as would be the case in particular if the series were convergent. Symbolically, the above relation is expressed as follows: (52) fix) - Oo + ajx + a,lx' + • • • . Several general observations are here desirable. First, a given function fix) can be represented asymptotically in but one way. In fact, we have from (51) (53) fix) = ao+ ailx + a.lx' + • • • + an-xlx--' + ""^'"^""^ ; lim €„(a:) = 26 See Ada Math., Vol. 8 (18S6), p. 29G. 2' In this definition no restrictions are placed upon (50) as regards convergence or divergence. However, in the usual applications the series is divergent for all values (positive) of x, but as an instance in which the contrary is the case we have X 3? X^ In the most important applications (cf. Chapters II and III) /(x) is a function (either given explicitly or else determined implicitly as a solution of a linear differential or difference equation) capable of analytic continuation into the complex field, being in fact analytic throughout the finite plane with the exception of points (finite or infinite in number) situated upon a finite number of straight fines radiating from the origin and having the point x = «= as a non-polar singularity. For further criticisms upon the definition of asj'mptotic series sec Thom6, Journ. fiir Math., Vol. 24 (1904), pp. 152-156; Van Vleck, The Boston Colloquium Lectures (New York, Mac- millan, 1905), pp. 77-85; Watson, Philosophical Trans., Vol. 211A (1911), pp. 279-313. Asy:mptotic Series 23 and in case we had also fix) = 60 + br/x + b,/x^' + • • • + 6n-i/a:"-^ + ^" ^^^'^'^ ' ^'"^ ^"'^^) = ^ •C a;z=-|-oo we should have (ao - bo) + (ai - fei) ^ + (a2 - 62) -^ + • • • + («n-i - 6n-i) -;ii:i . an-bn-\r enjx) — €„'(a:) ^ Whence, Oo = &o, as results from the last equation by placing x = + x> . Making use of this relation in (54), multiplying both members by x and proceeding as before, we obtain ai =61, • • •, etc. The converse of the above statement is, however, not true as appears directly when we note that if f(x) is represented asymptotically by (50) so also is, for example, the function f(x) + e~^.^^ Again, it is desirable for the sake of clearness to note that asymptotic series in general cannot be used for purposes of computation in the sense in which Stirling's series can be used to compute log r(.T). In fact, no information is at hand respecting the error committed by stopping at any preassigned term.-^ There are, however, numerous and important asymptotic developments^'' which, like the series of Stirling, are derivable by use of the Maclaurin Sum-Formula and for such the limit of error may usually be fixed by means of the formulas then present for the remainder. But in all cases, the asymptotic development furnishes information as to the behavior of the function when x is very large. Thus, the expressions ao, Go + ai/x, Go + ai/x + ai/x^, • • •, ao + ai/x + ai/x'^ + • • • + cim/x"' constitute a series of successive approximations to the value of /(.r) provided that X is sufficiently large. Furthermore, we have lim/(a:) = Oo z=+co (55) lim^a;[/(.r) - Oo] = cti lim x^'ifix) — ao— ai/x — ailx^ — . .. — a„_i/.r"-i] = o„. Conversely, when the behavior of f{x) for large positive values of x is known, the equations (55) serve to determine the coefficients ao, ai, 02, • • • of the corre- sponding asymptotic development if one exists. 28 By adopting a more limited definition of asymptotic scries than that of PoiNCARfi, Watson has obtained a noteworthy theorem upon this question of uniqueness. See Philosophical Trmis., Vol. 211A (1911), p. 300. 29 For noteworthy exceptional cases, see Stieltjes, Annales de I'Ecole Normale, ^'ol. 13 (18SG), pp. 201-202. 3" This is true in general of the developments considered in Chapter II. 24 The Maclaurix Sum-Formula 14. The following consequences of the definition (51) are especially note- worthy:^^ 7/ then (a) ib) fix) ~ oo + fli/.'K + aa/.v" + • • • , (p{x) ~ 6o + bi/x + bojx- + • • • Oi ± 6l , 02 =t &2 , fix) ± x^. Whence, , . M .r. rdx d M.r)|^.x--5j^ -.= --,; x>x, so that lim r]n{x) = 0. 26 The Maclaukin Sum-Formula In distinction, however, to the properties of convergent power series, the term by term derivative of the asymptotic development of f(x) will not neces- sarily be the asymptotic development of /'(a-). This is most easily shown by an example. Thus, (57) /(:r) = e"- sin (e^) ~ + " + ^+ • * ''' but since /'(.^) = — e""" sin {e"") + cos (e""), the expression hm/'(.r) is oscillatory so that not only does the term by term derivative of the series (57) fail to repre- sent /'(a-) asymptotically, but/'(.T) permits of no such representation whatever. However, if fix) ~ «o + - + ;^ + • • • and if /'(a) is known to be developable asymptotically, then „,, , ai 2a2 3cf3 (58) •^^^■^'^-.^-'^-"^ • In fact, if /'(.r) were developable asymptotically in any other way than (58) it would follow from (d) of the above results that /(.r) was developabl ; asymp- totically in two different w^ays. 15. In addition to the properties (a), (b), (c) and (d) of § 14 we note also the following general result: ''Let fix) = ao + tvix) ; ivix) ~ ^ + J + * ' ' and let Fif) he a function of x through f which, tvhen written in the form Fiao + w), is developable as folloics: Fiao + w) = Fiao) + F'iao)w + ^^ w' -\- ■ - - ^""^^ .F'^'-^'M „_, , F(")(ao) -f- 6n(2tO „ (n — 1)! nl u,=o ias happens in particular when Fiao + ?y) is analytic at xo — 0). Then we may write f(/)~f(«.)+'>|+---+f-:+---, where p\, jh, • • • , Pn are the coefficients of the successive powers of l/x ohtai?ied by substituting into (59) ( exclusive of the term " w" 1 the first n terms of the given asymptotic development of wix)." 32 Cf. BuoMwicii, " Infinite Series " (London, 1908), p. 334. Asymptotic Series 27 In fact, from (b) of § 14 we may write F{ao) + F'{ao)w -| 2]^^' + • • ' H ^^l — '^ F{ao) + — + -:$+ • • •, and hence (59) may be written in the form Fiao + w) = F(ao) + - + 1^+ • • • H ^n ^ 'T^^ ' ^1°^ '^nOx') = 0. If we now write €n{w)io'^ in the form and observe that lim en{w)w'^x'^ =■ a;=oo we obtain the desired result. 16. We note in connection with the definition (51) that we have supposed x real and positive. More generally, f{x) is said to be represented asymptotically by the series (50) throughout an infinite region T (usually a sector with center at a: = 0) of the complex plane when, for all corresponding x values, the equation (51) exists in which lim is substituted for lim . In the case frequently pre- \x I =00 a-= + «) sented of a single-valued function f{x) having an essential singularity at the point X = 00 , we note that the above mentioned region cannot completely sur- round the point a; = 00 , since we should then have lim f{x) = ao for all methods |x|=oo of increase of \x\, thus contradicting the hypothesis that the point a; = 00 is essentially singular. Again, if f{x) and the region T be given, we observe that the necessary and sufficient condition that f{x) be developable asymptotically throughout T is that there exist a set of constants oo, ai, CI2, • • •, On, • • • satisfjdng relations (55), it being understood that the values of x appearing in these relations are confined to T. In fact, if (55) exist we have (51) and conversely. The same relations (55), when employed as a sufficient test for the existence of an asymptotic de- velopment for f{x) throughout T, are usually difficult to apply and hence of little value in practice, since f{x) is not in general so given that it is possible to determine whether the indicated limits (representing ao, ai, a2, •••) exist. A sufficient test which has a wider field of applicability is supplied by the fol- lowing Theorem V.'^^ Let fix) he a function of the complex variable x analytic within and upon the boundary of a certain infinite region T of the x plane, the point .r = 00 , hoivever, being excluded. Also, let (p{x) = f{l/x) and let T' be the region {having 33 Cf. Ford, Bulletin Soc. Math, de France, Vol. 39 (1911), p. 348. Line 13 should here read " le point a; = 00 toutefois etant exclu." 28 The IMaclaurin Sum-Formula the point .r = u'pon its boundary) obtained from T by means of the transformation X = l/.r'. //, then, for values of x in T' the foil oiving limits exist: lim ^(.r), lim ^'(x-), lim ^"(.r), •••, lim ^^"^(a;), and are represented respectively by (p{0), ^'(0), • • •, (p''''\0), • • - {these values being assumed independent of the direction of approach of x to in T') we may write for values of x in T fix) - flo + ai(l/.r) + aoil/xY- + • • • + a.(lAr)- + • • • where cik = ^., ; /t = 0, 1, 2, 3, • • •, ?i, • • • . In order to prove this Theorem we shall begin by establishing the following Lemma in the general theory of functions: Lemma I. " Let (p{x) be a function of the complex variable x analytic within and upon the boundary of a certain region T' of the a:-plane, exception being made, however, of the point x = situated upon the boundary at which point ^"^(0), we obtain <^"(0) , , , <^("-"(0) ,.n-l where (62) r„(.T) = [{^—^y cp'^+'Kt)dt. In order to complete the proof of the Lemma it thus remains but to show that with r„(.T) defined as in (62) we shall have lim r„(.T) = provided always that X remain in T'. Now, for all values of t on the line of integration in (60) we have X — t X 1. Moreover, it follows from our hypotheses that we may find a positive constant M (independent of x) such that for all values of x in T' we may write | ^'^"+^^ (.r) | < M. Whence, if we place \x\= p we shall have for the given value of x |r„(.T)|< M jdp = Mp from which the desired result becomes evident. Theorem I follows as an immediate consequence of the Lemma upon sub- jecting the function /(.t) and the region T mentioned in the theorem to the trans- formation X = 1/x'. We note also that if, instead of having /(.r) defined throughout a complex region T, it is given as a function of a real variable x within the infinite interval (a, + °o ), we may obtain in like manner the following Lemma and corresponding Theorem : Lemma II. " Let (p{x) be a function of the real variable x which, together with its first n -\- 1 derivatives, is continuous within the interval (0, h), the end point X = being excluded. If, then, the limits ^(+ 0), ^'(+ 0), (^"(+ 0), • • •, (p("+^^(+ 0) exist, we may write for values of x in (0, 6) (+0), we may write for values of x in {a, -\r oo ) /(.)~a„+a.(l)+«.(y'+---+a,(^^)"+..., where Qi. aj (I2, \, ' ' ' dn, k, ' ' ' such that for values of 2 in A w^e have r/ N ^ / ^ I / N r , ai. A , ^2, A , , ttn, A + ^A.nCg) 1 F(Z) =/a(2) + <^a(2)[«0.A+-^H-^H 1 -n J; lim WA.nCz) = 0. |2|=00 Then, according to the definition of § 13 and the remarks of § 17, we may write for the indicated values of z F(z) ~ A(z) + «2.a, • • • (assuming that they exist) is usually one of considerable difficulty and, when regarded in a general sense, is one for which but fragmentary results exist at the present time. The known determinations appear to be either those for special functions of importance in Mathematical Physics, such as Bessel's function J„(2)/ or for certain types ot integral functions, notably those defined by infinite products." In the present Chapter it is proposed to show how the general theorems of Chapter I may be used, at least in certain cases, to make the above indicated determinations. In doing this we shall merely consider certain special functions F(z). No attempt will be made to obtain theorems of great generality, partly because of the difficulty of such an undertaking, but chiefly because of the bcliet that a few well-chosen illustrations suffice to adequately impart the spirit and possibilities ot the method employed. In each of the functions F(x) considered, 'See for example Lommel, "Studien iiber die Bcsscl'schcn Functionen " (1868), § 17. '^ See for exumplc Barnes, Philosophical Transactions, Vol. 199A (1902), pp. 411-500; ibid., Vol. 206A (1906), pp. 249-297. Each of these memoks contains an extended bibliography of the subject. See also Mattson, "Contributions h la Th^orie des Fonctions entiSres " (Thdse), Upsala, 1905. 31 32 Determination of Asymptotic Developments only the functions /a (2), ^^(2) and the first one of the constants a„,x which is not equal to zero are determined, since these three determinations constitute what is essential to the study of the behavior of F{z) for large values of |2;|. The method, however, permits equally of the determination of any one of the coefficients fl„, a- The functions F{z) considered fall into two classes: (a) those defined by infinite products and {b) those defined by infinite series. Under (a) we have eventually considered (§§ 24-28) the asymptotic behavior of the general integral function of order > — a problem to which considerable attention has been devoted in recent years^ and in connection with which we have entered into considerable detail owing to the importance of this and other analogous con- siderations in the general theory of functions. Under (6) we have eventually considered (§§28, 29) the asymptotic behavior of functions defined by power (]\Iaclaurin) series — a subject of evident importance owing to the essential role of such series in anahsis. The treatment for the latter is brief and indeed but fragmentary, yet it is believed that the most important known results (aside from those which concern the solutions of linear differential or linear difference equations)^ have been indicated. The determination of the asymptotic character of functions defined in other ways than as infinite products or infinite series might well have been considered also in the present chapter, as likewise the corresponding problem for certain noteworthy special functions.^ We have, however, limited ourselves in the manner indicated above, feeling that not all aspects of the subject could receive treatment within the limits of the chapter while those of the greatest permanence in the general theory of functions have been included, we believe, through the present selection. 19. Example 1. To obtain asymptotic developments for the function °° 1 ^^^ ^^'^=S(2M=T)H^^' We here choose a function which, as a result of the well-known formula^ tan 2; Y^ 1 ^' "="(2^+1)2^-22 * See note at the bottom of page 44. * See Chapter III. * For miscellaneous investigations of this description, see Barnes, Edinburgh Trans., Vol. 19 (1904), pp. 426-439; Proceedings London Math. Soc, Vol. 3 (1905), pp. 273-295; ibid., Vol. 5 (1907), pp. 59-116; Transactions Cambridge Philosophical Soc, Vol. 20 (1907), pp. 253- 279; Quarterly Journ. of Math., Vol. 38 (1907), pp. 116-140; Hardy, Quarterly Journ. of Math., Vol. 37 (1906), pp. 369-378; Littlewood, Transactions Cambridge Philosophical Soc, Vol. 20 (1907), pp. 323-370. ^ See, for example, Tannery's " Introduction h la Thdorie des fonctions d'une variable " (Paris, 1886), § 117. Special Functions 33 may be evaluated in the form TT e''^ — 1 (2) n^-i.7^x- and this fact will enable us to check our subsequent results. In order to obtain the asymptotic developments of F{z) as defined by (1), let us place ^'^'^^ ^ {2io + 1)2 + z" and regard z as having any fixed value z — j) -]r iq, i = V— 1, lying in a sector (center at z = 0) situated in the right half of the z complex plane and having neither of its bounding lines coincident with the axis of pure imaginaries. Then fz{w), considered as a function of the complex variable w = x + iy, satisfies the conditions demanded by Theorem IV {a = 0) of Chapter I, except that in case I g I > 1 the same function will present a single pole of the first order at the right of the pure imaginary axis, this pole being situated at the point iv = l{— 1 — iz) if g > 1 and at the point iv = |(— 1 + iz) if g < — 1. Thus we may apply the theorem, subject to the remarks of § 11, in order to obtain an expression for the sum x-l H fz(x); x>p. x=0 We shall now distinguish between the following four cases: (a) |g|< 1, (b) q>l,{c)q< -lAd)q= ± 1. In (ft) we may make direct application of the theorem. Taking m = 0, we thus obtain ^^^ £ {2x + 1)2 + z^ ^^'""^X (2a: + 1)^ + z^ ~ ^ ' {2x + 1)^ + z^ + "^^•''^' where (4) U^{x) = - * J g2^y _ I — ■ dy and (5) c. = 2(rT^) ~ "^^^^• In these results let us now allow x to increase indefinitely, observing that r dx If 2a: + 1 1^=" tt 1 1 f /o I i\2 I — 2 = TT ^rc tan = -. — arc tan - Jo (2.r + If -\- z^ 2z\_ z J^=o 43 2z z and that lim Q.z{x) = 0. We obtain a;=<» 4 34 Determination of Asymptotic Developments r^. . -^ , 1 1 1 4z 2(1 + Z-) 2z z (6) p r 1 1 1 dy + Vo L (1 + 2iyr- +z' (1 - 2i»2 + 2^ J e'^y - 1 * Upon developing the various terms of the second member in ascending powers of 1/z, we thus reach (Theorem V, Chapter I) the relation in which the coefficients 02, oa, cig, • • ■ may be evaluated to any desired point.'' In case (6), equation (3) and hence (6) also, will continue to hold true ac- cording to § 11 provided that we subtract from its second member the residue of the function /o^ M ""^^ [{2w + 1)' + zV'"" - 1] at the point w = — ^{l -\- iz) which (residue) is readily found (cf. (30), Chapter I) to be 7r/22(e"^ + !)• Since, for values of z within the proposed sector, this function is developable asymptotically in the form (50) of Chapter I with flo = «i =02= • • • =0, it follows that relation (7) holds true also in case (6). Similarly in case (c) we have equation (6) except (cf. § 11) that we must now subtract from its second member the residue of (8) at iv = ^{1 — iz) and also that of the function 2Trifz{iv) at the same point; i. e., we must subtract the ex- pression 2z(e-''' +1) ' 2z 2z(e'"' + 1) " Thus, as in case (c) we see that relation (7) again holds true. Moreover, the same relation continues in case (d) as appears by writing F{z) in the form 1 CO 1 1 + 22 ' „^ (2r^ + 1)2 + 22 and applying the method of case (a) to the summation here appearing; also recalling that in one and the same region there can exist but one asymptotic development for a given function. Similarly, if we note the effect in (4) of supposing the real part of z to be negative, we find that when z is situated in a sector lying within the left half of ^ It may be noted that by using a sufficiently large value of m in applying Theorem IV (Chap. I) we may obtain any one of these coefficients in a relatively simple form involving the Bernoulli numbers. Special Functions 35 the plane, relation (G) continues to exist provided that the term rj-iz be replaced by — 7r/4z. Thus in summary we may say that throughout any sector (vertex at z = 0) of the z plane tvhich does not contain portions of the pure imaginary axis, the function F(z) defined by (1) may be developed asymptotically in the form wherein the upper or lower sign is to be taken according as we are dealing with a. sector in which the real part of z is positive or negative. This result, which is at once seen to be consistent with the known relation (2), illustrates in simple manner the way in which asymptotic developments for a given function may be ascertained, at least in some cases, by means of the general theorems of Chapter I. This will be further illustrated in what follows> wherein we shall eventually consider cases of much greater generality.^ 20. In § 19 we have considered asymptotic developments of F{z) (cf. (1)) which are valid in sectors situated in the right or left halves of the z complex plane. We proceed to show how the same method may yield analogous developments holding for the upper and lower halves of the plane, exception being made natur- ally of those (pure imaginary) points corresponding to the values z = ± (2?i + l)i; n = 0, 1, 2, . • • at which F{z) becomes infinite. For this let us consider the function *(.) = F(i.) = t (2„ + \y. _ ^. . Regarding z at first as real, we place 1 {2W + 1)2 - z2 and again undertake to apply Theorem IV with m = 0. This can be done only in case ^z(w) is analytic in w throughout the right half of the lo plane. How- 8 In the special instance before us it may be shown that ao = 04 = ae = • • • = 0. In fact if we substitute in (7) the form for F{z) given by (2) we obtain 42Le'^^ + l J 2z Ll + e^'^^J z^ ^ z^ ^ where the upper or lower sign is to be taken according as the real part of z is positive or negative, and this relation is seen to be true when 02 = 04 = oe = • • • = 0. It is to be noted, however, that in general if a function is defined by a series of the type of (1) (cf. (12)) no formula analogous to (2) is at hand. The indicated method for determining the asymptotic development of the function, however, remains the same, thus leading to cocfTicicnts oo, Oi, 02, •••, which arc in general not all equal to zero. 36 Determination of Asymptotic Developments ever, we are concerned with large values of 1 2 1 , and whenever 1 2 1 > 1 it is evident that ipz{w) will have a pole of the first order within the indicated region at the point 10 = {z — l)/2 or w = — (z + l)/2 according as z is positive or negative. Let us first consider that z is 'positive. We proceed to apply the theorem, subject to the remarks of § 11. Since the residues r^, r^' of the functions at w = /3 = §(s — 1) are respectively ■K TT 2iz' 2i2(e""+ 1) so that ' P 2 * ^ Uze""^' + 1 we may write (at least when x > \{z — 1)) t^, {2x +1)2-22 ^''^ 4:iz e"'' + 1 (9) / ru^-l)-^ r-- \ dx 11 + e2 V Jo + 4._.He ) (2."+D^^2 - 2 (2.+ 1)2-.^ + "^^•^> in which Cz and l^zCr) are obtained by changing z^ to — s- in (4) and (5). But from elementary considerations, the third term in (9) reduces to i 1 (2+l)( 2a: +l-z) 4z ^^ (z- l)(2x+ 1 + 2)' Whence, upon allowing x to increase indefinitely we obtain (z > 1) tbr ^ _ ^ g""- 1 , 1 I 1 , z+ 1 ^V2j 4^.^ g- r. + 1 -^- 2(1 _ ^2) -+- 42 ^Og ^ _ ^ (10) _i rr 1 1 1 ^y ijo L (1 + 2^2/)2 - 2^ {l-2iyy- z'^je^"^ - 1 and hence (Theorem V, Chapter I) ^^^) ^(^)^4r2.'^-+^l + 2-^ + 2-+'--> where 62, &4, ' • • are determinate constants. This result may now be generalized to all values of z belonging to a sector S (center at z = 0) lying in the right half of the z plane, exception being made, as already indicated, of the points z = 2n + 1; w = 0, 1, 2, • • •. In fact, we have but to suppose \z\> 1 to have in (10) two expressions equal for positive values Special Functions 37 of z and each analytic throughout S and hence equal for all values of z in the same region.^ Moreover^ the last term in the second member (like the two preceding) is readily seen to be developable in ascending powers of I/2-, thus leading to a series which, in the sense of § 13, represents the same term asymp- totically for all values of z in S. Likewise, the same relation (11) is found to hold true for a corresponding sector in the left half of the plane, exception being made of the points z= - (2m + 1); n= 0, 1, 2, ••• so that, having replaced z by — iz, we may say in summary that throughout any sector {vertex at z = ()) of the z plane ivhich does not contain portions of the real axis, the function F{z) defined by (1) may be developed asymptotically in the form This result is again seen to be consistent with the known relation (2).^° 21. Generalization of Example 1. The method above illustrated for deter- mining asymptotic developments is in general applicable to functions F{z) defined by series of the form ^ ^ ,froX(n) + piz) where 2^(2) is an integral function of z and where \{n), ij.{n) are functions of n such that Theorem IV, subject to the remarks of § 11, may be applied to the expression ^'^''^ Mw) + p{z) in order to find for a given value of z the sum Z/.(.T). We observe in particular that by taking 2^(2;) = z'^ (q = integer ^ 1) the expression F{z) (or the sum of a number of such expressions) comes to include a wide variety of functions having radial clusters of polar singularities in the neighborhood of the point z = co — a characteristic common to many of the more important functions of analysis. In cases where fz{w) cannot be considered as a function of the complex ' It may be remarked that the last term in the second member of (10) is analytic throughout S {\z\ > 1) because the improper integral involved converges uniformly for values of z in any sub-region S' of S whose boundary does not touch the boundary of .S'. (Cf. Oscood, "Encyklo- padie der math. Wiss.," II, 2, § 6.) *" In view of the same relation it appears from (11) that in the present simple case we have bi = bi = b^ = • • • =0 and that the symbol ~ may be changed to =. Cf. note 8, p. 35. 38 Determination of Asymptotic Developments variable w = x -\- iy but is continuous in the real variable x we may frequently determine the desired developments by use of Theorems I, II or III of Chapter I (subject possibly to the remarks of § 10). The manner in which Theorem I may be thus used will be shown in the following example wherein an important type of function F{z) different from that of § 19 is taken. 22. Example 2. To obtain asymptotic developments for the function (12) /■(.) = n[i+^:]. As in example 1, this function may be evaluated beforehand and takes the form „ — T2 (13) ^(^) = S^" thus furnishing a check upon our subsequent results. We begin by writing (14) log F{z) = Elog [l + -H = Hm r Elog (x' + 2^) - 2 Elog a:] . n^l L ^* J x=ooL:>:=l x=l J From § 12 we have x-l - 2 Zlog a: - - 2 log r(a;) = - log 2ir - 2{x - ^) log x (15) -1 + 2x + coi(.r); hm a;i(a:) = 0. 2;=-)- 00 We proceed to apply Theorem I (Chap. I) with m = \ to the first summation in the last member of (14), taking for this purpose f{x) = log (x^ + 2-) and supposing for the present that z is real but different from zero. The theorem may be applied since the series (37) (Chap. I) becomes fl.Or) = E r I J72l0g O^-' + -') 1 0) (17) log F{z) = - log 2x2+ 7r2 - § log (l + -2 j + 2 (^1 - z arc tan- j - 12.(1). On the other hand, if z is negative we obtain log F{z) = - log (- 2tz) - ttz - I log (^ 1 + ^ j (18) . 1 X + 2 I 1 - z arc tan- 1 - 12.(1). We now observe that the expression 12.(1) is a function of z which is single valued and analytic in any region whose boundary does not cross the axis of pure imaginaries. Whence, within any region Ai situated in the right half of the z plane, equation (17) may be used, while similar remarks apply to equation (18) for values of z pertaining to any region A2 in the left half of the plane. More- over, if the boundaries of Ai and A2 are not tangent to the pure imaginary axis at 00, the function 12.(1) vanishes Hke l/z^ when |z|= ^ in Ai (or Ao) and is developable asymptotically by Theorem V, Chapter I, in powers of l/z^ within this region. It therefore remains but to apply the result stated in § 15 in order to say that throughout any sector {vertex at z = 0) of the z plane ichich does not contain portions of the pure imaginary axis, the function F(z) defined by (12) may he developed asymptotically in the form wherein the upper or lower sign is to he taken according as we have a sector in which the real part of z is positive or negative. 40 Determination of Asymptotic Developments This result is at once seen to be consistent wnth the known relation (13).^- 23. We proceed to show how asymptotic developments for the F{z) of § 22 may be obtained which will be valid in sectors that may include the pure imagi- nary axis. For convenience we shall convert this problem into the following: " To determine asymptotic developments for the function (19) F{iz) = $(2) = n[^-:^]' which shall hold good throughout certain sectors that include the real z axis." Considering at first that z has a fixed, positive, non-integral value > 1, we proceed (cf. (14)) to study the expression (20) H(z) = lim r Zlog (x'- - 2^) - 2 Elog a:] , x=xi L ^=1 •'■=1 J in which we agree to write log {x- — z-) = log (s- — x-) + iri whenever x < z. Then e'"^'^ = ^{z). In order to obtain a form analogous to (16) for the first sum here appearing, let us place This result is seen to be consistent with (13). 24. We proceed to the following more general problem: Example 3. Given or p = integer ^ 1 ^^ p ^ P < p+ 1 according as < p < I or p ^ I. To determine asymptotic developments for F(z). " We adopt the familiar notation exp x for e*. 44 Determination of Asymptotic Developments This problem, in view of the important role which the F(z) thus defined plays in the modern theory of integral functions, has already received considerable attention.^^ Our purpose here will be to deduce through a uniform method based on the fundamental theorems of Chapter I the known results together w4th others of a supplementary character.^^ ^Ye shall suppose at first that p is non-integral and > 1 (p < p < 2' + 1, p = integer ^1). Also, for the present z is to be regarded as having any fixed value (real or complex) except one of the following: 2^'^, 3^'", 4}'^, • -. The method then requires that we take for consideration the expression (cf. (20)) Elog (a:"- z) -o-Zlog.T + EE-( -^ I (^ = Ijp x=\ x=\ x=l v=l '^ X'*' / J in which the value to be assigned to log {x" — z) may for the present be taken in any manner consistent with the equation exp log (x'^ — z) = x" — z. Then exp H{z) = F{z) where F{z) is defined as above. We proceed to study the behavior of the first term appearing in brackets in (27) when x is large. Following the method of § 23, let us place f,{w) = log {W^ - 2), where ic" is understood to be so defined as to be real when w is real and positive and where the logarithmic function is understood to be rendered single valued in w throughout the right half of the u'-plane by means of a rectilinear cut ex- tending from the point w = 2" vertically downward to the point w = co , the value of z<' being determined in accordance with the following conventions: if 2 = r(cos (p-\- i sin (p) then z^ = r''(cos pep + i sin p(p) subject to the relation — 2ir < (p ^ 0. The function fz{iv) having been thus defined and defined uniquely for every value of w whose real part is positive, let us now impose for the present the additional condition upon z; viz., real part of z'' > 1; i. e., r" cos p(p > 1. Next, let us consider the complex integral X ^'^''^dw; 1, necessarily includes the point iv = z'' within its interior, it being here understood, as in Fig. 2, that the point I is the Fig. 3 one on the left side of the cut corresponding to A and that the closed loops BC and GH include respectively the points w = 2, 3, 4:, - - • , q and w = q-\- \, g + 2, • • •, {x — 1) where q is the integer for which q < real part z'' < q -\- V^; also that the curve DEF forms a circle of arbitrarily small radius ^ surrounding the point w = z''. Corresponding to relation (25) of § 23 we thus obtain x—l /• E log {X'' - 2) = I log (1 - Z) - fi,(l) + log (W'^ - Z)dw - \ log (.1- - 2) + 12.(.t) - {^^^ dw where M indicates an integration over the path IGFEDBx,^' where L indicates an integration over the path ADEFI in which, however, the points A, I are now supposed to be taken at an infinite distance along the cut, and where ^zix) is given by the formula 1^ In case real pari zp = q = an integer, the indicated loop HG, instead of containing w = q in its interior, will have this point upon its boundary. To obviate the difficulty thus arising, let it be understood in this case that the cut does not extend vertically do\\'nward from the point w = ZP but first extends an arbitrarily small distance to the right of this point and then vertically downward as before. The reasoning which follows will then apply. 1' In case zp is real and > 1 the path M becomes the curve, in part rectilinear and in part semicircular, \FECx of Fig. 2; while if imag. part zp < the path M may be taken as the straight line Ix (Fig. 3). 46 Determination of Asymptotic Developments or (29) ^^(-)=-^i log[^ ^_.^^._J ^,^, it being understood that the integrand of (29) is so defined as to be equal for all values of x and y to the integrand of (28). Now, an integration once by parts shows that /r dw log {w" — z)dw = IV log {w" - z) - aw - <^z J ^^ _ ^ ' Whence, r C dw . ^^ . , log (26"'' - z)dw = X log (.1-'" - z) - ax - az _ - log (1 - 2) + a. We have now but to recall the formula (15) to see that the first two terms in the square bracket of (27) combine into the following: Z log iw" - 2) - 0- Z log .T = -7> log 2tv -\ log (1 - 2) 1=1 x = l - (30) - fi^(l) + a;(.r) + a + (.r - §) log ( 1 - ^;^) -x,^.+«.(^)-j; ^(w) We turn next to consider the third term appearing in square brackets in (27). By use of the well-known relation^^ 11 , 1 , n^~' , « / s I r^d^ V(irt p> 0, !-» = 1 + 21 + 3-,+ •■■ + (;^3T)-. + (-^+'' 1. We now proceed to study in further detail the expression B{z) and for this it is desirable to remove for the present all restrictions as regards 2, thus enabling us to determine certain functional properties of the same expression. We turn first to the expression ^.(l) which appears in B{z) and which by reference to (29) is seen to be defined by the relation (40) ^,,, • ^^ [ i^ + wr-z l dy For a given value, real or complex, of z this 12.(1) evidently has a meaning unless z be such that the equation (1 ± iyY = z has a real solution in y. In order to determine the values of z for which this happens, let us place z = r(cos (p -\- i sin (p) and make the conventions already indicated as to the meaning of z''. For the exceptional values in question we must then have 1 ± iy = r''(cos pep zt i sin pep) so that the same values are those lying on the locus of the equation r*" cos pep = 1; — 2t < ep ^ 0. Whence, if r be large the same values will tend to have an argument of the form — {2n + l)7r/2p wherein 7i is a positive integer for which the same argument lies between — 27r and 0. Again, if the locus just mentioned be drawn, the z plane is thereby divided into portions in each of which 12.(1) is a single valued, analytic function of z, since within any sub-region T' lying wholly within such portion the convergence of the integral in (40) is readily seen to be uniform. Moreover, if arg z has any value other than one of the exceptional type just mentioned, we may write lim 12.(1) = 0. In fact, upon reference to Z I =00 Theorem V, Chapter I, it appears that under such hypotheses we shall have where the coefficients oi, 02, • • • may be obtained by expanding the integrand of (40) in ascending powers of 1/z and integrating term by term. General Integral Function 49 Secondly, we turn to the expression S{z) defined by (33). Since w" -z ^ ^ivF' w^^iw" - z) ' we may write r dw f. z" ^^ r dw 2)' Whence, recalUng that M extends from w = 1 to w = x, we obtain the following relation : where N represents the path obtained from M by supposing a; = + oo . In the consideration of this expression we have thus far considered that real part z'' > 1 and from what has already been noted it follows that if we have also imag part s^ < we may replace (42) by (43) Siz) r^ z" ^^^ r dx 1 ''lhl-(Tv ^' X x'-^ix'^-z)] The form of (42) may also be simplified when i7nag part s*" > {real part z*" > 1). In fact, we may then write (44) r ^j^ = r ^ r ^^ where the last symbol represents an integration in the positive sense about the circle. Moreover, the last term of (44) is readily evaluated and found to be equal to 2Tripz''-P-\ Thus, when imag part z" > (real part > 1) we may write (45) sw-[|:j^„-.-f,-.,(^]+2.,v. These facts being premised, let us consider the properties of the right member of (43), all assumptions as regards 2 being laid aside for the moment. Evidently, the expression in question represents a function of z which is single valued and analytic in any region T which does not cut the portion of the real z axis extending from z=lto2:= + oo. Moreover, when 1 2 1 < 1 we find upon expanding in ascending powers of z that the same expression is developable in the form (40) is such that (o) when considered as a function g{ic) of the complex variable w = X -{■ iy it is single valued and analytic throughout all portions of the w plane lying to the right of (or upon) the vertical line lo = a — ^ -\- iy except for a finite number of poles situated at the points lo = Xi, X2, • • • , X*, • • • , Xn ; X( =f in- teger ^ a-° and (6) is such that to an arbitrarily small positive quantity e there corresponds a positive constant K (independent of x and y) such that g{x ± iy) 9(.x) < K exp ey for all values of .t ^ a — ^ and for all positive values of y sufficiently large, then the function f(z) defined by (47) when 1 2; | < r may be extended analytically throughout the whole z plane with the exception of the positive half of the real axis, and throughout this region will be defined by the equation (48) /(.) = -^J_^ ^;^- dy - Z r, in which, if we place z = r(cos (p -\- i sin (p) it is supposed that we write (_ 2)a-.i-f. = exp [(a - i + iy) log (- z)] = exp [(a — ^ + iy)(\og r -{- i(p -\- irr)] and take — 27r < ^ < and in which r< represents the residue of the function (49) ^g(^)(- ^)'^ sin TTW at the pole w = X<." For the proof of this lemma let us at first suppose for simplicity that g{w) has no poles at the right of (or upon) the vertical line w = a — \ -\- iy and let us regard z for the present as having any fixed value. The lemma then results from a consideration of the result obtained by integrating the function (49) " Cf . LiNDELOF, "Calcul des residus" (Paris, 1905), p. 109; also, Ford, Journ. de Math., Vol. 9 (5) (1903), p. 223; also. Bulletin of Amer. Math. Soc, Vol. 16 (2) (1910), p. 507. 2" This condition is fulfilled from the fact that g{n) has a meaning when 71 = a, a + 1, a + 2, • • • . Otherwise the given series (47) would lose significance. 2' This condition is satisfied in particular if constants Ci > and C2 ^ Ci exist such that Ci < \g{u-)\ a and where j is any positive quantity, arbitrarily large. Upon applying (30) of Chapter I to the result of such an integration, we arrive in the first place at the relation f'a^ V ^ ^ n 1 r g(^)(-z)" ', (oO) 2^ 5'(w)z" = W-- I ■ dw. Supposing at first that z is real and negative, we proceed to study the integral here appearing in further detail. First, along the side of Cn upon which w = x + ij we have dio = dx and sin TTiv = sin 7r(.T + ij) = sinh 7r;(sin irx coth wj + i cos ttx) so that if we call the contribution from the side in question I, we may write J = (- ^y' r~'- g{x + ij){- zY ^^ r 2i sinh irj .'tn+s, sin wx coth irj + i cos irx Whence lim 7=0 provided that j=ao (51) lim e-'^gix + ij) = 0; x ^ a - ^. Similarly, we find the same result for the contribution arising from the side of Cn upon which w = x — ij provided, however, that (52) lim e'^^gix — ij) = 0; x ^ a — I. We observe that both conditions (51) and (52) are satisfied in the present case as a result of (6) of our hypotheses. Next, let us consider the side of C„ upon which w = | + 2/i + iy. Here w^e have div = idy, sin iriv = cos iiry = cosh wy so that having taken i = <» , the contribution in question becomes /=^- z)i+^ n g(^ + 2n+iy){-z)'" _^ 2 Xoo COshTTW ^' and it follows from (6) of our hypotheses that the improper integral here appearing has a meaning (z real and negative). Moreover, it follows likewise that if l^l < r we shall have lim J = 0. n=oo Whence, if we now take account of the contribution arising from the remaining side w = a — ^ -{- iy of Cn, noting that we here have sin ttw = (— 1)""^ cosh iry while the integration takes place from ?/ = + °° to y = — oo , we may write (53) !:,(„),. = (=i)! r .(a-» + «>)(-.r-'^-- ^ ^ ^^^a"^ 2 J_«, coshx?/ 52 Determination of Asymptotic Developments This relation must hold good as we have indicated, for all values of z which are real and negative and such that |z| < r. But the first member represents a function of the complex variable z which is single valued and analytic throughout the circle of convergence of (47) while the second member, with the conventions introduced in the lemma as regards the meaning of (— 2;)°"^+'^ represents a function of z which is single valued and analytic throughout the whole z plane except for the positive half of the real axis. In fact, for all values of z in a region T which does not cut or touch the positive half of the real axis we shall have from the indicated conventions — t < (p < w so that upon introducing (6) of the hypotheses it appears that we may choose e so small that the improper integral in (53) will converge uniformly for all values of z in T. Whence, the same integral will have the analytic properties just indicated, and we reach in summary the lemma for the case in which g{w) has no poles to the right of the line 10 = a — ^ + iy. That the lemma holds true in the more general case follows at once upon noting that relation (50) then continues {n sufficiently large) provided we add to its first member the expression TO Er,. Returning to the second member of (43) which is defined when 1 2 1 < 1 by the series (46), let us now apply the above lemma to the latter series, taking for this purpose g(iv) = l/(p — iv). Since the residue of the function 7r(- zr (54) (p — w) sin irp at the pole iv = p is — 7r(— 2) ''/sin Trp, it thus appears that for all values of z except those real and positive we may write the expression in question in the form sm Trp where «(p + ^-^y)cosh7r2/^^' it being here understood that the expressions {— zY and (— 2)""=+''^ are to be interpreted in accordance with the conventions stated in the lemma, i. e., if 2 = r(cos (p -\- i sin (p) with — 2ir < (p '^ 0, then (— 2)" = exp p(log r + i(p + iir) = r''[cos p(

1, imag part z'' < we shall have (55) S{z) = '^^^^+R{z), ^ ^ sin 7rp while throughout any similar region T2 in which real part z'' > 1, imag part z" > we shall have S(z) = ''':~^^' + 2Tipz' + R{z). sm 7rp Hence, according as z lies in Ti or T2 the expression B{z) defined by (39) takes the form Biz) = exp C(z) or B(z) = exp [C(z) + 27rip2;''], where c(z) = i;r(^)5+l^'-^iog(i-^)-o.(i) + ij(.). We note also that since R{z) is equal to l/27ri multiplied by the result of integrating the expression (54) from ?/= — 00 to y = -\- co along the ^ine 2V = — ^ + iy it follows that we may replace R{z) by a similar expression R{z) in which the path of integration is w = — k — ^ -\- iy (k = arbitrarily large positive integer) provided this R{z) be increased by the sum of the residues of the function (54) at the poles iv = — 1, — 2, - ■ ■ , — k. Moreover, since these residues form the first k terms of a series of the form (56) ffo + — + -| + • • • ; fli, 02, • • • constants as regards z z Z" while lim z'^'Riz) = it follows that the original expression Riz) is developable I 2 , = <» asymptotically in the form (56) (arg z 4= 0). It follows, then, upon reference to (38) and to the properties which we have now estabhshed for 12^(1) and R{z) that we shall have the following relation in which the upper or lower of the double sign ± is to be taken according as z is confined to T2 or Ti: Upon observing that when imag part 2" > the function exp iriz'' is develop- able asymptotically in the form (56) with ctq = 0, oi = 0, • • •, while the same is true of the function exp — iriz" when imag part z'' < 0, it appears that the above relation may be simplified into the following holding good for values of z in regions of either type Ti or T2 : ^, , 2sinxz'' [i-J^\-'', i-zYl "4 Determination of Asymptotic Developments This relation, as we have noted, holds true only when real part z'' > I. We now proceed to determine an analogous relation for any region Tz in which real part z'' < 1. If this assumption be made at the beginning, the cut in the iv plane falls entirely outside the rectangle bfgk so that we at once obtain (32) except that the last term of the second member is lacking. Moreover, the expression S(z) takes the form (55) so that, upon writing log (1 — 2) = log (— 2)+log (1— (l/z)), we have H{z) = - ^ log 27r - ^ log (- 2)" - i log (l - i ) - fi.(l) (59) t^l \pj P Sm TTp and hence 1 rpfy\z' 7r(- 2)M (60) F{z) ~ ,^ , exp E r - -+ • • Before summarizing the preceding results into a theorem it is desirable to note certain corresponding results which may be obtained when z'' is confined to the real domain (!, + «») — a case not included in the above discussion. If this assumption be made at the beginning the corresponding Fig. 3 becomes that represented in Fig. 2, except that the cut extends from the point to = z'' instead of from the point w = z. Thus we obtain equation (32) as before with S{z) defined by (33) in which, however, the path M is now understood to be IFECx of Fig. 2. We arrive, therefore, at (38) in which A{z) is defined as before, while B{z) is defined by (39) with S{z) given by (42) wherein N repre- sents the path IFEC + <». In this form S{z) is now developable (as was (43)) when 1 2 1 < 1 into the form (46) from which we find as before that unless (p = arg z = the expression S{z) is given by (55). In other words, S{z) will be given by (55) when

~2^- p P P P P Moreover, when cp = 0, S{z) preserves a meaning as appears from (42), provided that z =1= 1, so that the same expression may be obtained from (55) when z == r is large by placing therein z = r(cos v? + i sin 0): with the assumption that p is such that p < p < p -\r I. If, having placed z = r(cos <^ + ^ sin (p) we agree that z'' and (— zY shall be defined respectively by the equations gP = r''(cos p

/2^ L .t^ \P/^ SlUTTpJ In the following figure the sectorial regions indicated, I and II, represent those in which for large values of |s| the first or second of the above forms holds good respectively, while the dotted lines represent the special directions along which the third form ai)plies. It is to be understood that the last radial 56 Determination of Asymptotic Developments line drawn is that upon which (p = — 97r/2p, but that a complete figure would contain all similar lines upon which tp = 2k -\- 1 2p tt; h = 0, 1, 2, and (p "> — 2t, the scheme of alternate division of the plane into sectors of types I and II being carried forward up to and including the last sector thus obtained. Fig. 4 Upon noting that for values of z which are real and positive {z — r) we have , x(— z)'' . , cos pir -{- i sin pir — TTiz'' + -:77 = — TTir'' + irr'' z-zr~r = Trr'' cot px, sm Trp sin pT it appears that the above theorem is consistent with certain results of Hardy to be found in the Quarterly Journal of Mathematics, Vol. 37 (1905), page 158 (later corrected on page 373). For values of z for which arg z = (p ^ the theorem is not altogether consistent with the results of Barnes in the Philosophical Transactions, Vol. 199vl (1902), page 470, since an equivalent to the first of the forms above is there assigned to F{z) for all values of z such that ^ 4= ( | s 1 sufficiently large). It is to be observed that both Barnes and Hardy take for discussion the function F(— z) instead of the F(z) employed above. 25. In the discussion of the function F(z) of § 24 we have thus far supposed p < p < p + 1 where p is any integer ^ 1. The corresponding results for cases in which < p < 1 may now be readily supplied, it being understood that F{z) assumes the first of the forms given at the beginning of § 24. Proceeding as in § 24, we obtain equation (27) as before except that the third term in square brackets is lacking. Whence, equation (32) continues except that the terms involving the function f are absent, while instead of (33) we have S(z) = lim 0- 1 — s -^ ^=00 L JmIV — zJ Thus it appears at once that the theorem of § 24 holds true lohen < p < 1 General Integral Function 57 provided that the term .7 . 7 , y=l \PJV there appearing be then omitted. 26. We proceed to consider the remaining cases —viz., those in which p = p = an integer. The function F(z) is then defined by the second of the forms appearing at the beginning of § 24. Equations (27) and (30) are now obtained as before, but instead of (31) we write Moreover, the last sum here appearing is evidently of the form c + log X + 02 (.t); lim 02(.t) = 2=00 where c represents Euler's constant. Instead of (32) we thus obtain in the present case H{z) = - |log 27r - I log (1 - 2) - fi,(l) + Z i{1, we may write when 1 2 1 > 1 (66) S{z) = h TrizP + 2P log 2 + r(2), where r(2) is an expression developable asymptotically in the form (56). The form (66) for 1, imag part z^ < 0. The corresponding form for regions T2 in which real part 2^ > 1, imag part 2^ > is obtained (cf. (44), (45)) by adding 2iriz'^ to the right member of (66). Thus, instead of (57) we reach in the present case (07) F{.z)~^^^^\eriz'' + 'tAlY^+"^{c-l) + z'\oiz\ ■\2irz^ L v=\ \P / V P J where = or = 1 according as z is confined to Ti or T^. Upon observing that when imag part 2^ > the function exp iriz'^ is develop- able asymptotically in the form (56) with a^ = ai = a^ = • • • = 0, it appears General Integral Function 59 that (58) becomes replaced in the present case by „ . . 2sin7r2P ["^ ^ f v\z'' , z^ ^ ^ , 1 F{z) -^ 2 exp Zn - -+- c- 1 +2Plog2 . This relation holds true, then, whenever real part z^ > 1. In case real part 2^ < 1 the equations corresponding to (59) and (60) become respectively H{z) = - 2^ log 27r - - log (- z)^ - i log (^1 - ^^ - 12,(1) + mz^ + |:r(^)^+^(c-l) + 2Mog2+r(2), Finally, in case z^ is real and positive, i. e., in case (p has one of the arguments 27r 47r Gtt ~7' ~7' ~7' '"' we find by reasoning analogous to that at the close of § 24 that S{z) vAW be given by (66) and hence we shall have (67) in which ^ = 0. In summar}^ we arrive then at the following Theorem II.-^ Given the typical integral function F{z) defined in Theorem 1 together with the assumption, that p = the integer p. For values of z of large modulus lying within sectors of the type 4A;+3 ^ ^ 4fe+l \k = {), 1, 2, 3, 2p 2p [ - 27r < v? < 0; (p = arg z we shall then have F(z) -^ , exp Xn-)-+-{c- l) + 2Mog(-2) ^2T{-zy L-1 VP/^ P feV /J f being the symbol for the Riemann ^function, and c representing Euler's constant, while for values of z of large modulus lying within sectors of the type _ 4A;+1 4k- 1 U' = 0, 1, 2, 3, • . . 2p '^ < "^ < 2p "" \-2w< 0. Barnes-^ has also considered the corresponding problem for certain type functions whose order is eqnal to zero, but we shall confine ourselves to the case treated above. Asymptotic Developments of Functions Defined by Pow'er Series 28. The results thus far indicated in the present chapter are but indirectly applicable to the determination of asymptotic developments for functions defined by power series. This subject, however, is one of evident importance. We shall now point out a general theorem in this field, resulting from the lemma of § 24. Theorem III. If the coefficient g{n) of the poiver series (68) 2Z^(w)2"; r = radius of convergence > 0, may he considered as a function g{ic) of the complex variable w = x -\- iy and as such satisfies the following conditions: (a) is single valued and analytic throughout the finite w plane except for a finite nuviher of singularities situated at the points w = w\, W2, ' • • , Wp, none of which coincide with one of the points w = 0, 1, 2, 3, • • • , and (b) is such that to an arbitrarily small positive constant e there corresponds a positive constant K {independent of x and y) such that g(x ± iy) g(.x) < K exp ey for all values (real) of x and for all positive values of y sufficiently large, then the function f{z) defined by (68) (|sl < r) will he such that for all values of z lying in any sector {center at z = 0) that does not include the positive real axis we may write ,.Q, -., , f^ g(-l) g(-2) g(-3) (69) f{z) ^ -l^rm ; -^ -^ • • •, OT=1 2 2 2 where rm represents the residue of the function wg{w){- zr (70) Sm TTW at the point w = Wm- In order to prove this theorem we observe that for all values of z except those real and positive we may at once apply the lemma of § 24 with a taken as an arbitrarily large negative integer: a = — /, and write (71) Z g{n)z- = i: g{n)z- + f{z) = - Z r^ + et{z), n=—l n=—l m = l 23 See Philosophical Trans., Vol. 199A (1902). pp. 46G-468. Functions Defined by Power Series 61 where €i{z) vanishes to as high an order as the {I + |)th when | z | = co . Whence follows the indicated result. For example, let us consider the function (72) f{z) = E " ; P 4= integer < 0. Here we may take q(w) = and the residue of (70) at the pole iv = p is readily found to be 7r(- zY n sin Tp Whence, throughout any sector such as indicated in the above theorem we shall have (73) f{z) ~ - -^^y - l^^i^, - (^ + 2).^ • • ■ • This result ceases to hold when p = a negative integer since the expression then has a pole of the second order at w = p. Such cases may, however, be treated by the same theorem. Thus, in particular, when p = — 1 we obtain directly lo g (- 2) ri = — z and hence, instead of (73) ,, , log (- 2) 1 1 1 (74) /(^)-^V--^^-2?-3^---- This result may be verified by noting that when p = — 1 the equation (71) gives j^r \ log (1- 2) m = — ^ — while the power series appearing in (74) converges when | s | > 1 to the value I log (1 — 1/z) so that (74) gives the same form for/(z). 29. Generalizations of Theorem Ill.—li for a given series (68) the function giw) is not single valued throughout the w plane, but contains q branch points w = 'Wi, ibi, • • • , Wq, conditions (a) and (b) remaining otherwise the same, the theorem continues true provided that, after rendering giiv) single valued by means of q cuts extending vertically downwards^'^ to infinity from the points 2^ Since the series here appearing is convergent for \z\ > I tlic symbol ~ may be changed to =. 25 More generally, in any direction tending to infinity in the right half of the plane or vertically upwards or downwards. 62 Determination of Asymptotic Developments w = Wm', (w = 1, 2, • • ■ , q) respectively, we subtract from the second member of (69) the expression 9 m=l where am represents the loop integral (assumed to exist) of 1 g{w){- zr 2i sin irw taken in the positive sense from the point w = Um — ioo to the same point after surrounding the (one) branch point w = fCrn- This result, in fact, appears directly upon reference to the demonstration of the theorem. We note that in case the point w = I'Cm coincides with a point of the type IV = u'm mentioned in the theorem, the corresponding value of rm is to be neglected, the term am then being evidently the only one of the two to be retained. A particular type of function f{z) to which Theorem III and these supple- mentary remarks apply is the following, discussed by Barnes :^^ f r fl\ _ V ^''x(^^ + d) = constant #= or neg. integer, h (2; ^J - £. (n + ey ' ^ = constant, where x(l/2) is regular at the origin. Besides this, Barnes considers the corre- sponding problem for certain special types of functions for which condition (b) of Theorem III is not fulfilled. Of these latter may be especially mentioned the function „ y^ 2" 6 = constant 4= or neg. integer, J^^{z;d) - Z, (^4_0)3r(,,_^ 1) ; ^ = constaiit, for which it is stated-" that for all values of z of large modulus we may write F,(z; d) ^ + ai(.r)y("-" + a2(.T)2/^"-'^ + • • • + an{x)y = 0, wherein the coefficients ai, a^, • • •, «« are assumed to be developable for large positive values of x either in convergent or asymptotic series of the form arix) r , flr. 1 I ar.2 . "I ^ ^ ttr, o+~+"^+---h r=l, 2, •••,n. h being zero or some positive integer.^ In this equation the point .r = qo is in general a so-called " irregular point "^ so that the usual " normal solutions " about the point x = co , as provided by the well-known theories of Fuchs, come to involve power series in \jx that are divergent for all values of x.^ Never- theless, the same solutions continue to satisfy the equation formally^ and it can be shown that they represent asymptotically, in the precise sense of § 13, certain actual solutions. In fact, we may begin by citing the following note- worthy theorem first established rigorously by Horn:^ " If for the equation (1) the roots ??u, W2, • • •, 7n„ of the characteristic equa- tion — i. e., the algebraic equation ' The integer fc + 1 is termed the rank of (1) at a; = . See for example Horn, "Gewohn- liche Differentialgleichungen bcheber Ordnung" (Leipzig, Goschen, 1905), p. 187. 2 For an exposition of the definitions and basal theorems in the theorj^ of linear differential equations, one may consult Picard's "Trait6 d'Analyse" (1896), Vol. 3, Chap. 11. 3 Cf. PiCARD, I. c, § 22. * Cf. PiCARD, I. c, § 23. « Cf. Acta Math., Vol. 24 (1901), p. 289. 64 Real Variable 65 (2) m" + ai, om"-i + • • • + On. o = 0, are distinct from one another, equation (1) possesses n linearly independent solutions yi, y^, • • • , yn valid for large positive values of x which are developable asymptotically in the forms (2') yr ~ e'^'^h^^ E ^" ; r = 1, 2, • • • , n, where /r(a:) is a polynomial of degree ^* + 1 in x, the coefficient of whose highest power in x is mrjik + 1), while pr and Ar, j are constants^ with Ar, o = 1."^ If in this theorem the restriction be removed that the roots of the charac- teristic equation be distinct — i. e., if multiple roots be present — the theorem fails and we at once encounter a problem for which no general solution has as yet been obtained. However, Love^ has recently made a noteworthy advance in this direction, his theorem (which manifestly contains the above as a special case) being as follows: If, other conditions remaining as stated above, " the characteristic equation has I roots mi, TO2, • • • , mi, occurring respectively rii, ri2, • • • , ni times (rii + na + '•'-}- ni = n) and such that no multiple root of the characteristic equation is also a root of the equation (3) ai, im"-i + a2. im"-^ + h fln. i = 0, then the equation (1) possesses a fundamental system of solutions yr,q (r = 1, 2, ' ' ■ , I; q = 1, 2, ' ' ' , Ur) developable asymptotically in the form r ir— l 00 J where fr.qix) is a certain polynomial of degree nr(k + 1) in .r'"'', the quantities Pr,q and Ar, q, i, j arc determinate constants, and Ar, q.o,o — !•" Love has furthermore considered in detaiP the equations (1) of the second and third orders, including the cases in which (3) is satisfied by a multiple root, ^ The precise values of the coefficients of frix) and of the constants pr, Ar.j may be deter- mined by the method of undetermined coefficients after substituting y^ in (1). A similar remark should be understood with reference to the/r,<,(a;), pr.q, etc., that follow. ^ Historically, the first form of equation (1) to be studied in this connection was that taken by Poincar6 in which Ci, oj, • • •, a„ are rational fractions, thus possessing no other singularities than poles at x = «=. See Ada Math., Vol. 8 (1886), pp. 295-344. 8 Cf. Annals of Math., Vol. 15 (1914), p. 155. " Love does not use, at least directly, the method common to the greater part of Horn's work, viz., that of successive approximations, though the latter could doubtless be employed to the same ends. His method rests rather upon certain general studies of Dini to be found in Vol. 2 (1898) of the Annali di Mat., pp. 297-324, wherein the equation (1) of the nth order is first con- verted into a VoLTERRA integral equation of the second kind containing n arbitrary functions, termed "auxiliary functions," and the latter (equation) solved by the usual process of iteration, thus yielding forms of solution for the original equation (1). Tlirough the arbitrariness existing 66 Asymptotic Solutions of Differential Equations and has arrived at complete results for these orders.^" Thus, for n = 2 we have the following :^^ " In the differential equation 2/" + h{x)y = suppose that b{x) is a real or complex function developable asymptotically for large real positive values of x in the form Hx) ..«[^. + ^+...], where )!: is or a positive integer. Then, for the same values of x equation possesses two hnearly independent solutions yi, y^ such that (a) if ho =t= 0, i. e., if the roots mi, m% of the characteristic equation m^ + &o = are distinct, we may write ^, ~/'^'>a:''^[l+^'+ •••], r= 1,2, where TiirX^^^ , ar.-fca:^ /'■(^) = ^ipT "^ ^ !-•••+ «r.-l.T; 1,2, (Jb) if 6o = 0, 6i 4= we may write 2/.^^'-^vTi+^^+--- +4(^^.0+%-+ • where (c) if ^- = 6o = ^1 = we may write in general yr'^x''^^l+^+ •••], r= 1,2; (d) but if p2 = pi or, in general, if p2 — pi is a positive integer we have in the choice of these auxiUary functions, the resulting solutions, though frequently complicated, are of great flexibility and it thus becomes possible to adapt them to a wide variety of investi- gations, as DiNi himself has abundantly shown in a series of papers in the Annali di Mat. extending over the years 1898-1910. In the case of studies such as are being considered in the present chap- ter, the method readily provides actual solutions that are valid for large (positive) values of x and thus the problem becomes merely that of showing that the auxiliary functions may be chosen in particular in such a way that these solutions are developable asymptotically in the sense of § 13 . 10 Cf. Am. Journ. of Math., Vol. 34 (1914), pp. 165-166. " For the sake of completeness the case of unequal roots, though covered by the above mentioned theorems, is included in the statement. 12 It will be obscr\'ed that (6), (c) and (d) relate to the cases in which ?«i = W2. If in (c) or (d) the series for bix) converges for all \x\ > R then x = <» is a " regular point " of the differ- ential equation and hence in the results for yi and 2/2 the sign fo may be changed to = ; lx| > R. 12 Real Variable 67 2/2 ~ 2/1 log X + x""- yl2, + -^ + • • • •" The complete result for the equation of the third order is as follows : " In the differential equation y"' + h{x)y' + c{x)y = suppose that h{x) and c{x) are real or complex functions developable asymp- totically when X is large and positive in the forms where A; is or a positive integer, and suppose that h'{x) also has an asymptotic development. Then for the same values of x the given equation has three linearly independent solutions yi, yi, yz possessing asymptotic developments as follows: (a) If the roots mi, mi, mz of the characteristic equation m^ + hum + Co = are distinct, we may write where ,/A)^P.[^14.^i_|_ ...J. r= 1,2,3, irirX^'^^ ar,-kX^ (6) If Ml 4= m2 = Ws we may write in general where /i(.r) has the same form as in (a) and (c) But if in (6) ps = po, or in general if pa — P2 is a positive integer, we have 68 Asymptotic Solutions of Differential Equations 2/3 ~ 2/2 log X + e^'^'h'' \Az, o + -^' + ' ' ' J ' where /i (a:) aiid/2(.r) have the same form as in (a). (d) If 7?zi = 7712 = W3 and either Ci ^ or 6i = Ci = 0, C2 =[= we may- write where (e) If c\ = 0, 6i =}= 0, 2/1, 2/2, 2/3 h^ve expansions of the same form as in (6). (/) If I' = 6i = Ci = C2 = we may write ^2/1 log a: + x^' ^2. o + -|^ + • • • , £2/1 log2 .T + .T^Uog a: [ 53. + ^^ + • • • ] + .T"' [ ^3, + ^ + • • • ] . 13 While the complete results for the equation (1) of order n ^ 4 have not as yet been obtained, a careful examination of those just given for 7i = 2, 3 throws light upon what the corresponding forms may be expected to be. Moreover, in connection with this question the following result should be noted:" " Let T(.r) be one of the system of functions ^^^' .^'" .T(logx)i+'" .T log a:(log log .r)'+'" *"' ('' > ^^ and put Xoo T{x)dx. 1' It will be observed that (6) and (c) refer to the case ?«i 4= w?2 = m% while (d), (e) and (/) refer to the case m\ = m-i = mz. If in (/) the series for h{x) and c(x) converge for all lx| > i? then X = 00 is a regular point of the differential equation and hence in the results for yi, 7/2 and yz the M may be changed to =; |x| > R. " Obtained by Dini for the case in which the roots of the characteristic equation all have the same real part, and partially obtained by him when this restriction is removed {Annali di Mat., Vol. 3 (1899), p. 136. The result has recently been established in its entirety by Love in the American Journal of Mathematics. Complex Variable 69 Suppose now that in the differential equation (4) 2/(") + [ai + ai(.T)]2/(«-i) + [a^ + «2(a:)]2/("-2) +... + [«„ + a„(y)]y = q the functions ai{x), a2(x), -- -, an(x) together with their 2n — 1 derivatives are continuous when x is sufficiently large, and suppose that the characteristic equation (5) M" + aiM""' + h fln = has I different roots jui, 1x2, • • • , m occurring wi, 712, • • •, ni times respectively (ni + n2 + • • • + w; = n) and let n' be the largest of the numbers Ui, 112, " -,711. If one of the functions t(x) exists such that for sufficiently large values of x (6) |ar«(.T)|^^J^; r= 1, 2, ...,7i; s = 0, 1, ■ ■ •, 2n - 1, then for the same values of x the equation (4) has n linearly independent solu- tions T/i, k{x) expressible in the form Vi. k(x) = a;^-ie-'^[l + e,-, ,(x)]; i = 1, 2, - - -, I; k=l,2, -- -, m, where €», A:(a;) vanishes at infinity to at least as high an order as that of Tiix), while further yi'Mx) = x'-'e''^'[txi' + ri.fc. s(x)]; 5 = 1, 2, . . ., n, where lim ^ = 0." x=ao It is to be observed that for the special case t{x) = Ifx^ this result relates to an equation of the form (1) (wherein k = 0), and furnishes the " dominant terms " of developments for the corresponding solutions yi, k{x). Doubtless by a sufficiently critical examination of the form of e,,fc(a;), these developments could be identified with asymptotic developments in the precise sense of § 13. For the type of equation considered, the result is seen to be in every sense general so far as the possibility of multiple roots in (5) is concerned, except for the restrictions (6). These latter when interpreted with reference to (1) mean that (7) ar.a=0; r = 1, 2, 3, --^n; 5 = 1, 2, 3, • • -, 2?i' - 1 and hence come to impose unfortunate restrictions. However, the result is of decided value in showing that all further studies upon the problem in hand may be limited to those cases (assuming multiple roots present in (2)) wherein (7) are not satisfied. Complex Variable 32, Passing to the corresponding studies upon (1) when the independent variable x is allowed to take on complex values, the existence, form and range of the asymptotic solutions have been completely discussed by Birkiioff in case the coefficients ar{x) (r = 1, 2, • • •, ii) are developable in convergent series 70 AsYAiPTOTic Solutions of Differential Equations (|a;| > jB = constant sufficiently large) and under the assumption that the roots of the characteristic equation (2) are distinct.^" Corresponding results when multiple roots are present in (2) do not appear to have been thus far ob- tained. Birkhoff's essential result may be summarized as follows: " Representing by vii, m^, • • • , tUn the n (distinct) roots of (2), let there be drawn from the origin (2=0) the N = n{n — \){k -\- I) raj'S ("critical" rays) determined by the equation real part of [{vig — m<)a:'=+^] =0; s =^ t. Let the angles which these rays make with the positive real axis in the order of their increasing magnitude be denoted by n, T2, • • -, r^r and place Ty+i = ri + 27r. Then, corresponding to the sector Tm = arg x < tm+i there exists a set of fundamental solutions pr {r = 1, 2, •••, n) of (1) developable asymptotically in the forms (2)' where fri^), pr and Arj continue to have the meanings there indicated. The set of solutions satisfying (2)' in the sector (jm, r^+i) differs at most by one solution from the set satisfying (2)' in the adjacent sector (r^+i, 7^+2)."^® Linear Difference Equations 33. If instead of (1) we take for consideration the linear difference equation (8) yix + h) + ai(x)y{x + ^ - 1) + a2{x)y{x + A - 2) + • • •+ an{x)y(x) = wherein the coefficients oi, 02, • • • , On are assumed to be developable for large positive values of x either in convergent series or asymptotically in the forms (8)' ar{x)^x^'^^ar,o + ^ + ^+ •••]; r= 1,2, --^n, 1^ Trans. Am. Math. Soc, Vol. 10 (1909), pp. 463-468. Birkhoff considers, instead of (1), the system of n ordinary linear equations of the first order: (A) ^= t,c'iiix)yi a = 1,2, ■'■,n), in which for |a;| > R -we have a.-,- (a;) = ai^xt + a.-, y^'x'-i + • • • + ai/«> + ai,f«+i) 1+ •■■ {i,j = 1,2, ••■,n), X the characteristic equation then becoming |a,-,- — 5,-,a| = 0; 5,-j =0 if i =t=j; S.y = 1 if i = 3. The equation (1) may be transformed into a sj'stem of the form {A) by placing y\ = a;"*;/, 7/1 = x<"~i>*y', •••, y„ = x*2/'"~^> in which case we find q = k. Thus, whatever appUes to {A) applies to (1) as a special case with q = k. The important case in which the coefficients ar(x) of (1) are rational polynomials was dis- cussed in a series of earlier papers by Horn whose results are summarized by Van Vleck in the Boston Colloquium Lectures (1905), pp. 85-92. 1^ For the precise nature of this dependence, see Birkhoff, I. c, p. 468. Linear Difference Equations 71 k being zero or a positive integer, we have, corresponding to the first result cited in § 31, the following: " If the roots mi, m^, ••-,?«« of the characteristic equation (9) TO" + ai,om"-i + h fln.o = are distinct and no one of them equal to zero, equation (8) possesses n linearly independent solutions yi, y^, • • - , yn valid for large positive values of x which are developable asymptotically in the forms (9)' yr - [r(;r + l)fm/x<'^ Z -^fr' ; r = 1, 2, . • . , n, j=0 X where ^r.o = 1."^^ In case (9) has multiple roots, or a zero root (an,o = 0) the principal results thus far obtained appear to be those of Norlund who employs asymptotic " faculty series " and allows the independent variable x to range over complex as well as real values. Using his notation and including for the sake of complete- ness the case of distinct roots, his results are as follows '}^ " Given the linear difference equation k (10) Z P^{x)u{x - i) = 0, 1 = where the coefficients are faculty series of the form Pi{x) = co^') + — _|_-y + (a:+ l)(a: + 2) (11) (0 {x+l){x-r2){x-\-3) + 7— r-rT^V7^v7r-^^+ •••; i = 0,1,2, -••, k all of which converge throughout the right half of the x plane.^^ Suppose first that the roots ai, a^, az, • • •, a^ of the characteristic equation (12) co^o^z*^ + Co">2^-^ + V co^^) = 0; Co^"^ 4= 0, Co^^'^ + are distinct. Then there exist k solutions U\, ii2, - • -,Ukoi (10) such that through- out the sector — (7r/2) + e < arg x < (7r/2) — € (e arbitrarily small and > 0) we have nQ^ ■ . r(.T + 1) (13) ^.— «/r(.r-p,+ l)^^^^^' where py is a constant and M2> • • • J Mm such that the total number of roots which are finite and different from zero in the corresponding characteristic equations thus obtained is exactly the order k of (10).^° If, whenever a multiple root occurs in one of these charac- teristic equations, the corresponding conditions under (1) are satisfied, then there exists a system of fundamental solutions of (10) each of which is asymp- totically represented within the sector — (7r/2) + e < arg x < (7r/2) — € by a series of the form V>''-{x)af^s{x). Exceptions occur, however, when some of the numbers iXr are not integers, since the coefficients in the above-mentioned difference equations are then no longer developable in faculty series of the form (11). For example, suppose jur = a rational fraction j^jq. We may then put x = yz, u(x) = v(z) and derive from (10) a difference equation for v{z), thus demonstrating the existence of solutions expressible asymptotically in the forms r " Norlund, Ada Math., Vol. 34 (1911), p. 16 "(i) "'"'*•©• Linear Difference Equations 73 Important studies of (8) when x is complex and under the assumption that the roots of (9) are different from zero and distinct and that the coefficients ar{x) are rational fractions developable in the forms (8)' (wherein the series would then converge for all \x\ sufficiently large), have been made also by Galbrun^^ and by Birkhoff,^^ with the essential result that there exists a system of funda- mental solutions G(x) = yi, ?/2, ys, • • • , yn developable asymptotically in the respective forms (9)' throughout the right half of the x plane, and at the same time there exists a second system H{x) = yi, 7/2, • • -, yn of fundamental solutions developable likewise in the forms (9)' but throughout the left half of the plane. Moreover, the elements of the system G{x) when considered in the left half plane possess asymptotic developments other than (9)' whose forms change as arg x passes through any one of certain radial directions (" secondary critical rays ") lying in the second and third quadrants,^^ while similarly the elements of H{x) when considered in the right half plane are developable asymptotically in forms differing from (9)' and changing as arg x passes through certain radial directions situated in the first and fourth quadrants. Returning again to the case in which x is regarded as real and positive and assuming further that it is confined to integral values, we have, corresponding to the last result stated in § 31, the following i^'^ " Let t(x) be one of the system of functions (3)' and put 00 Ti{x) = S r(.Ti). Xl=Z+l Suppose now there is given a difference equation [ao + aQ{x)]yix + n) + [ai + ai(x)]yix + ?i — 1) + • • • (14) + [«n + an{x)]y{x) = 0, whose characteristic equation aoM" + aiix''-^ + . . . + a„ = has I different roots )Ui, 1x2, - - • , Hn occurring n\, ni, - - - , ni times respectively (jii -\- 712 -\- ••■-{• ni = 11) and let n' be the largest of the numbers ni, n2, • • • , nu 21 Acta Math., Vol. 36 (1913), pp. 1-68; also Comvt. Rend., Vol. 148 (1909), pp. 905-907. 22 Trans. Am. Math. Soc, Vol. 12 (1911), pp. 243-284. As in his studies on linear differential equations (cf. footnote, p. — ), Birkhoff considers a system of linear difference equations of the first order. In order to identify the forms (9)' with those occurring in his results, it suffices to observe that r(x + 1) ~ x'+^'h-'(^co + ^ + §+•••) • See for example Horn, Math. Annalen, Vol. 53 (1900), p. 191. 2' For the precise statement, see Birkhoff, I. c, p. 277-278. See also p. 279, lines 1-7. 2* Cf. Love in Afn. Journ. Math. Obtained earlier by Ford in case all roots of the charac- teristic equation have the same modulus {Annali di Mat., Vol. 13 (1907), p. 328). 74 Asymptotic Solutions of Differential Equations If a function t{x) exists such that for sufficiently large values of x t(x) \cxrix)\^^;^^; r = 0, 1, 2, ..-, n, then for the same values of a: the equation (14) has n linearly independent solu- tions ?/,-, k{x) expressible asymptotically in the forms Vi, k{x) ~ .r^-Vi11 + ^i. k{x)]; i = 1,2, '-'J; k = 1,2, ■'-, iii, where e;, u vanishes at infinity to at least as high an order as that of Ti(a')." Summary 34. A comparison of the results noted in §§ 30-33 would indicate that the study of the asymptotic solutions of either the differential equation (1) or the difference equation (8) is already in a fairly satisfactory state provided the assumption be made throughout that the roots of the characteristic equation are distinct, but much remains to be done in those cases where multiple roots are present. In fact, it is only for the equation (1) of the special orders n = 2 or n = 3 that we find what could be described as a complete discussion, and even this has thus far been carried out only for the real variable x. CHAPTER IV ELEMENTARY STUDIES ON THE SUMMABILITY OF SERIES 35. Introduction. — The divergent series (1) 1- 1 + 1- 1 + 1- 1+ ..• was regarded b}^ Euler^ as having the sum | on the ground that the expression 1/(1 + X) gives rise by division to the series (2) 1- x+ x" - x' + x^ - x^ + • • •, so that in particular (placing x = 1) one must have (3) i= 1- 1+ 1- 1 + 1- 1+ •••. In general, the " sum " of a series (convergent or divergent) was taken to be the number most naturally associated with it from the standpoint of mathematical operations. This conception, however, naturally led to inconsistency. Thus, by developing the expression (1 — .t")/(1 — a;"") into the form (4) 1 - a;" + a;"* - .t"+^ + x-"" - • • • , and noting the result when a: = 1 we obtain for the series (1) the sum n/m instead ofi The notion of sum as thus loosely conceived was eventually replaced by the exact definition of Abel and Cauchy according to which the sum of any series (5) ao + ai + a2 + as + • • • is taken to mean the limit (6) s = Hm (oo + ai + 02 + • • • + On). Series for which this limit exists were termed convergent, all others divergent. Of the two classes of series thus arising, the former occupied almost exclusively the attention of the immediate successors of Abel and Cauchy and to such an extent that all divergent series came to be regarded as of questionable value and indeed of doubtful significance. It is a noteworthy fact, however, that Abel and Cauchy themselves never ceased to regard divergent series with much interest and with the belief that such series should by no means be banished from analysis for the mere reason that they fell outside the pale of the particular 1 For a more extended historical account, see Borel, " Lemons sur les Series Divergentes" (Paris, 1901), Introduction. 75 76 Studies on Summability definition (6). Each felt on the other hand that the subject presented a rich field for further research. Only since the time of Weierstrass has the question thus arising — viz., whether any numerical significance can properly be attached to a divergent series — been scientifically attacked and in large measure answered. The avenue of approach has been chiefly through the so-called boundary-value (Grenzwert) problem in the theory of analytic functions.^ Thus, Frobenius^ showed in the first place that if 00 (7) Z dnX"" n=0 be any power series having a radius of convergence equal to 1, then (8) lim 2^ a„.T" = lim -j—. , ar=l— On=0 n=oo 71 -J- i where Sn = Oo + oi + 02 + • • • + Qn- This was shown to be true, at least, whenever the limit indicated on the right exists. Now, the first member of (8) is naturally associated with the series (in general divergent) 00 (9) Z an, SO that it becomes natural to associate with the latter the sum 50 + 5i + 52 + • • • -\- Sn (10) s = lim n-\- 1 whenever this limit exists. Formula (10), regarded as a general formula for defining the sum of any given divergent series (9), finds additional justification in the demonstrable fact that for any convergent series (9) the sum as defined by either (6) or (10) is the same — i. e., formula (10) is consistent. Moreover, this selection for s is seen to bear an interesting relation to the early statement of EuLER noted above respecting the particular series (1), since, when applied to (1), it gives at once 5 = |. In the present chapter certain general studies are first undertaken (§§ 36-40) upon a few of the well-known, standard definitions for the " sum " of a diver- gent series. The definitions selected (which include (10) as a special case) are subjected in turn to a number of tests which it is believed any such definition may well be asked to satisfy, and the results attained are summarized in § 41. 2 For a description of this problem see Jahraus, " Das Verhalten der Potenzreihen auf dem Konvergenzkreise historisch-kritisch dargestellt," Programm des Kgl. humanist. Gymnasiums Ludwigshafen a. Rhein (1901), pp. 1-56. See also Knopp, "Grenzwerte von Reihen bei der Annaherung an die Konvergenzgrenze," Dissertation (Berlin, 1907). 8 Journ.fiir Math., Vol. 89 (1880), p. 262. Generalities 77 The underlying principles guiding the development of these §§ are stated in the Preface and hence need not be repeated here. In the latter part of the chapter the essential properties of "absolutely summable " series are considered (§ 42) and this is followed by a few supple- mentary theorems and remarks on the theory of summability in general, proofs being suppressed when reference can be readily made to them elsewhere. 36. Definitions of Sum. — Let any given series (convergent or divergent) be represented by (11) f.Un n=0 and let us place n Sn = 2-^ llri' n=0 If (11) is convergent let its sum be indicated by S, if divergent let the sum as- signed to it by whatever manner be indicated by s. The definitions for s to which we shall confine our attention^ are as follows: (I) s = \\m-f-(r)'y r = fi^'ed integer ^ (Cesaeo),^ n=oo ^n where (12) S (r^ - , \ rs .1 '^' + ^^ . 4- I Kr + 1) • • • (r + n - 1) '^- ni J. (,) _ (r+l)(r+2) ■•• (r+n) n\ Under (I) is thus included as a special case corresponding to r = 1 the definition (10). The least value of r for which the second member of (I) exists is called the degree of indeterminacy of the series (11). * We have confined the attention to what may be called the older and best known forms of definition, (I) and (II) being connected with the early studies of Holder and Cesaro upon the boundary value (Grenzwert) problem for functions defined by power series (see § 35), while the remainder, especially (III) and (IV), are connected with the independent and now classical studies of Borel upon divergent series. A form of definition prominent in the more recent literature, especially in England, is that of Riesz {Compl. Rend., July, 1909) : s = lim X) Wf ( 1 - ~ ) ; r = integer ^ 0. n=iio u=u \ ' * / There should be mentioned also the following definition of De La Vall6e Poussin (Bulletins de la classe des Sciences de V Academic Royale de Belgique, 1908, pp. 193-254) : , - lim ( V A--^ n(n - 1) • • . (n - fc + D \ ' - i™ T" + h (n + i)(n + 2)...-(ir+T) 'V • For a general study of possible forms of definition, see Silverman's Thesis "On the definition of the sum of a divergent series" in the scientific publications of the University of Missouri for April 1913, pp. 1-96. ' 5 Bidlctin des Sciences Math. (2), Vol. 14 (1890), p. 119. Chapman has extended the defini- tion to include fractional values of r (Proc. London Math. Sac, Vol. 9 (1911), pp. 369-409). (ID where Studies ox SuMiL\BiLiTY s = lim Sn^''^; r = fixed integer ^ (Holder),^ = ^^^ (So''^ + ^1^°> + • • • + Sn^'^), Sn . *n (2) = 71+ 1 (>•) = 1 (5o"^ + 51^1) + • • • + 5n«), (^o^--^) + 51^'-" + • • • +*n^^^0 (HI) n+ 1 s = lim e~''5(a) (Borel),^ where 5(a) is defined by the following series (assumed convergent for all values of a) (13) 5(a) = i:^ a". (IV) ,=0 n : e'" u (a) da (B orel) / where u{a) is defined by the following series (assumed convergent for all values of a) n{a) =^ t'^.a'^. (14) e~''Up{a)da; p = fixed integer ^ 1, (V) where Wp(a) = (wo + '^1 + • • • + Vp-i) + {lip + Up+i + • • • + «2p-i)a + {U2p + • • • + UZp-lW + (VI) s= r e-''Up{a)da,' Jo where (15) Up{a) = 2 2/„a' „=o (np) ! ' p = fixed integer ^ 1 . 37. Consistency of the Above Definitions. — It is at once to be assumed that any tenable definition of sum for divergent series must be such that in the case « Math. Annalen, Vol. 20 (1882), pp. 535-549. ^Cf. "Lemons," p. 97. 8Cf. "Lemons," p. 98. s Due to LeRoy. Cf. Annates de la Faculle des Sciences de Toulouse (2), Vol. 2 (1902), p. 217. Consistency of Definitions 79 of a convergent series it gives s = S. This property of a definition is called its consistency}^ We proceed to establish the consistency of all the above definitions by a uniform method based upon the following general lemma in the theory of limits.^^ Lemma. — Let so, Si, s^, • • • , Sn, • • • be a sequence of quantities (real or com- plex) such that lim Sn = I and let ao^^^ a/^^ 02^^\ • • •, On^^^ • • • be a sequence 71=00 of positive quantities (weights) dependent upon a parameter p (independent of n). Also let it be supposed that the expression Sp = .Z^ Ctji Sn has a meaning for every value of p in a given sequence P of positive elements which increase indefinitely to + oo. If, then, p be allowed to increase in- definitely ranging over the values in P we shall have lim Sp = I provided that p=+» (A) lim '^^^— = 0, «=o where m is any fixed positive integer (independent of p and n). Proof. — We have by hypothesis Sn = 1+ e„ ; lim €„ = and it suffices to K = 00 show that lim Dp = 0, where p=X) ^ P 00 ''• By writing n=0 n=0 n=mH-l and then placing 5„ = / + fn in the last term here appearing we obtain m 00 E(^n-/)a„^'^>+ Z €„fl„(^) P 00 • n=0 loCf. BuoMwiCH, "Infinite Series" (London, 190S), § 100. " Cf. FouD, American Journ. of Malh., Vol. 32 (1910), p. 320. As here generalized, the lemma was first- obtained and applied to the discussions of the present chapter by Meni)enh.\ll in his thesis entitled " On the Characteristic Properties of Sum-Formulaj in the Theory of Divergent Series," University of Michigan, 1911. 80 Studies on Summability "Whence, if we indicate by g^ a positive quantity such that gjn^\si\;i = 0, 1, 2, • • •, m, we may write \Dp\^ {gm + \l\) n=0 ZaJ^' This relation holds good for any preassigned value of p belonging to P and for any preassigned arbitrarily large positive integral value of m. The same having been once established, let us now choose an arbitrarily small positive quantity e and then take m so large that ] e„| < «; n = m + 1, to + 2, • • •. We may then write £ \en\aj^' — n — 1) . a„(p> = -^— ^— ^ — -^ f -' when n < p; {p — n)\ a„^p) = 1 ivhen n = p; aj^^ = when n > p. Then S^ = Sp'^'^Dp^''^ where Sp^'^ and Dp^'^ are given by (12). Condition (A) of the lemma is satisfied since ZaJ^' Zy iiS Ir^ " ll'S (r+l)(rV2) ■■'{r + p) n=0 = Hmr p=a) L P + rp r+p {r -\- p){r -{- p — 1) rp{p — 1) • • • (p — m + 1) + (r + 2^)ir+ p — I) • ■• ir+ p Thus we have the desired result : lim Sp = lim Sn = S, ^1 = 0. - m) J p=00 provided the latter limit exists, i. e., when (11) is convergent. Consistency of Definitions 81 The consistency of (II) follows directly from that of (I) if we make use of the following established result: "If the limit s defined by (II) exists for a given value of r then the limit s defined by (I) exists for the same value of r, and conversely. Moreover, the two limits s are the same." In view of this result it appears that formulse (I) and (II) are coextensive both in applicability and in the values of s which they associate with a given series (convergent or di- vergent). As the proof of the indicated result is lengthy, it will be omitted here.^^ To show the consistency of (III), let P be taken as the continuous domain p ^ and let aj^^ = p^'/n !. Then Sp = -^-^ = e-Ps{p). Condition (A) is satisfied since in „m (16) lime-^E— ,= 0. p=oo n=0 ''i • Thus the lemma yields the desired result : (17) lim e~^s{p) = lim e'^sia) = lim Sn = S. p—oo a=oo n=ao In considering the consistency of (IV), we first note that when (11) is con- vergent, lim Un = 0. Whence, if we apply the lemma of § 37 with s„ = w„ and n=oo fln^^^ = p"/n!, noting also relation (16), we obtain (18) lim e'^uip) = lim e"'u{a) = lim Un = 0, p=ai a=co n=« where u(a) has the meaning given in (14). Now, from equation (17) together with [e~''s{a)]a=o = uo, we may write r d r d S — Uo — I ^[e~°-s{a)]da. But ^[e-"5(a)] = e-V(a)-5(a)], where -^s{a) = s'{a) = 5i + 52o: + *3 .yj + • • • . Whence, if we note that d a^ ^u{oc) = u'{a) = s'ia) — s(a) = Ui + i^a + U3^^-{- ■■■, "See Ford, I. c, pp. 315-326. Also Schnee, Malh. Annalcn, Vol. 67 (1909), pp. 110-125. In view of this result we shall omit the detailed discussion of (II) throughout the present chapter, all statements respecting it being identical with those obtained for (I). 82 Studies on Summability we have /»« (19) S - uo= e-''u'{a)da. Jo Whence also, upon integrating by parts, S - Wo = U"" / u'(a)da +1 e"' j u'{a)da Ida e-^iiiia) - uo] \ +1 e-^iuia) — Uo}da. Introducing (18) together with 11 = I e~'^iioda Jo we reach the desired relation : (20) I e-''u(a)doi = S. Jo Definition (V) is at once seen to be consistent, for when (11) converges to S so also does the series {llO + Wi + • • • + Up-l) + («p + Up+l + • • • + «2p-l) H- {U2p + W2H-1 + • • * + UZp-l) + • • •, and by applying (IV) to this series we obtain the desired result: f e °'iLk{ot)da = S. Likewise, the consistency of (VI) may be shown by use of (20) for it is merely the application of this equation to the series «o + + + h + wi +0+0+.--+W2+0+-.-, wherein p — 1 zeros are inserted between each term and the preceding term in (11). 39. The Boundary Value Condition. — It is well known that two definitions of sum, both " consistent " (§ 37), do not necessarily give the same sum to a given divergent series. In other words, consistency alone is not an adequate principle upon which to base a scientific theory of summation because it does not insure uniqueness of sum.^^ A theory free from this objection may be " See remarks in Preface. It would appear that many of the formulae for sum suggested within recent years have been obtained from considerations quite regardless of the question of uniqueness. The Boundary Value Condition 83 attained if (having demanded consistency) we confine the attention to those series (11) for which the corresponding power series^^ (21) f(x)^f:2lr.X- n=0 has a radius of convergence equal to 1 and then agree to retain those definitions of sum for which (B) s = lim fix). a:=l-0 This procedure is in Hne with the historical genesis of the theory of summability and allows the theory a well-defined usefulness in the study of analytic functions.^^ Indeed, if a general, self-consistent theory is to be formulated, it would seem that it should contain (B), or an equivalent condition, though such a condition evidently tends to hmit the immediate range of appHcability of the theory to a particular class of series (11) (cf. Preface). Having assumed, then, that the series (11) is such that the power series (21) has a radius of convergence equal to 1, we shall undertake to determine in the present § those definitions of sum which satisfy (B). Definitions having this property we shall speak of as satisfying the boundary value condition. We begin by showing that definition (I) satisfies (B), i. e., (22) lim flunx^ = lim Sn^^yDJ^\ x=l—On=0 n=oo whenever the latter limit exists. This may be done as follows by the aid of the lemma of § 37. Let the Sn of the lemma be taken as SJ''^IDJ''\ Then place x = 1 — 1/p so that as x ranges from a to 1 (0 < a < .t) the quantity p ranges from 1/(1 — a) to + oo; also take aj^^ = DJ'\1 - l/^;)". The expression Sp of the lemma then becomes n=0 \ PJ ji=0 '■• It is to be observed that tliis scries is formed by supplying the successive powers of x into (11) beginning with x°, thus excluding, for example, the series (4) in connection with the study of (1). This choice oi fix), though arbitrary, is evidently the most natural and the one most likely to result in a theory of summability having useful supplemental relations to the boundary- value problem. 15 Some sum-formulae, such as (IV), §36, not only satisfy (B) when applied to series (11) for which (21) has a radius of convergence equal to 1, but they have the further property that they preserve a meaning in certain regions in which |x| > 1 and in these regions furnish the analytical continuation of (21) (cf. § 44). 84 Studies on Summability so that 00 lim Sp = lim X w„a:". p=«) 2=1—0 n=0 Let us now confine ourselves, as may be done without loss of generality, to values of p pertaining to the sequence P = 1, 2, 3, • • •. Condition (A) of the lemma is now satisfied, since which expression is evidently equal to zero since the denominator has a meaning for all 2^ > but becomes infinite with p, while the numerator remains finite as p = 00. Applying the lemma, we may therefore write (22) as desired. We turn next to definition (III) and shall show that (B) is again satisfied, i. e., 00 (23) lim Xlw„a:" = lim e~''s{a), Z=V—0 n=0 a=oo whenever the latter expression has a meaning. For this purpose we first note that for any series (11) (convergent or di- vergent) for which the second member of (22) exists we have in the notation of §36 (24) ^[^~"*(«)] = e-'^Wifii) - s(a)] = e-^u'ia); [e-«5(a)]„=o = ^o and hence ^ [e-''s(a)]da = Wo + I e~"w'(a)c^a. Conversely, it appears from the same relations (24) that for any series (11) for which the last member of (25) exists, the second member of (23) exists also and we have relation (25). This premised, let us return to the series (21). Since this series is convergent when I a: 1 < 1 it follows from the consistency of definition (III) that when 0< a;< 1 ^UnX"" = Hm e~X(«)> ji=0 a=oo where s^(a) represents the function s{a) corresponding to the series (21) . Whence, upon applying (25), we have also The Boundary Value Condition 85 (26) X]«n.^■" = Uo-\- I e'^u' {ax)da; u'{ax) = ■^u{ax). Assuming for the moment that the integral here appearing converges uniformly for all values of x in the interval aOwe now have, using (25), 00 /»oo hm ^UnX'"' = 2 0, with which the proof of (29) becomes complete. Definition (V) does not in general satisfy condition (B), as appears from an example. Thus, let the series (21) be and take p = 2. Then Up{a) = and hence s, as given by (V), is equal to zero. But, lim E (- 1)M- = Y^-r = h a-=l-0 n=0 ■>- I ■•• That definition (VI) satisfies condition (B) may be readily inferred from reasoning similar to that followed in connection with (IV). Thus, from the consistency of the definition we have 00 /»oo <31) limX!wna:"= lim e-''Up,xici)da, 1=1— On=0 x—l—oJo ■where Upon placing x = 2" the second member of (31) takes the form /^oo <32) lim 1 e-'Upiazjda, »=l-0»/0 w^here Up is defined by (15). Now place az = y and subsequently replace z by 1/(1 + 6). Expression (32) then takes the form roo e-ye-'"Up(y)dy. . - Since the integral /»« e-yUp{y)dy f has a meaning by hypothesis, we may show by means of Abel's test, as in con- nection with (IV), that the integral in (33) is uniformly convergent for all values of 6 in the interval < 6 < b, with which the proof is at once completed. I'Cf. Bromwich, I. c, p. 121. Fundamental Operations 87 40. Fundamental Operations. — Besides being consistent and satisfying the boundary value condition (i?)/^ it is evidently desirable that the sum assigned to a numerical divergent series (11) shall, at least so far as possible, be one for which the usual operations applicable to convergent series are preserved. The operations of this type which we shall consider are the following: (C) If s represents the sum of the divergent series (11) by a given definition, then the series (34) z2un; k = positive integer n=k shall have a sum 5^^^ by the same definition such that (35) 5(^> = 5 - (wo + ^1 + • • • + w,_i). Conversely, if the series (34) has a sum s^^^ by a given definition then the series (11) shall likewise have a sum s by the same definition and relation (35) shall exist. {D) If with a given definition of sum, the two divergent series: (36) E Un, Z 'Vn »i=0 n=0 have respectively the sums Si, S2, then the series Zl (Un ± Vn) w-0 shall possess by the same definition the sum Si ± 52. (E) With the hypotheses stated in (D) the series (37) tw., n=0 where shall have the sum *i*2 (at least after certain additional conditions have been placed upon w„ and v„ analogous to those imposed when two convergent series are multiplied together). We begin by showing that definition (I) satisfies condition (C). For this it evidently suffices to suppose Jc = 1, since a repetition of the reasoning leads from this to the general result (35). 18 For reasons stated at the beginning of § 39 we shall continue throughout the present § to regard the given series (11) as belonging to the class for which the corresponding power series (21) has a radius of convergence equal to 1. This hypothesis, however, plays no part in the deductions about to be made. 88 Studies on Summability Placing [Sn= Uq + Wi + • • • + W„, [ (Tn = Wi + «2 + • • • + lin+l, , rjr + 1) , , r(r + 1) • • • (r + n - 1) iSn^^'' = 5„ + r5„_i H 2] *"-2 + • • • + ~l -^0' n (.) _ (r+l)(r+2) ••• (r+n) ^" ~ n! we are to show, then, that if the hmit 51 = Urn ,S/'-^/Dn^^^ n=QO exists so also does the limit 52 = Km iSj^^Dn^'-^ n=ao and that Si = uo + 52, with the corresponding converse statement. Since Sn+i = uo + cr„ we have r(r+l) , r(r+l)--(r+n-l) where 1 I _L ^(^ + 1) - - r(r+l) ••• (r+7^-1) _ ^ (,) Whence, The desired result (both direct and converse) now follows upon noting that As regards definition (III), it appears from an example that this does not always satisfy (C). Thus, consider the special series (11) for which Uq, Ui, u^, • • • are so determined that sin (e") = Uo + {uo + ni)a + {uq + Wi + W2) 2]+ * " '• For this series we have s = lim e"" sin (e") = 0; 5^^^ = lim e-" ;psin (e") — i/o^" = lim cos (e") — ^0, a=« a=oo Loo; J „=«! SO that although s exists, the same is not true of s'^^\ Fundamental Operations 89 In this connection we may, however, estabUsh the following noteworthy result : " If the series 00 (39) Ew„; p = 0, 1, 2, 3, .••,^• are each summable by (III) to the respective values s, s^'^^, s'-'^\ ■ • •, s''''\ then relation (35) is satisfied." In fact, with Sn and (Xn defined as in (38) and with 2 s{a) = 5o + 5iq: + 52 ^ + • • • , it follows that lim e~'^'u{a) = so that we have as desired 5^^^ = s — Uq. In order to prove the more general case we have evidently but to repeat the same reasoning k times. Definition (V) does not always satisfy condition (C) since, as we have just shown, it does not do so for the special case in which p = 1. Likewise, the same is true of definition (VI) (which reduces to (IV) when p = 1), but we here have an alternative result similar to that indicated above. We turn then to condition (D). This is evidently satisfied by two series summable by any one of the definitions of § 36 and, therefore, needs no further comment. As regards condition (E), it is obviously necessary to impose further condi- tions than that of the mere summability of the two series (36) in order that (E) be satisfied, at least in generab since even in the case of two convergent series such supplementary conditions are required. We here have, however, the following noteworthy result of Cesaro relative to series (36) summable by (I) : " The product series (37) of two series (36) whose degrees of indeterminacy (§ 36) are respectively r and s is summable and has a degree of indeterminacy no greater than r + 5 + 1."^^ Conditions under which condition (E) will be satisfied by definition (IV) will be considered in § 44. 41. Summary of Results. — The principal results of §§ 35-40 may be sum- marized into the following statement: Let (42) Zun be any divergent series such that the corresponding power series 00 " We omit the proof of this well-known result. The same may be supplied from Bromwich, I. c, § 125. For Cesaro's original proof, see Bulletin des Sciences Math., Vol. 14 (1890), pp. 118, etc. Absolutely Summable Series 91 has a radius of convergence equal to 1. Also, let (I), (II), (III), (IV), (V) and (VI) represent the six definitions for sum indicated in § 36. //, then, we represent by (A) the condition of consistency (§ 37), by (B) the boundary value condition (§ 39) and by (C), {D) and (E) the conditions of § 40 carried over from the theory of convergent series, the relation of the various definitions to these conditions appears in the following table wherein the * when placed in any square indicates that the corresponding definition and condition are compatible: I II III IV V VI A * * * * * * B * * * * ^ C * * D * * * * * * E Fig. 5 Moreover, the squares corresponding to (III, C), (IV, C) and (VI, C) may also receive the *providedlpne substitutes for (C) the following slightly more restrictive condition : {Cy If the series 00 X)w„; p = 0, 1, 2, 3, • ••, A; n=p are each summable in accordance ivith a given definition of sum to the respective values s, 5(», s^'\ s^'\ •••, s^^) ^(fc) = s — (^u^ -^ u^ -\- . . . -j- Uk_i). then 42. Absolutely Summable Series. — A noteworthy class of divergent series (11) for which conditions (A), {B), (C), (D), (E), of § 41 are all satisfied when we adopt the definition (IV) of sum, has been pointed out by Borel and made the object of especial study throughout his investigations." Such series are called abso- lutely summable and are defined from the fact that not only the integral /^oo (43) s = I e-''u{a)da Jo is supposed to exist, but also each of the integrals e-''\u^^\a)\da; p = 0, 1,2,3, " Cf. "Legons," Chapter III. 92 Studies on Summability wherein u^^\a) denotes the pth. derivative of the (integral) function ii{a) (cf. (14)). Absolutely summable series, as thus defined, being but special series summable by definition (IV), at once satisfy conditions (A), (B) and (D), as shown in earlier §§. It therefore remains but to consider such series with reference to conditions (C) and (E). Now, if the series (11) is absolutely summable, it follows from definition that both s and s^''^ exist. Whence, by the results obtained in § 40, we have relation (35). In order to complete the proof that (C) is satisfied, we must now show that if the series (34) is absolutely summable, so also is (11) and that with s and 5^*^ defined as before, relation (35) exists. For this let us first consider the case in which k = 1. Place23 (p{x) = I \u'{t)\dt^\ 1 u'{t)dt We thus have (p{x) ^ \u{x) — Wo I and hence |w(.r)|^ (p{x) + |wo|, so that the integral I e~''\u{x)\dx Jo must converge whenever the same is true of the integral (44) I e-'u{x)v{y)dxdy Ur»A /»\ "1 /^oo /»« e~'^u{x)dx I e~H{y)dy = I e~''u{x)dx I e~yv{y)dy = SiSi. Now, in case u{x) and v{y) are always positive the indicated double integral when extended over the triangle OA'C has a value lying between the corre- sponding integrals taken over the squares OABC, OA'B'C , and since the latter each approach the limit Sis^ as X = co, the integral over the same triangle will also approach the limit S1S2. On the other hand, if u{x) and v{y) are not always positive, the absolute value of the difference between the integrals over OABC and OA'B'C may be made arbitrarily small by taking X sufficiently large, as we shall show presently, thus again rendering the integral over the triangle OA'C equal in the limit to s\Si. In order to show this, let us represent by I{S) the integral in question when extended over any given area S. Also let G{S) be the corresponding integral when the absolute value of the integrand is used. We then have (45) \I{OABC) - 1(0 A' C) I = \IiCBC') + I(AA'B) \I(CBC)\+\IiAA'B) I < GiABCCB'A'A). Since ABCCB'A'A = OA'B'C - OABC and since the integrand of G(S) is always positive, the last member of (45) may be written in the form (46) /^2A /^2A /»A /^A I e~^\u{x)\dx I e~"\v{y)\dy — I e~^\u(x)\dx I e~"\viy)\dy. Jo Jo Jo Jo 94 Studies on Summability Moreover, since the series (36) are by hypothesis absolutely summable, each of the iterated integrals in (46) approaches the same limit when X = co, so that the expression (46) itself approaches the limit zero. We may therefore in all cases write (47) S1S2 = Yimj J e~''^'^^hi(x)v{y)dxdy, A=oo where the integration is performed over the right triangle OA'C, the length of whose side is 2X. This result being premised, let us now introduce into the second member of (47) the new variables ^, rj defined as follows: x -\- y = ^,y = ^rj or x = ^(1 — 77), y = ^^• We then have^^ dydx = d^drj = ^d^drj, dx dx a| dv dy dy d^ d-q so that the integral in question becomes (48) e-^^d^ u{^{l - v)mv)dr}. Concerning the limits of integration here, we wish to integrate over that area in the ^, rj plane which corresponds to the area of the triangle OA'C in the X, y plane. Now, the three sides of the triangle are respectively x = 0, y = and X -\- y = 2X, and our first problem is to determine what these bounding lines become in the ^, 77 plane, it being understood as indicated above, that the equations of transformation are .T = ^(1 — r]),y — ^V' Evidently, corresponding to a: = we have the two lines ^ = 0, rj = 1, while corresponding to y = 0, we have the two lines ^ = 0, rj = 0, and corresponding to x -]- y = 2\ we have the one line ^ = 2X. The area bounded by these four lines is that of the rectangle whose vertices (in the ^, 77 plane) are (0, 0), (2X, 0), (2X, 1), (0, 1). Whence, the limits of integration are as indicated in (48). The series for u(x) and v{y), being power series are absolutely convergent. Hence, by the rule for the multiplication of two such series, it appears that the expression u[^{l — 'r])]v{^r]) may be expanded into a series whose nth. term is (49) rE^r^^r,— (l-T?)". r=o^!(^ — r) ! Moreover, this series will be uniformly convergent as regards tj throughout the " See, for example, Gotjrsat, "Cours d'Analyse," Vol. I (1902), p. 298 Absolutely Summable Series 95 interval < 77 < 1 since for all such 77 values the term (49) is less in absolute value than ^ .t^r!(n-r)I and this expression is the nth term of the (convergent) product series obtained by multiplying together the (convergent) series ** I -)/ I ^ 1 11 I »=o nl n=o nl The integration with respect to 77 in (48) may therefore be performed term by term upon the series whose nth. term is (49), thus giving fum - v)mv)dv = i:^"E ..y'""!., fr-^a - vYdv. Jo 71=0 r-Qi-V'' ')-Jo But r n-m ^r, rl(n-r)l Thus we have /»! 00 1 w[^(l - v)H^v)dv = Z) Wn t/0 n=0 (n+1)!' where Wn has the meaning used in condition (E) (§ 40). The integral (48) thus becomes /•2A Jo where and accordingly we have the equation (50) 5i52= re-nva)d^. Jo The second member of this equation is seen to be the sum of the series (51) O + W0+W1-] , so that our final result will now follow as soon as we show that under the existing hypotheses the series (52) Wo + ici + «'2 + • • • is summable by definition (IV). 96 Studies on Suivimabilitt We may in fact show that the series (52) is absolutely summable. Moreover, since we have shown that absolutely summable series satisfy condition (C), it will here suffice to show that the series (51) is absolutely summable — i. e., that the integrals re-f|irW(?)M^; A; =0,1,2,3, ••• converge. The proof of this presents no difficulties and will therefore be omitted.^^ 43. Uniform Summahility . — Following analogy with uniformly convergent series, Hardy-^ has proposed the following definition of uniform summability for divergent series, basing the same on the form (IV) (§ 36) of definition of sum: Definition I. If (instead of the series of constant terms (11)) we have the series (convergent or divergent) in which each term Unia) is a function of the (real) variable a, this series is uniformly summable throughout the interval jS < a < 7 if for these values of a the integral 00 /»00 ^Un{a) = I e~''u{x, a)dx Jo converges uniformly, wherein t(x, a) = J2un(.ci) ni. I* Upon the basis of this definition the following theorems analogous to those encountered in the study of uniformly convergent series may be established:^^ Theorem I. " If all the terms Un{a) are continuous functions of a and IXl Sw„(q;) n=0 is uniformly summable, and Y.Un{(x)—. n=o ni uniformly convergent for any finite value of x, in an interval ((3, 7), the sum of the first series is a continuous function of a throughout the interval." Theorem II. "If 00 is uniformly summable in {ao — ^, cxq-\- ^) a7id 26 Cf. Bromwich, I. c, pp. 282-283. " See Transactions Cambridge Philos. Soc, Vol. 19 (1904), p. 301. 28 Cf . Hardy, I. c. Uniform Summability 97 n=0 "• • uniformly convergent for any finite value of x, the series n=0 may be differentiated term hy term for a = ao" Theorem III. "If 00 (53) Sunioc) 71 = is uniformly summable in (/3, 7) and I] Wn(a)— j 71=0 n i uniformly convergent throughout the domain (0, X, /3, 7) however great he X, the series may he integrated term hy term over (/3, 7)." Extensions of Theorem III to cases in which (53) fails to be uniformly sum- mable in the neighborhood of a finite number of isolated points within (/3, 7) and to the case in which /3 = 00 have also been obtained. It would appear, however, that with the indicated meaning for 00 Sw7i(a), 7!=0 Theorems I, II and III together with their generalizations relate in substance to the properties of definite integrals of a certain prescribed type rather than to the subject of infinite series, the latter appearing merely in the role of suggesting the type in question. For this reason the notion of " uniform summability," at least as formulated upon the basis of definition (IV) (§ 36), together with the resulting theorems appear somewhat artificial. This seems less true, however, in case definition (I) (or (II)) is adopted. Thus, confining ourselves for sim- plicity to the important case in which r = 1, we then have the following Definition IIP A series (convergent or divergent) (54) JlnM) n=0 in which each term Un{oc) is a function of the (real) variable a, is uniformly summable throughout the interval /3 < a < 7 if for these values of a the ex- pression 5o(o;) + Si{a) + • • • + Sn{a) 71+ 1 lohere Sn(oc) = Voia) + Ui{a) + • • • + Un((x) converges uniformly to a limit U(a). " Cf., for example, C. N. Mooue, Transaclioiis American Math. Soc, Vol. 10 (1909), p. 400. 98 Studies on Summability The theorems corresponding to I, II and III now become considerably more direct. Thus, corresponding to Theorem I we evidently have the following: " If all the terms ?/„(«) of the series (54) are continuous and the same series is uniformly summable throughout the interval ((3, y), then its sum U{a) is con- tinuous throughout (j3, 7)." The corresponding forms for Theorems II and III can be at once supplied. Supplementary Remarks and Theorems 44. From §§ 41-43 it may be concluded that of the six definitions of " sum " in § 36 those deserving of especial emphasis are (I) (Cesaro) or its equivalent (II) (Holder) and (IV) (Borel). We now add certain noteworthy results respecting (I) and (IV), omitting proofs in cases where suitable references can be given. 1. If a series (convergent or divergent) is summable by Cesaro' s method for a given value of r (cf. § 36), it is summable by the same method for all larger (inte- gral) values of r. In fact, with *S„^''^ and D^^''^ defined as in (12), we have the identities ^.C+l) = So('-> + ^i^'-) + So^^^ + • • • + Sn^^\ D,^r+1) = J)^ir) _^ J)^ir) _^ J)^^ir) + . . . + J)n^^\ and since by hypothesis lim Sn^^'^Dn'-''^ exists, it follows from a well-known theorem due to Stolz''" that lim Sn^'^^^/Dn^'^^^ also exists and has the same value, provided however that as n increases »Sn^''^ eventually does not oscillate but is such that lim »S„^'"^ = ± co — a condition here fulfilled because by hypoth- n=oo esis lim 1 cannot be summed by Cesaro's method. Thus, in particular Cesaro's formula cannot serve to prolong analytically the power series (56) outside its circle of convergence. In fact, placing w„ = a„a;o" we have and hence Whence T Ufl lim = .To n=oo ^n — 1 — — = Xo-\- €„; hm €n = 0. Un = WoGto + eOCa-o +62) • • • (a:o + €„). Now, having chosen an arbitrarily small positive quantity 77, we have | Cn | < ^ for all n > a determinate value n^, and hence I .^0 + €n I > I Xo I — I r? I ; n > nr,. Thus, as n increases indefinitely the expression w„ becomes infinite to as high an order as that of (|a-o| — It?!)"*. But for a sufficiently small choice of 77 we have |a:o| — [r/l > 1, since by hypothesis |a-o| > 1. Thus (55) cannot be satis- fied for any value of r. In contrast to this result, we have the following important theorem arising when, instead of the definition (I) of sum, we adopt the definition (IV) of Borel. 4. Let fix) = 12 Ctn-T" be any power series having a radius of convergence equal to 1. //, then, the series 71=0 is summable by definition (IV) (§ 36) so also is the series 00 S aniTo" n=0 provided xq lie within the polygon formed by tangents to the given circle at the points {assumed finite in number) upon the circumference at ichich f{x) has singularities. Moreover, f{x) may be extended analytically to all such points xq by mean^ of the sum formula in question, i. e., 100 Studies on Summability /(•^o) = I e~'^u{aXQ)da Jo where , , ^aniax^Y u{oiX^ = 2^ ^ — . n=0 ^ • r/ie summahility at xq will be absolute (§ 42) anc^ i^ will be uniform (§ 43) throughout any region situated ivholly within the indicated polygon (polygon of summability).^^ 5. Absolutely convergent series are absolutely summable, but series that are merely convergent may not be absolutely summableP 6. // but one of tico series is absolutely summable while both are summahle by definition (IV) {BoreVs integral) to the respective limits Si, S2, then the product series (cf. (37)) is summable by the same definition to the value Si, S2, but not neces- sarily absolutely summable. ^^ 7. If two series are summahle by definition (IV) {BoreVs integral) to the values S\, S2 respectively, then the product series (cf. (37)) whenever summable necessarily has the sum sis^.^^ 8. If the coeficients ui, U2, u^, ••• of the divergent series (11) are such that the expressions Eo = Wo, El = Uo-\- ui, E2 = Uq + 2Ui + U2, /r js Es = Wo + 3wi + 3w2 + W4, w(n -1) , , , , En = wq + nui H ^-j — u^-f- • • • + nw„_i + w„ all vanish after a certain point : n = m, then the series may be summed by definition (IV) {BoreVs integral) and the sum will be — ^ A- ~ -i- ^ -\- I -^"t *~ 2 '2- 2^ 2"'''"^ — i. e., the sum will be given by summing the series by Euler's well-known method for converting a slowly convergent series into a more rapidly converging one.''® '2 Proof of the various statements here made is readily supplied from the remarks of Brom- WICH, I. c, § 113. 33 Cf. Hardy, Quarterly Journ. of Math., Vol. 35 (1904), pp. 25, 28. 3< Cf. Hardy, I. c, pp. 43-44. 3* Cf. Hardy, I. c, pp. 44-45 " Cf. Bromwich, I.e., § 24. Supplementary Remaeks 101 This result evidently becomes of especial significance for all series (11) of the form «o — «! + 052 — «3 + • • • ; flm positive for which the successive differences between the quantities Qq, Oi, oz, • • • all eventually vanish — e. g., the series 1-2+3-4+5- .-., wherein the quantities ^o, Ei, E^, etc., become J^o = 1, ^1 = - 1, E2 = Es= ■•■ = En^O, and hence * = 2 ~ i = i- The proof of statement (8) may be readily supplied when we make use of the Lemma of § 37. Thus, in the notation there employed, let us take in the present instance Sn 2 ^ 22 ^ 23 ^ ^ 2"+i ' "" ~ ~^ • Then 7 T Eq El , Em and condition (A) of the lemma is at once seen to be satisfied (cf. (16)). Application of the lemma thus gives I = lim e-^P Z '^H^ Sn = lim e-'"5(2a) = lim e-"5(a), p=co n=0 ni a^s-i) the points (q in number) at which f{x) becomes infinite in the (closed) interval (— tt, tt), assuming at first for simplicity that Xi ^ — TT, Xq ^ TT. Having chosen an arbitrarily small positive quantity CO, let us also suppose at first that the value x = a — e lies within one of the following intervals: (8) (— IT, Xi— 0)), (.Ti + CO, a'2 — co), •••, (a:g+co, — x), i. e., let us assume that a: = a — e is not one of the points at which f{x) becomes infinite. We may then express the integral in question in the form (9) 1 f(x)cp(n, X - a)dx ^ S + R, where + + • • • + + ]f{x)^{n, X - a)dx, (g ^ q) and + + • • • + ) f{x)2, s be the fluctuation of f{x) in the 5th one of the intervals 62, • • •, let Dq+i,s be the fluctuation of /(.r) in the 5th one of the intervals dq+i. Finally, let us form the sums Proof of Theorems io7 (10) hilDr,,, hJlD,,,, ..., 5,+ii:Z) «— 1 s=l .-1 «=1 3+1, 8- Since /(.t) is integrable over each of the intervals (8), it follows that we can make our choice of the integer p so large that each of the sums (10) will be less in absolute value than the preassigned quantity p already mentioned. At the same time, y may be chosen so large that each of the integrals (11) ( \f{x)\dx; ,= 1,2,3, ...,9+1, where the integration is performed over any one of the intervals 5„ will like\^^se be less than p. In what follows the quantity y will be understood to be any special one determined according to the two conditions just indicated. Returning to the expression S, let us now consider the first of the integrals of which it is constituted. Calhng x = ^,_i, x = ^, the values of x corresponding to the end points of the 5th one of the intervals 5i, we have 'J-n s=iJ^,_i [ (p = (p{n, x — a). Now, introducing the constant B defined in (III), we may write f'f' n„, - 1< 01 < 1, a' < a < h'. n > n„, a' n^, where M represents the upper limit of |/(.r) | in the intervals (8). Similarly, all the g -\- \ constituent integrals of S, except the last, may be thus treated, thereby leading to the equation (14) S = P + /^a—e I / • (pdx, where for all values of n greater than some value independent of a we have I P I < 2gMp(x + gBp ^ 2qMpa- + qBp. Let us consider finally the integral appearing in (14). For this we first note that the interval of integration consists of a portion (or at most the whole) of the interval {xg + co, Xg+i — co) belonging to the set (8). Let us suppose that rjr < oi — € ^ r)r+i where rjr and r]r+i are the values of x corresponding to the extremities of the rth of the p divisions of length 8g+i into which we have already divided the interval {xg + co, Xg+i — co). We may then write I / • (pdx =1 / • (pdx +1 / • (pdx. The last integral here appearing is less in absolute value (cf. relations (III) and (II)) than (15) B r ' \f(x)\dx < B P" \f(x)\dx < Bp where p has the meaning already given. Proof of Theorems 109 Again, let there he I {I "^ p) of the divisions 8g in the interval {Xg + co, r]r). Then, treating the first integral in the second member of (15) as we did the first integral in S, we obtain (cf. (13)) I / • (pdx < 2Mh + Bp^ 2Mp(7 + Bp; n > n„, where n^^ is independent of a. In summary, then, we have the following result: Let xi, xi, x^, - • ■ , Xs, • ■ - , Xq', (Xs > Xs-i); {xi 4= — TT, a:, =# x) represent the q points within the interval (— TT, tt) at which /(x) becomes infinite, and let a be any value such that a — e (cf. (7)) lies within one of the intervals (— TT, Xi — Oi), (.Ti + CO, .T2 — Co), •■•, (Xg + CO, Tt) ; 03 arbitrarily small and positive and also such that — x < a' ^ a ^ b' < tt. Then, corresponding to an arbi- trarily small positive quantity p and a second such quantity a, we may determine a positive value n^ independent of a and such that £ f(x)cp(n, X - a)dx < 2pM(q + 1)(t + B{q + l)p; n > n. Since B, q, M and j) as well as n^ are each independent of a, it follows that for all the indicated values of a the first integral in the second member of (7) converges uniformly to zero when n = oo . It remains to show that the same is true when a — e pertains to one of the intervals of the following set : {xi — CO, Xi-\- co), {X2 — 0}, X2 + co), ' ", (x g — CO, X q -\- oi) ; 03 arb. small and positive. The desired result follows by reasoning directly analogous to the preceding after rewriting (9) in which S and R are, however, defined as follows: + + • • • + f{x) ai, where ai has the meaning given in the Theorem. Thus, we reach the desired result respecting the third term in the second member of (7) and similarly, we reach the indicated result for its last term. 47. We turn to the proof of Theorem II. It is our purpose here to show that relations (I) and (III) of § 46 together with the following suffice for the proof: (II)' Having placed (19) Hn,t) =-^. 12^{n,t), n -\- 1 n=0 where (p(n, t) is the trigonometric expression (5), we may write for a given value of the positive quantity e and all subsequently chosen sufficiently large values of n £ \^{n, t)\dt< C; where C is a constant (independent of both n and e). In proving Theorem II we shall therefore substitute relation (II)' for relation (II) of § 46, but we shall employ relations (I) and (III) as before. Assuming first that a has any special value such that — it < a' ^ a ^ b' < tt, we have from (7) ^, ^ [■^o(a) + si{a) + . . . 4- Sn{a)] = I f(.v)^{n, x — a)dx (20) + I f{x)^{n, x—a)dx-\- I f{x)^(n, x — a)dx + 1 f{x)^(n, x — a)dx, where $ is given by (19). 112 SUMMABILITY OF FoURIER SERIES AND AlLIED DEVELOPMENTS Now, the fact that

for ip, and since satisfies relations (I) and (III) it follows precisely as in the discussion in § 46 that the first two integrals on the right in (20), when considered for values of a such that — tt < a' ^ a: = 6' < tt, converge uni- formly to zero as n = co, provided merely that the integral (4) exists. The third term of (20) may be written in the form /(a - 0) j ^{n, t)dt + J [/(a + - /(« - 0)Mn, t)dt, provided that f(a — 0) exists. When n = the first term here appearing approaches the limit |/(q; — 0) since, as already pointed out, $(??, t) satisfies (I). As to the second term, we may choose e so small that throughout the interval — 6 < i < we shall have \fia -{- t) — f{a — 0)\< 0)• + r /(tt + 0«p(^i, 0^^ + r /(- vr + Ov(^, 0^^ We may now show that as n = oo the limit approached by each of the first two integrals here appearing is 0. In order to do this it will suffice, since the integral (4) exists, to show that the property just indicated is true of each of the integrals J /(tt + t)cp{n, t)dt, J /(- TT + t) the expression (25) approaches the limit |/(7r — 0). Likewise, as n = oo the last term of (23) is seen to approach the limit |/(— tt + 0). In case x = — ir (instead of x = tt) we have the following equations corre- sponding to (22) and (23) : Sn(- Tr) = j V(- ^ + t) no- provided a and I are assigned values consistently with the relations a' < a. tie I «/ I provided a and I are assigned values consistently with the relations a' < a < b'; e'^t^b — a. It may be added that in case one confines the attention to the convergence of the integral (27) for special values of a (thus not considering questions of uniform convergence) it suffices that relation (I) shall be satisfied for each special value oi a {a' < a < b'). Similarly, the con- stants A and B of (II) and (III) may then depend upon a. Formation of General Theory 117 Proof.— The proof of this theorem is readily supplied upon referring to the methods employed in § 46 for the study of the integral (6). We shall therefore merely indicate the essential steps. Representing the integral (27) by *„(«), we first write (cf. (7)) m ice shall have for all subsequenthj chosen sufficiently large values of n OT' j \Hn,a,t)\dt< C where C represents a constant independent of n, a and e. Also, let fix) be any function which satisfies condition (A) of § 51 together with the following : {BY When considered in the neighborhood of the {special) point x = a (a' < a < b'), the limits f {a - 0), f{a + 0) exist. Then ive shall have for the (special) value of a mentioned in (B)' (31) hm I f(x)^{n, a,x- a)dx = \ [f{a - 0) + f{a + 0)1. ^ Moreover, if {instead of condition {B)') f{x) is continuous throughout the interval (a', b'), the points x = a', x = b' included, we shall have uniformly for all of the same values of a lim I f{x)^{n, a, x - a)dx = /(a). 118 SmiMABILITY OF FoURIER SERIES AND ALLIED DEVELOPMENTS The proof of this theorem, Hke that just indicated for Theorem I, is at once suppHed upon following the steps indicated in § 46 with reference to the special integral (6) there occurring. We therefore omit it. 53. As a generalization of the Theorem III of § 46 we have the following Theorem III. Let (p(n, a, t) he a function of the real variables n, a, t loliich, lohen considered for the special values a = a, a = b (b > a) satisfies the following four relations in ivhich n is restricted to positive integral values and in which e repre- sents a positive quantity ichich may he taken arbitrarily small: (Da.i lim I (p{7i, a, t)dt = — | when a— b-\-€^t^ — e, n=oo t/0 lim J (p(7i, h, t)dt = ^ when e '^ t ^ 6]— a — e. (II)a,b Relation (II) of § 55 is satisfied ivhen a = a and t lies in the interval ^ i ^ e; also when a = h and t lies in the interval — e ^ t ^ 0. I I (p{n, a, t)\ < B ivhen a— b-\-e'^t^ — e, ^ 1 ! 'o \ Jo Je Jt)~a-e / Xb—a—e /»0 f{a + i)(p(n, a, t)dt + j f{h +tMn,h, t)dt + 1 /(a + t)(p{7i, a, t)dt, Jo from which we deduce the indicated result as before. 54. Again, we have (cf. the remarks in § 49 on Theorem IV) the following Theorem IV. Let (p(n, a, t) he a function of the real variables n, a, t which, when considered for the special values a = a and a = b satisfies relations {T)a,b, (III)a,6 and (IV) of § 54 and also the following : The integrals (II)'a,6 j |$(n, - 1, t)\dt, j |$(n, 1, t)\dt (€ > 0), when considered for all values of n sufitciently large remain less that a constant (independent of e). Also, let f(x) be any function which satisfies condition (A) of § 51 and is such that the limits f{a + 0) , /(6 — 0) exist. Then we shall have X>> pb f{x)^{n, a,x — a)dx = hm I f{x)^{n, b, x — b)dx = H/(&-0)+/(a+0)]. 55. Besides the relations given in Theorems III and IV concerning the func- tions (p{n, a, t) and (p{n, b, t) (which relations are satisfied in particular by the function (5) pertaining to Fourier series, with a = — tt or a = tt) it is important to note certain others which we shall find fulfilled by some of the functions (p{n, a, t) met with in the succeeding pages but which are not fulfilled by (5). These relations together with their effects upon the limiting values of the integrals I f{x)(p{n, a, X — a)dx, I /(.r)$(?i, a, x — a)dx *Ja Ja we now summarize in the following four theorems : 120 SUMMABILITY OF FOUBIEE SeEIES AND AlLIED DEVELOPMENTS Theorem V. Let (p(n, a, t) he a function of the real variables n, a, t which, when considered for the special value a = a satisfies the following three relations in which n is restricted to positive integral values and in ichich e represents a positive quantity which may he taken arbitrarily small: (I)a lim I a), Gi being a constant (independent of t) . (II) a Relation (II) of § 51 is satisfied when a = a and ^ i ^ e. (Ill)a \ a), G2 being a constant (independent of t) . (11)6 Relation (II) of § 51 is satisfied when a = b and — e ^ t ^ 0. (111)6 \J a where is defined by (30) . Theorem VIII. Let (p{n, a, t) be a function satisfying relations (X)b and (111)6 of Theorem VI but, instead of (II)^, the following : {ll)b Relation (II)' of § 52 is satisfied when a = b, it being understood that the integration there appearing is then taken from — e to instead of from — e to e. Also, let f{x) be any function ivhich satisfies condition {A) of § 51 and is such that the limit f(b — 0) exists. Then tve shall have lim r fixMn, b, x - b)dx = 6^2/(6 - 0), n=oo tJa where is defined by (30) . The first of the Theorems V, VI results directly upon writing Xb—a / /"e nb—a \ f{a + t)ip{ii, a, t)dt = ( I + 1 )/(«+ t)^{n, a, t)dt; e > and then applying to each of the last two integrals the methods already used in § 48 for the study of similar integrals. Theorem VI likewise results upon writing (32) Snib) = r /(6 + tMn, b, t)dt =( f + f ^ )f{b + tM7i, b, t)dt. Ja-b \ J-e Ja-b J The proofs of Theorems VII and VIII being likewise readily supplied, are suppressed. 56. We proceed to make certain observations which will prove useful in applying the general theorems of §§ 51-55 to special integrals (27). (1) If in applying Theorem I of § 51 it is found that for some special value of t different from zero, ^ = ^1 4= say, the function (p{n, a, t) becomes infinite or otherwise is of such a character that uncertainty arises concerning any one of the relations (I), (II), (III) when t = t\, then the theorem will still hold good provided that it can be shown that the integral h= { \f{a+t) 0, approaches (n = 00 ) uniformly the limit zero for a < a' ^ a ^ 6' < 6, or else is such that for the same values of a and 122 SUMMABILITY OF FoURIER SeRIES AND ALLIED DEVELOPMENTS for all (positive integral) values of n the same integral approaches uniformly the limit zero as ^ = 0. An examination of the method used in proving Theorem I shows at once the correctness of this remark. More generally, in case of uncertainty of any kind in the behavior of f{a + t)(p{n, a, t) for the value t = ti ^ (a — a(n, a, t) shall satisfy relations (I) and (III) of Theorem I together with (II)' of Theorem II. This follows from the fact that the conditions placed upon (p{n, a, t) in Theorem II are there inserted merely that $(w, a, t) may have the properties just indicated, the latter being those upon which the proof in reality depends. Similarly, in using Theorems VII and VIII the conditions stated relative to (p{n, a, t), (p{n, b, t) may be replaced by the same conditions referred to ^{n, a, t), $(n, b, t). (3) Assuming that relations (II), (III), {A) and (J5) of Theorem I are satis- fied, let us suppose that instead of relation (I) we have the following :^^ (I)' lim I ip{n, a, t)dt = n=oo c/q I ~" 2 H~ x(oi, t) ivhen a — a ^ t ^ — h + x(aj when e ^ t ^ b — a, where x(a, t) is any function of a and t such that (a) Having given an arbitrarily small positive quantity a, one may determine a positive quantity ^ dependent only upon a such that " As in (I) of § 51, it is here to be understood that the convergence (n = oo ) is uniform for the indicated values of a and /. Formation of General Theory 123 a' ^ a ^ h', xia, t)\< a when i _ t < / < t (6) The partial derivative dxfdt exists whenever a' ^ a ^ b', a — a '^ t b — a and for the same vakies of a and t is such that dx dt < D = constant independent of a and t. Under these conditions it is easily seen that the function (p{n, a, t) — dxl^t comes to satisfy relations (I), (II) and (III) of the theorem of § 51 from which it follows that for a fixed value of a such that a' ^ a < 6' we may write - I /(^) -^1 dx + hm I f(x)(p(n, a, x - a)dx = - S^^ • tJa L '-'^ J(=x— a n=co *Ja ^ Moreover, if (instead of condition (B)) f{x) is continuous throughout the interval a' ^ x ^ b' , the end points x = a', x = b' included, and has hmited total fluctuation throughout an interval {a\, bi) such that a < ai < a' < b' < bi < b, then for all values of a in (a', b') the equation will hold true uni- formly, it being understood that the right member is then replaced by /(a). Analogous remarks relative to Theorems (III), (V), (VI) are readily supplied. Ill The Calculus of Residues as Applied to the Series Developments for AN Arbitrary Function.^^ The General Problem of Sturm 57. A comparison of the developments occurring in mathematical physics for a function /(.t) of one real variable x shows that they are ordinarily of the form (33) J f{x)F{x)Hi{\n, x)dx J f{x)Fix)H2{\n, x)dx Ih{\n, X) -^^h 1- H2(\n, X) ~^^ F(x)Hi'(Kn,x)dx F{x)H2\\n, x)dx I f(x)Fix)Hm{\n, x)d:i *J a j' F(x)HJ{\n, x)dx + • • • + i/„.(X«, x) where //i(X„, x), H2(kn, x), • • •, HmO^n, x) are m functions of x and of a certain parameter X which takes different values from term to term in (33) according to some given law, and where F{x) is a function of x only which is finite throughout the interval (a, b). Thus in the case of a Fourier series we have m = 2, i/i(X„, x) = sin nx, -^2(Xn, x) = cos nx; and a = — t, b = ir, F{x) = 1. Again, in deahng with the usual expansion of f{x) in terms of Bessel's function of order zero, we have m = 1, //i(Xn, x) = Jo(Xn, x), a = 0, b = 1, F{x) = x, Xn being one of the roots of the transcendental equation Jo{x) = 0. It is to the important developments (33) that we shall hereafter devote our attention. The first n terms of (33) when considered for any particular value of x such as X = a may evidently be put into the form b f{x) u{z) 2 — X„ [e{z)w\z){z - \n)% = e\\n)[l0\\n){z - X„)^] + e{K)[w'{z)(z - ^n)']',. W'(X„)2 u'Oinf ' Calculus of Residues 127 relations (43) and (44) may be written in the form (45) ±.rm,,^±eiKi 2inJc„u{z) „=iw(X„)' ^^^^ 2-KiJcyiz)'^^ ntlU'(Xn) u' (KnY \' It is desirable to note also that if in (46) we substitute d{z)^{z) for ^(2) we obtain so that if \l/' (Kn)u' (hn) — \p(XnW{Xi) = we shall have eWnrnXn) 2*'(X.)^ 59. We proceed to apply the results in (45) and (47) to the sum (37) which defines the integral (36) whose properties are desired in order to investigate the convergence of (33). Let us suppose that for the given value of a we can construct a function 6{z) which shall be analytic throughout the finite 2 plane and such that its value at the points Xi, X2, • • • , X„, • • • shall be given by the equation ^ //,(Xn, a) f F(a + t)H,(Kn, a + t)dt (48) OiK) = E 'y^ u'(K). As a result of (45) we shall then have (49) r (^) + F,{x)]H{z, X) = 0, where K(x), F(x) and Fi{x) are functions of x only, while v{z) is a function of z only. In such cases the developments (33) assume the form (54) JlqnH{\n,x), 71 = 1 where (55) Qn = r f{x)F{x)H{\n, x)d:^ f F{xW(\n, X)dx We first proceed to note certain general consequences which flow from the above restrictions upon H{z, x). From (53) we have (56) ^(^(•'^)^^— ) + {^(•^)KXJ + F^{x)]H(K, X) = 0, 1^ It is to be observed that if in (49) the function d{z) has singular points within C„ the formula continues true provided that the sum of the residues of the right integrand at such points be subtracted from the second member. Similar remarks evidently apply in (51) if 0{z)i(/{z) has singular points within C„. 1* Cf. DiNi, " Serie di Fourier, etc.," §§ 90-96. The problem here presented has been the subject of numerous and extensive researches in recent years, but usually under the assumption (not here introduced) that the differential equation (53) in terms of whose solutions the proposed development is to be made, shall have no singular points within the (closed) interval (a, b) for which the same development is to hold. But this assumption unfortunately rules out some of the most important special developments, such as those in terms of Bessel functions and Legendre functions. For summary' remarks upon the more recent researches of this character, see Bocher's address before the International Congress of Mathematicians at Cambridge in August, 1912, § 11, Problem of Sturm 129 (57) ^(^(•^)^^^^) + {F(^>0 ^-^'(X^, x) dH(Kn, x) 1 - A (a) H(Kn, x) ^ H(Km, x) — = 0, provided m 4= w- Moreover, among the different ways in which this relation may exist is that of supposing that for every value of n we have the following two equations simultaneously: (60) K{x) ^ h'H(Kn, x) = when x — a, K{x) -~^ hH{\n, x) = iDhen x = b, h and h' being any real constants, including the limiting values A = ± °o , h' = dz ^ corresponding to which the same equations become //(X„, a) = and //(Xn, 6) = respectively. We shall hereafter confine our attention to the cases in which relations (60) are satisfied. Furthermore, if K(a) =# 0, K{b) 4= 0, we shall suppose that the transcendental equation v{z) = whose roots deter- mine the quantities Xi, X2, X3, • • • is taken in the one or the other of the two following manners: (61) u{z) = [/i(.T) -f^ - h'Hiz, X) ]^ = 0, (62) u{z) = [/v(a') ^"^ - hlliz, x) ^ = 0, 10 130 SUMMABILITY OF FOXJUIEE SERIES AND AlLIED DEVELOPMENTS thus rendering one of the two relations (60) satisfied at once. Similarly, if K(a) = 0, K(b) 4= we shall use (62). In this case it is to be observed that we have merely to place h' = (u{z) having been chosen as indicated) to have equations (60) satisfied ichatever the solution H{z, x) of (53) chosen to be used in (54). Likewise, when K{a) =}= 0, K{b) = we shall use (61) in which case the solution II{z, x) of (53) to be used in (54) may be chosen arbitrarily. Finally, if K{a) = 0, K{b) = the equation u{z) = may be taken arbitrarily together with the solution II {z, x) without destroying the coexistence of (60). 61. We add that if the solution II{z, x) considered as a function of the two variables z and x is finite and continuous together with its first and second partial derivatives : dH/dx, dH/dz, d'^H/dxdz for all real values of x such that a ^ x ^h and for complex values of z in the neighborhood of each of the points z = X„, and if the equation (61) {h' finite or infinite) is satisfied identically for all values of 2 in these regions, then it is easy to evaluate each of the integrals (63) f F{x)H\\n, x)dx, which appear in the coefficients g„ of the series (54). In fact, if we change \m to z, as we may now do, and integrate from a to a; (a < X < h) we shall have by (58) and (61) J^Fix)H(z, x)H{\n, x)dx = ^^^^ } ^^^^ ^K{x) [h{z, x) dll(\n, X) dx dl dx HQ^n, x) — ^^ — j J , and this holds true for any value of z in the indicated regions. Whence, upon allowing z to approach the value Xn we obtain under the present hypotheses \dH{\n,x)dH(\n,x) £F(xWiK,x)dx = ^^[Kix){ d\n dx - //(Xn, X) d\ndx J J ' where if desired X„ may be changed to z for values of z in the indicated regions. Passing now to the limit as a: = 6 we obtain in which as above we may replace Xn by z provided z has values in the indicated regions. Pkoblem of Sturm 131 Finally, by use of the second of equations (60) we may write (64) in the following form when h is finite : (65) £n^)u^,^^,^),.-^[m^.,^)\k'^^-^ -^w^l^^ll- (66) In like manner, if A = ± co so that i/(Xn, 6) = then (64) may be written dH{\n, x) dH{\n, x) £ F(x)H'{\n, x)dx = -^,^^^^^ ^^Kix) d\n dx 62. Expressions (64), (65) and (66) thus enable us to find under special con- ditions the value of the integral (63). Among the cases in which the same special conditions cannot be satisfied, the following are to be especially noted. If, as we have supposed, F(x) and //(X„, x) are real when x is such that a ^ X ^ b and if in this interval F{x) does not change sign, then the integral (63) cannot be equal to zero. Whence, under these conditions (64) cannot be used if K{b) = {h finite or infinite) or if (67) H(Kn,b) = {h finite) or if (68) veH(K,x)i^ remx.,x)^ L 5X„ J, ^ o*^ L dx i ^ {iitnjimte) or (as appears from (65)) if (69) L ^-axr^ - ^^-""^ -^Kdx^ i = ^^'fi'''^'^' 63. Returning then to the series (54) and assuming that the quantities Xi, X2, X3, • • • are taken as the positive roots of the equation (62) while the equa- tion (61) shall be satisfied identically for all values (real or complex) of z in the neighborhoods of the same values; assuming also that the partial derivatives of H{z, x) exist and satisfy such other conditions as we have imposed in § 61, we may say that unless K{b) = or one of the conditions (67), (68) or (69) is satis- fied, we shall have for such developments when h is finite uiz) = [k{x)^--^^ - hlliz, x)'^^, J F(x)IP{\n, x)dx = -^-^ [/i(X„, .r) I a' dllO^n, X) din (70) -^(^)-ax^„aV--|Jr"7(>o''^'-'^- On the other hand, if A = ± <» , we shall have u{z) = II(z, b), - Rn, 132 SUMMABILITY OF FOURIER SERIES AKD ALLIED DEVELOPMENTS (71) f n.yHKK, .) sin 2 u (z) = — 2 sin z + cos z -j- p cos z = — 2 sin 2 + (1 + p) , ^"(2) = — sin 2 — 2 cos 2 — sin 2 — p sin 2 = — [2 sin 2 + ii{z)], and hence (33) j w'(Xn) = - ~ [X„2 + P(P + 1)], L u"(Kn) = — 2 sin X„, so that (84) I sin^ Xnir dx = - Jo X„2 + p(p+l)' Let us now avail ourselves of formula (77).^'' In order to do this we are first to determine the function \{/{z) according to the condition A possible choice of 1^(2) is xpiz) = 2^ + p(p + 1) since from (83) we have u['(Knl = 2X„ Assuming that \l/(z) has been chosen in this manner, we now have to deter- mine a function ^(2) according to the condition (78) which, by means of (84) becomes in the present instance sin Xno: I sin X,i(a + t)dt e'(K) = 2[x.2 + p(p + 1)] -^^ '/'(Xn) 2 sin X„a I sin X„(a + t)dt. Jo 20 DiNi has shown through an elaborate investigation that this formula will always lead to de- cisive results whenever the solution H{z, x) has the special form II (zx); that is, when the variables 2 and X enter only through their product. (Cf. "Serie di Fourier, etc.," §§ 97-109.) The well- known developments in terms of Bessel functions form a special class of this kind. 136 SUMMABILITY OF FoUllIER SERIES .^J^D ALLIED DEVELOPMENTS Hence, let us take 6{z) such that e'{z) = 2 sin za j sin z{a + t)dt = I [cos iz — cos (2a + t)]dt. Jo Jo In particular, let us take e{z) = j I [cos tz - COS (2a + t)z]dzdt = I — ~ ^ dt. Formula (77) thus becomes r If z' + p(p + 1) r [sin tz sin (2a +0^ 1 ,, , Jo ^^'^' "' ^^'^^ == 27^ X [.cos2+2'sin.]2 i L ~1 2a + ^ J ^^^' («^) =2^r^^x c„ [2; cos 2 + 2? sin z] 1 + a;i(2;) sin tz dz Q^ [cos z + ^2(2) sin 2]^ t _ 1 r _d^ r [1 + coi(2)] sin (2a+02 j 27ri Jo 2a + ^ Jc [cos 2 + 0^2(2) sin 2p ' where the contour C„ is so taken as to inclose the roots Xi, X2, • • • , Xn and only these roots of the equation ^(2) = and where, in the last two integrals we have placed for simplicity y(p + 1) ,. V ^^1(2) = ^2 ' '^'2(2) = -. We observe at this point that in applying Theorems I and II of §§ 51, 52 to the function (p{n, a, t) of the present development, the values of a and t with which we shall bi concerned are such that r < a' < a < '/ < 1, (86) \ - a^t^l - a, [0 < a' < a ^ 2a + t ^ 1 + a < 1 + b' < 2. Returning then to (85), let us take as the contour Cn the rectangle formed in the 2 plane {z = x -\- iy) by the lines z = x -\- ij, z = x — ij, z = iy, z = k ■\- iy\ j being any positive quantity arbitrarily large and k being any positive quantity lying between X„ and X„+i. Now, the function appearing in the integrand of (85) is an odd function of 2 which remains finite in the neighborhood of the point 2=0 since p ^ — 1. Whence, the portion of the integral in question due to integration over the 2/-axis is equal to zero. Upon the sides which are parallel to the a:-axis we have dz = dx. Whence, considering first the side upon which z = X -\- ij the last integral of (85) extended over this side becomes J_ ndt r 27rWo ^Wo {1 + coi} {sin Ax cosh Aj-{- i cos Ax sinh Aj] where yl = 2a + ^, Di= cos x cosh j — i sin x sinh j + C02, D2 = sin x cosh j + i cos X sinh j. Certain Sine Developments 137 Now, the functions wi = o)i(z), coo = ^2(2) are each less in absolute value than a constant (independent of z) provided \z\>Q= fixed mimber > 0. Thus, we have but to make use of the well-known properties of the hyperbolic functions to see that if we place j = + co the expression above wall approach uniformly the limit zero for all a and t satisfying relations (86). Similarly, we reach the same result for the last integral of (85) when extended over the side upon which z = x — ij. Turning now to the first integral in the second member of (85) extended over the sides upon which z = a; ± ij, we note that and hence, sin tz sin tx . . . sinh tj — — = — - — cosh tjzti cos tx - — - — sin fe , . . . • 1 . —z — = X COS tix cosh tj ± ij cos tx sinh tij, where ti and ^2 are values lying between and t. Moreover, for all values of t under consideration in (86) we have | ^ | < 1 so that if we place j = -\- as before, the first integral in the second member of (85), like its last integral, will approach uniformly the limit zero for all values of a and t concerned in (86). We turn then to the consideration of the last member of (85) when extended over the side of the rectangle C„ which is parallel to the ?/-axis. Here we have z = k -{- iy, dz = idy and, having taken i = + 00 , we see from what has just been said that for all values of a and t in (86) this member reduces to 27r Jo J- 00 1 + ^1 sin iz dy [cos s + a;2 sin 2]^ t (87) _Jl r ^^ r U + ^i) sin(2a+02 27r Jo 2a+ tj_^ [cos z -\- 0^2 sin zf ^' in which it is understood that z = k -\- iy. Now, it sufiices for our purpose to examine the behavior of (87) as k = 00 and we may take for k any number which, at least for all values of ii greater than some fixed value, increases indefinitely with n without at any time being a root of the equation m{z) = 0; i. e., of the equation E = cos z + coo sin s = 0. Thus, we may take k = mr, in which case jB^ = [cos (mr + iy) + (^2 sin {mr -\- iy)]- = coslr y[l + 10)0 tanli y]- and hence 138 SUMM ABILITY OF FoURIER SERIES AND AlLIED DEVELOPMENTS (88) 1 If . 2 ^ tanh^ y + 2iooi tanh^ y E'= ^^YTyV "''"'''' ^^""^y ~ ''' 1 + 2io^, tanh y - <.,' tanh^ y 1 cosh^ y tanh w 5 1 + 7— -^ + r.(, where, upon recalHng the form of 0^2(2), we have 7 = — 2ip and therefore inde- pendent of z, while 5 depends upon z but has a modulus less than a certain fixed number M for all values of | z | > a fixed number ko. Thus expression (87) becomes (89) where y^'IK^+^^^^'^^+W 27r sin tz dy z^ J t cosh^ y -2^1 2^+lLV'+^'"^^^ + ^0 cosh^^ ^^' 1 + - tanh ?/ + -J = [1 + coi] 1 + -tanh 2/ + -J , 2 z" L ^ ^" J so that 5i like 5 has a modulus less than some constant Mi when 1 2 1 > a fixed number = ki. Considering now the terms in (89) which have 2- in their denominator, we see that for all values of a and t in (86) these terms approach uniformly the limit zero as A; = CO . Thus, since (90) sin (2a + 0^ cosh^ y ^ sm {2(x + t)k — cosh^ y + sinh (2q; + Qy cos (2a + i)k r9 i cosh^ 2/ where < a' ^ 2q; + i ^ 1 + 6' < 2 we have, however great 1 2 1 may be, sin {2a -\- t)z so that cosh^ y 1 r dt r* 5i sin (2a + t)z -Jo 2a+U-co22 <2, 27r Jo 2a + ^ J_oo 2" cosh^ ?/ In like manner, noting that sin tz sin tk dy . 1 C^K.. r dy sinh ty Ml a'k' (91) cosh the limit of the first term in the last member as ^ = oo is /i(0)/4 (of. Appendix, Lemma II). But and hence as k = the term just mentioned approaches the limit ^ when ^ > 0. Again, by breaking up the integration in the last term of (94) into that from t = to t = 7] plus that from t = rj to t = t (rj arbitrarily small and > 0) and obser\ing that the function /i(^) has limited total fluctuation in the neighborhood of the point t = 0, it follows that the same term approaches the limit zero as k = 00 (see Appendix, Lemmas I, III). Likewise, if i < we obtain lim Zi = — ^. A-=oo The second and third integrals of (93) may be reduced respectively to the forms yi r' sin kt , T" k^ y tanh y cosh ty ~ 2Tk X ~kr "^^ J_„ FT? ^osh^^^ '^^' r' 7 7 r" ^'" y tanh y cosh tiy I cos kt at I 7^—; — , , ^, ay, Jo J-^k^+y cosh^w ^' where tx is a quantity lying between and t. Since we have alwaj^s P , sin kt ^1. -T:r- ~^.\fy~- "^y (101) , 2 f" , , cosh $y , 3 f" ,, , cosh By , ^-eL-'^y^ J3ii?i ^y-elj ^y^ ^o^y ^y- Whence, upon combining (100) and (101) we obtain P ^ nosh 02/ 1 f" sinh 02/ 2 T* cosh 0y sinh 2/ 4 f " , , cosh dy , 6 r=° , , cosh 6y . + 6^ L ^^y^ c^sh^y ^y ' e-^ L ^^^^ cosh^^ ^' and this equation at once gives (99). Certain Sine Developments 143 Similarly, we may find an analogous form for the integral f »/ — c ^ sinh dy ^ Thus, for the function /i(0) where /i(t) is defined by (73) we may write ,^^^ .. sinr f°° cosh(2-T)y ^ .. f ^ sin r f °° cosh (2-r)y ^ ] _3 p cosh 2y _3/ r°° dy p tanh^ y \ ~ 4 J_„ cosh-* y ^ ~ 4 \ J_„ cosh^ y J_^ cosh^ y ^ J = I ( [tanh yr^ + i [tanh^ 2/]^„ ) = 2. As to the fifth integral of (93) when a = 1, the values of t to be considered are as before those for which ^ t ^ — 1 and for these we have |2a+ t\= |2 + ^!^2 instead of [2 + ^| < 2. The reasoning employed in studying the corresponding integral when < a: < 1 can not therefore be employed. However, if we break up the integral in question into that from t = to t = ei plus that from t = ei to t = t (ei arbitrarily small but > 0) the last of the two integrals thus obtained will have the limit zero as A: = oo since for the values of t concerned we have |2q: + ^|^2— ei<2; while the first of the same integrals may be made arbi- trarily small with ei since by placing tanh y cosh (2 + t)y cosh {2 + t)y ^^^y = *'<2') -^i^lT • ^(^/^ = *•"'•> 2' and applying (99) we see that the integral L y cosh (2 + t)y , -00 ^'^ + 2/^ cosh^ y remains less in absolute value than a constant independent of ei for all values of t such that — €i ^ ^ ^ 0. Similarly, when a = 1 the sixth term of (93) may be neglected in the limit as A: = CO . Thus condition (1)6 of Theorem VI, § 55, becomes satisfied in which in the present instance we have 6^2= — ^ ~ 2 — — 1(q:= 1, a=0, 6= 1). Relations (II);, and (111)6 of § 55 as well as (11)6' are now readily seen to be satisfied (as in the studies already carried out in connection with (93)) so that by virtue of the general theorems of §§ 51-55 we reach in summary the following 144 SUMMABILITY OF FoURIER SeRIES AND AlLIED DEVELOPMENTS Theorem. Iff(x) remains finite throughout the interval (0, 1) with the possible exception of a finite number of points and is such that the integral (102) r\f{x)\clx exists, then the series (103) 2 qn sin \nX in tvhich On = 2 -v o i" — 7 — ; — Tn I f{x) sin \nxdx: p = constant =}= — 1, Xn + p(p + 1) ' I f{x) sin \nxdx; Jo \n being the nth positive root of the equation z cos z -\- p sin 2 = 0, loill converge at any point x (0 < .r < 1) in the arbitrarily small neighborhood of which f(x) has limited total fluctuation, and the sum will be H/(-^-0)+/(.r+0)]. Moreover, the convergence loill be uniform to the limit f{x) throughout any in- terval (a', b') enclosed U'ithin a second interval (ffi, bi) such that < ai < a' <. b' < bi < 1 provided that f(x) is continuous throughout (a', b') inclusive of the end points X = a', X = b' and has limited total fluctuation throughout (ai, bi). Also, if f{x) remains flnite throughout the interval (0, 1) with the possible ex- ception of a finite number of points and is such that the integral (102) exists, then the series (103) ^vill be summable (r = 1) at any point a: (0 < a: < 1) at which the limits f{x — 0), f{x + 0) exist and the sum will be i [/(.r-0) +/(.!• + 0)]. Moreover, the summability ivill be uniform (§ 45) to the limit f{x) throughout any interval {a', b') such that < a' < b' < 1 provided that at all points ivithin {a', b'), inclusive of the end points x = a', x = b', the function f(x) is continuous. Under the same conditions for f{x) when considered throughout the whole interval (0, 1), the series (103), when considered for the value x = \, will converge to the limit /(I — 0) provided f{x) is of limited total fluctuation in the neighborhood at the left of the point x = \ and will be summahle (r = 1) to the limit f {I — 0) ichenever this limit exists. 65. It may be observed that in the exduded case for which p = — 1 the methods which we have followed may be readily altered so as to yield corre- sponding results. In this case the integrand of (85) has a pole at the point 2 = so that this point should be excluded from the contour Cn- Supposing this to have been accomplished by means of a small semicircle extending to the right of z = 0, we may then take as Cn the resulting contour in part rectangular Developments in Bessel Functions 145 and in part semicircular. If the integrations be now carried out as before over the respective portions of Cn, that arising from the semicircle will be equal to — ^r where r represents the residue of the integrand of (85) corresponding to the pole 2=0. Except for this auxiliary term, the reductions are the same as before, so that in applying the general theorems of §§ 51-55 we encounter an application of the remark (3) in § 56. A similar instance will occur in connec- tion with the developments in terms of Bessel functions, to which we now turn, and in that case we shall elaborate the consequences at some length, though such studies will be omitted for the sake of brevity in connection with the present series (103). 2. The Developments in Terms of Bessel Functions. 66. As a second application of the general results obtained in §§ 51-55 we shall now consider certain developments in terms of the function P^(z) defined by the equation where Jy{z) represents Bessel's function of order v. The developments in question are closely related to the well known developments for an arbitrary function in terms of Bessel functions and at once yield, as we shall show, results of considerable generality concerning the summability and convergence of the latter. For the function P^(2;) as thus defined, we have, when v 4= negative integer. P (^^) = '^^^a^ = ^ r 1 v ""> j_ "^^^ (zxY 2T(z/+l)L '"''- ' "" (104) {zxY ~2T(z.+ l)L 22(z.+ l)"^24-2!(^+l)(^+2) {zxf _ 26.3!(^+l)(^ + 2)(^ + 3) while the equation (53) becomes ^ -^+22a;2''+ip^(s.r) = dx or, placing for brevity P^(2;.r) = P, d-P dP (105) ^^ + (2, + 1) — + 22,.p ^ 0. Taking a = 0, b = 1, the development (54) in terms of the functions P,,(X„.r) becomes (106) llqnP.iKx), 11 146 SUMMABILITY OF FOURIEK SERIES AND AlLIED DEVELOPMENTS where (107) ?n = (KnX)dx Equations (60) become Jo x-''+'PJ'{\nX)dx (108) ,^.2.+i p^(X„.r) - hT^iKx) = ichen x = 0, ox 3,2.+i p^(X^3.) _ /iP^(x,,r) = when x = 1. ox Of these the first is seen to be satisfied identically for all values of z if we place /?' = and assume j' > — 1, while the second gives as the equation u{z) = (cf . § 64) of the present developments, dPM u{z) = z dz - liPXz) = 0. We may therefore apply (64) and write dP dP jr'.-.p.w.. = ^['/-'^-p^l or smce we have dP zdP dx dz a^P z d^P d-P dzdx Jx=i dx X dz ' dzdx x dz^ ' dPY_l^dP z ^d^P (109) L by (105). Thus, \i h = ± CO so that u(z) becomes simply P^iz), we have (110) f x-^'+'PJ'iKx)dx = i (£ ^.(2) )]^^ = h^'O^nY and, since we then have by (105), it appears that if we wish to apply (77) in the study of the function (p(n, a, t) of the present developments we may take at once \l/(z) = l/s-""*"^ and ^(2) such that (111) d'{z) = 2z'"'+'P{az) r (« + ty-''+'P{(a + t)z}dt; P = P,. Jo Developments m Bessel Functions 147 On the other hand, if h be finite so that u{z) = zP'{z) — hP(z), we shall have by (109) (112) J\'-'+'F-(Knx)dx = ^^^ {h(2v + h) + X„2}. Now, we have by (105) u'{z) = zP"iz) + (1 - h)P'iz) = - {2p+ h)P'(z) - zP{z), i2v-{- 1)(2j/+ h) u"{z) = - (2^ + h)P"{z) - zP'iz) - P(z) = ^ ^ ^^ ^ ^ P'iz) Whence, - zP'{z) + {2v + A - l)P(s). so that An ^"W = ^^¥ {^(2^ + 1)(2^ + /^) + (2^ - 1)X„2}, u'{Kt ^ 2{/i(2^ + /i) + Xn'l r a:2''+ip2(X„.T)(/.r w''(Xn) 1 Klv + l)(2j' + /O + (2^/ - l)Xn2 2j/ + 1 , 2X„2 W'(X„) Xn /i(2j' + /l) + Xn' Xn ' ^(2j/ + A) + Xn^ * Thus, in order to satisfy the conditions relative to -^{z) in the present case we should take it so that lA'(z) 2j. + 1 2z2 i^iz) Z ' ]l{2v^ /0 + 22- Let us therefore take ]i{2v + A) + z2 i^iz) z ,2»'+l in which case it appears that we may take d(z) as before, viz., such that equation (111) is satisfied. Now we have from (105) with a similar equation obtained by replacing a by o: + i^- Hence, [{a + ty - a'']z'''+'P(az)P{{a + t)z] 148 SUMMABILITY OF FoUKIEK SeRIES AND AlLIED DEVELOPMENTS Placing for convenience a -\- t = (3 and letting accents represent differentiation with respect to z, we may therefore write r z'''+'Piaz)P{^z)dz = J^,[P'(az)P(l3z) - P'{^z)P(az)] Jo pa. so that both when h is finite and when infinite we may take (113) e{z) = 2z^"+' X'^^^- [P'(az)P{(3z) - P'mP(az)]dt. Upon noting the analytic properties of the functions \p(z) corresponding to the two above mentioned cases and of the function d(z), it appears, upon applying (77) that the integral (36) of the present developments will be given by the ex- pression H /,2.+i r P'(az)Pm - P'mPiaz) 1 n 8^"+^ r (114) -: ^2 Jt P\z) dz or according as ii{z) = P(z) or 2i{z) = zP'{z) — hP(z). It is to be noted also that in the developments (106) we shall have by (110) and (112) 2 r^ Qn = p//^ s2 I f{x)x^''+^P(KnX)dx or 2x ^ r^ ^" " [h{2p +h) + \J]P'{K) Jo /(^)^"'"^'^(^-^)^^ according as the quantities Xi, X2, • • • are the roots of P{z) = or zP'(z) - hP{z) = 0. These results premised, let us now consider the rectangle in the z plane whose vertices are the points z = ij, z = k -\- ij, z = k — ij, z = — ij, j being any positive quantity arbitrarily large and k being any positive quantity lying be- tween Xn and Xn+i where Xi, X2, • • • represent the successive positive roots of the equation P{z) = or zP'{z) — hP(z) = according as we are dealing with (114) or (115). From the boundary of this rectangle let us exclude the point 2=0 by means of a small semicircle of radius rj and let us now take the resulting contour in part rectangular and in part semicircular as the contour of integration Now, the function appearing in the integrand of either (114) or (115) is an odd function of z and hence the two portions of these integrals extended along the y axis mutually destroy each other, while in either case the portion extended Developments in Bessel Functions 149 along the semicircle may be made arbitrarily small with rj unless in (115) we have ^ = 0. In this exceptional case the integrand of (115) has a pole of the first order at the point 2=0 and hence, upon applying Cauchy's integral the- orem, the value of the contribution to (115) arising from the semicircle in question becomes - {2p + 2) r ^^''+^dt = q;2''+2 - /32''+2. Jo (117) In order to discuss the remaining portions of the integrals (114) and (115) we shall now make use of the following established result :^^ " Representing by J^(z) Bessel's function of order v we shall have when V > — ^ and z has any value except zero whose real part is positive or zero (116) JM = - -i-[^i(z)e^(-«2.-l)/4].) _ ^^(^Sjg-i(.-[(2.-l)/4].)-|^ '\2Trz where and where, when lz| is sufficiently large, these expressions *Si and *S2 may be expanded into the forms (118) ^f^ "^^ r(^ + i + n) 1 ^'^'^ = h iXM^I)i>TI^^) (2^- + ^'^'' '^' in which m is any positive integer and in which the expressions 9i(;:, v) and 62(3, v) become infinitesimals of order as high as the 7?zth when |s| = 00 and, at least when v > -\- ^, possess first derivatives which as | z | = 00 become infinitesi- mals of order as high as the {m + l)st." Placing — i = e'"'''^ in (116) we obtain JXz) = -7^[5i(z)e*^^-f^'''-^^/''^'^)+ 52(s)e-«^-«'^-^^/'i'^i Whence, upon expanding and making use of (104), we have ^^(^) = 2^)>^ [ {^^(^) + ^2(^)} cos (2 - ^tt) + i{Sr(z) - S,{z)] sin i^z - — ^tt)], 21 Cf. H. Weber, Math. Annalen, Vol. 37 (1890), pp. 404-416. The facts which we shall state regarding the derivatives of 61(2, p) and 02(z, p) are not explicitly obtained by Weber, but follow at once from his analysis. 150 SUMMABILITY OF FOUEIER SeRIES AND ALLIED DEVELOPMENTS SO that by (118) we may write when v > ^ and when the real part of z is positive (or zero) (120) ^.(2) = '\^^^^^H^)L^^'^ '^^'''^^ cos\^z-~'^j—Trj . / 2p-\-l \ + ^{Z, v) sin I Z ^ TT I where the functions e{z, v) and f(z, v) become infinitesimals of at least the second and first orders respectively as 1 2 1 = co and possess first derivatives which as I z I = 00 become infinitesimals of at least the third and second orders respec- tively. Moreover, by use of the relation Py{z) = {2v + 2)P^i{z) — z^P^^iz), we may readily show that (120) holds true for all values of v for which P^(z) has a meaning — i. e., unless 1/ is a negative integer. Furthermore, since P'{z) = — zP^^i{z) we see that unless j^ is a negative integer we may write (121) p/(2) = _ y^l^-—^^Y [l + y]{z, v)] sin [z - -^^-^-^J ( 2?/+ 1 \1 + Q{Z, V) cos I Z ^ TT J J where 77(2, v) and ^(2, v) have the properties mentioned above for e(2, v) and f (z, v) respectively. Equations (120) and (121) having been obtained, we return now to the dis- cussion of (114) and (115) when the indicated integration is extended over the portions of C„ remaining after removing the semicircle of radius 77 and the portions of the y axis. Placing for brevity 21/ +1 /2 a=--^-7r, c=^-, we have by (120) and (121) for all values of z upon these portions of C„, unless a = or i8 = 0, (122) P{pd) = , w+(i/2) [{l + €(az)} cos {az - a) + ^{az) sin (az - a)], \CX.Z) (123) P'{az) = , .,!■(! ;2) [ U + r){az) \ sin {az - a) + e{(xz) cos (az - a)], (124) Pm = !^o/2) [{l + f(/32)} cos (iSz - a) + r(i32) sin {^z - a)], (125) P'ifiz) = , ~,!a.) [ { 1 + 77(i3z) } sin (^z - a) + Sm cos (/32 - a)], z{pz) Developments in Bessel Functions 151 and, excluding the case in which a = 0, we observe that in applying Theorem I of § 51 to the integrals (114) and (115) in question the values of a, t and fi = a ■}- t with which we shall be concerned are such that < a' < a < 6' < 1, — a = t ^ 1 — a, 0 ^ = constant, are less in absolute value than certain con- stants independent of a, (3 and s. For the same values of 2 we have also P(z) = -i;:f(i]^) [1 + e(2)][cos (2 - a) + u{z) sin (2 - a)] zP'{z) - hP{z) = ^-^^) 1 + 77(2) + 1 r(2) I [sin (2 - a) + ZJ(2) cos (2 - a)], where ,(. 0(2)4-^1 + 6(2)] f . r(2) -f s 2 ^ ^^ i + 7?(2) + ^r(2) Whence, upon recalling that a — /5 = — i, we see that whether we are dealing with (114) or (115), the portions of the integral arising from the part CJ of C„ now under discussion will be of the form 1_ p/^Y+g/^) dt r sin [(a + /3)2 - 2a] dz Developments in Bessel Functions 153 ."+(1/2) dt r (?2(z) sin [(a + ^)z - 2a] 1 f7j8\''+(i/2) ^^ /> E' ^2 '^ 2Tri TTlJo \«/ a + jSJc^/ JS2 l_ /-Y/3y+a/2) ^^ /^ g4(2) cos [(g + ^)2 - 2a] + 27rJo UJ a + ^X/ ^^ where the functions ^1(2;), ^2(2), 93(2) and q^iz) (Hke the functions ^1(2), 2^2(2), etc.) may be put into the forms 61/2^ 01/2^, di/z + 61/2^ and /1/2 + gi/z^ respec- tively and where E = cos (2 — a) + co(2) sin (z — a), (129) E = sin (z — a) + £0(2) cos (2 — a), according as we are deahng with (114) or (115). Considering first the portion of Cn consisting of one of the Hnes parallel to the X axis, we readily obtain as in § 64 the fact that for all values of a, /3 and t in (126) each of the integrals in (128) when extended over the line in question approaches uniformly the limit zero as j = 00 . Thus we have merely to con- sider (128) in which z = k -\- iy and CJ is understood to extend from y = — to 2/ = + °° along the line 2 = ^ + iy. Now, from the manner in which k is to be chosen, we see from (129) that we may take ^ = nr + a or ^' = mr/2 + a ; (n = positive integer) according as we are dealing with (114) or (115), In either case, equations (129) are such that 1 7,5 -^= 1 + -tanh y + -, t," Z Z" where y is independent of z while 5 depends upon z but has a modulus which for all values of 2 under consideration is less than a certain quantity M. Thus, (128) may be written in the form ({^+|)cos[(a + /3).-2a]|^, + + where 62 has the properties mentioned above of 61 154 SuMMABiLiTY OF Fouhier Series and Allied Developments Considering first the terms of (130) which have z- in their denominator, we have but to refer to the discussion of similar terms in (89) in order to see that for all values of a, /3 and t in (126) these terms have uniformly the limit zero when ^- = CO . The same is true also of the term X\a) a + I3j_^z cosh2 y ^' since we have cos tz = cos tk cosh ty — i sin tk sinh ty and we know that when |/| < 1 (as is the case in (126)) the integrals r" coshj^ r°° I sinh ty\ J_«, cosh^ y ^' Xoo cosh2 y ^ have a meaning. Thus, (130) reduces to .+(1/2) /-» / y \ sin fe tm '£('*>"") dy 27r Jo \ « / J-oo \ ^ / i cosh- y where e/ depends only upon a, jS and t and is finite for all values of these quan- tities in (126) and where A(a, t, k) and also dA(a, t, k)/dt depends upon a, /3, t and z but when considered for all values of a, /3 and t in (126) may be made (uniformly) as small as we please in absolute value by taking k sufficiently large. Upon placing z = k -{- iy and recalling the values which k may assume; also placing for convenience a + /3 = 2 — r and dropping those integrals which vanish identically since they are relative to odd functions of y, we thus obtain (131) in the form 1 rf^y-^^"^hmkt^ r cosh ty, 2^1 [a) ~r"^U-«c^sh^^^ _yi r' f^Y^^"^^^lM n y tstnh y cosh ty ~ 27r Jo U J t J_„ (F + f) cosh2 y "^^ yi r' (^Y^^m p fctanhysinh/ y^ + 2^Jo Uj ^«^"J_„(F+F)icosh^/^ i- r /^V^^"'^sin^ r°° cosh(Q; + /3)y ^■27r4_,U/ a + rJ-. cosh'y "^^ (132) r /^V^^"'^sin_fcr r°" y tanh y cosh [a + f>)y L-.m) a^rL^k'^f cosh^?/ ^^ Developments in Bessel Functions 155 i_ r /A*""^''^ /Cos^j r°° fccosh (a + i5)y 27rJo(,_„)\Q;y q: + |8 J_«, (^- + ^z^) cosh^ ?/ the upper or lower sign being taken according as we are dealing with (114) or (115). The expression (132) may, moreover, be used to determine the value of the integrals (114) and (115) corresponding to the case a = I. In fact, when a = 1 and (3 and t are confined by (127) we readily see that each term of (132) con- tinues to have a meaning. From the properties already found of the integrals in (93) it now appears that the second, third, fifth, sixth, seventh and eighth integrals of (132) when con- sidered for all values of a, ^ and t in (126) or (127) have uniformly the limit zero as ^* = °o , while if we treat the first integral as w^e treated the first integral of (93), remembering here that lim (^(aY^^'^'^-^ = 1, we find that when k= ^ this integral behaves precisely as the indicated integral of (93) — i. e., approaches the limit i or — I according as ^ > or i < — 0. Similarly, the integral J_ r /^y+a/2)sin^T r°° cosh (a + /3)y 27ria-„)U/ a + ^ .L cosh^ 2/ ^' like the fourth integral of (93), has uniformly the limit zero if a' < a < h' while if a; = 1 it has the limit ^. Whence, if we are dealing with values of a, (3 and t satisfying (126), the ex- pression (132) converges uniformly to the limit | or — | when k = <=o according as ^ > or ^ < — 0, while if a = 1 and /3 and t have values consistent with (127) the same expression has the limit or 1 according as we are dealing with (114) or (115). Thus, exception being made of the case h = in the integral (115), the inte- grals (114) and (115) satisfy relation (I) of Theorem I, § 51 provided, however, that t has only those values for which — a-\-^^t = l — a; ^ > 0. ]\Iore- over, when a = 1, relation (1)6 of Theorem VI, § 55 is satisfied for the same values of t and in this relation we have in the present instance G2 = or G2 = 1 according as we are dealing with (114) or (115). Again, if ^ = in (115) the limit approached by this expression as k = <» (a' or / < 0, it being understood as before that — q;+s=< = 1 — «• Likewise, if a = 1, other conditions remaining the same, the limit approached 156 SUMMABILITY OF FOURIER SeRIES AND AlLIED DEVELOPMENTS by (115) as A: = 00 will be - 1 + {or'^^ - fi-''+^). In both the cases which thus arise when ^ = we evidently meet with an application of the third general remark of § 56 and we shall make this application presently. Turning to the other relations of Theorems I and VI of §§ 51 and 55, we see that in the present developments the function (p(n, a, t) is equal to nQQ^ 1 ^"'+' r P'(az)Pm-P'mP(az) in the case of (114) while for (115) the same function reduces to or 1 /S^-'+i r P'((xz)P(l3z) - P{^z)P'{az) , (135) _ (2. + 2) ^^- + - ^-, l^^ p-(.)^ d^' according as h =|= or h = 0. Now, for values of a, jS and t in (126) we may transform (133), (134) and (135) by use of expressions (122), (123), (124) and (125) and thus we find that, exception being made of the term — {2v + 2)^-"^^ in (135), these expressions all reduce to the sum of the derivatives with respect to t of the expression (132). From this it follows directly upon using the lemmas of the Appendix that the above expressions satisfy relations II and III of Theorem I, § 51 ; also that when a = 1 conditions (II) 6 and (111)6 of Theorem VI of § 55 are satisfied, it being understood throughout as before that we are dealing only with values of t such that — a+^^f^l — o;;^>0. Moreover, if we affect each of the terms of (132) by the operation 1 n -E, understanding that absolute values are taken under the various integral signs, it appears as in the study of (93) that when — a+^^i^l — a (^>0) relation (II)' of § 52 is satisfied, as also (11)6' {h — 1) of § 55. It remains, then, merely to consider the integrals (114) and (115) when t takes values such that — a^t -^ — a-\- i, (^>0) and for this it becomes necessary, as already noted, to use some other expressions for P{^z) and P'ifiz) than (124) and (125), since /3 now takes values indefinitely near to zero. Considering, then, that t = — a is one of the exceptional points of the type mentioned in remark (1) of § 56 it will now suffice for the application of Theorems I, II, VI and VIII of §§ 51-55 that such additional conditions be placed upon/(.T) that when either of the expressions (133), (134) or (135) is multiplied by /(a + t) the absolute value of the product, when considered for values of t such that Developments in Bessel Functions 157 — a ^ t ^ — a-\- ^ and for all values of n, may be made uniformly small with ^, this being true when a' < a < b' and when a = 1. Let us now divide Cn into two portions C„" and Cn" the first of these com- prising that portion of the line z = k -{- iy for which \y\< t], where t) is an arbitrarily small positive quantity and the second comprising all other portions of Cn'. As regards the expressions (133), (134) and (135) when the integration is performed over Cn", we have but to make use of the well known formula 1 C PJz) = T^ I sin-" (p cos (z cos (p)d(p; v > — h to see that when |jS|< ^ the same expressions (C„' now replaced by Cn") are each of the form (3^'"^^G{a, /3, n, ^, 77) where G{a, /3, n, ^, i)) is less in absolute value than a constant independent of a, (3 and n. In order to study the same expressions when the integration is performed over Cn" we first make the following observations: Let us write (120) in the form / 2.+ 1 \ \7r 2"+(i/2) (136) P,{z) = ^J- .+(1/2) '■^(^' ^)^ so that / 2?/ + 1 \ A{z, v) = {1 + e(s, v)] + r(2, v) tan Is ^ — tt I. For all values of z (real part > 0) lying upon Cn" and of modulus greater than some fixed value zo > we see that A{z, v) remains less in absolute value than a constant Mx. Moreover, li v {v ^ neg. integer) has any value except one of the form |(1 d= 4w); n = Q, 1, 2, 3, • • •, the same expression when con- sidered for values of z (real or complex) as near to zero as we please remains less in absolute value than a constant il/2 provided v ^\. In fact, it appears from (136) that as z = 0, A{z, v) will tend to zero like 2"+^^^-^ since from (104) we have lim P^z) = o.-p/.. I IN ; p > - I. z=0 2T(j/ + 1) ' Whence, if p has any value = — 2 except one of the form |(4?i -\- 1); n = 0, 1, 2, • • •, we may write for all values of z upon Cn"' ( 21/ +1 \ cos Is T TV J PM = ^H^iT^) B{z, p), where B(z, p) remains less in absolute value than a constant (independent of z). 158 SUMM ABILITY OF FoURIER SeRIES AND ALLIED DEVELOPMENTS Similarly, if v has any value > — | except one of the form §(4w — 1) ; n = 0, 1, 2, • • •, we may write for all values of ^ upon C„"' sin / 2.+ 1 \ where B{z, v) has the properties just mentioned. It follows that for all values of z (real part > 0) upon C,/" and for all values however small of the positive quantity j8 we may write, provided v ^ — ^ cos i^^-H^^) (137) P.,(/3^) = (^^).+(i/o) B(^z, p) or sm / 2.+ 1 \ [^z-^^.j (138) P.m = (^zy+ai2) -^i^^' ^)' where for the indicated values of z and /3 the expression B(^z, p) remains less in absolute value than a constant independent of both j3 and z and where the first form can be used in all cases except when p = ^(4?i + 1); ?i = 0, 1, 2, • • •, while the second form can be used in all cases except when p = ^(4/1 — 1); n=0, 1, 2, •... By means of the relation we now obtain, as formulae corresponding to (137) and (138), sin (r 2.+ 1 \ I\'m = ^.-(l/2),.+ (l/2) B{^Z, P), (139) (r 2v+1 \ cos \pz J — ''^ I Pv (P2) = oi'-(l/2)2"'+(l/2) -o(p2, V), where B{^z, p) has the properties given in connection with (137) and (138) and where the first or else the second formula (and in general both) can be used for any given value of i' ^ — |. Now, if we use in (133), (134) and (135) the forms (137), (138) and (139) (thus confining ourselves to an integration over Cn") we find as before that by taking j = oo the complex integrals become simply those arising when, in- stead of Cn" , we take as path of integration the line z= k -\- iy, it being under- stood that the integration now consists of that from y = — ^ to y = — t] Developments in Bessel Functions 159 together with that from y = rj to y = go . This statement, as in the former case, is seen to be true either when a' < a < b' or a = 1. The resulting complex integrals thus take a form analogous to (132) involving real integrals of the form where the expressions (pi{y), '-('/2) appearing on the outside) are always less in absolute value than a constant independent of a, /3 and k, it being understood throughout that a' < a < b' or a = 1 and 1/5 1 < ^. Thus the expressions (104), (105) and (106) when considered for values of jS such that |i3| < ^ are of the form jS"-^^"^^ H(a, j(5, n) where H(a, /5, n) is less in absolute value than a constant independent of a, fi and n. It follows therefore (considering the forms which we have now obtained for the expressions (133), (134) and (135) when the indicated integration is performed either over Cn" or Cn") that we shall be able to apply Theorems I, II, VI and VIII of §§ 51-55 to the present developments if we demand (in addition to the conditions placed upon /(a-) in the same theorems) that the function /(/S)/?""*""'^^ be integrable in the neighborhood at the right of the point /? = 0, it being under- stood also that v > — ^. In other words, we need merely make the additional demand that x'''^^^''^^f(x) be integrable in the neighborhood at the right of the point a; = 0. Upon applying Theorems I, II, VI and VII of §§ 51-55 and remarks (1) and (3) of § 56 we thus arrive in summary at the following result: " If f(x) remains finite throughout the interval (0, 1) with the possible ex- ception of a finite number of points and is such that the integrals (140) I .r""^'-''"^ \f(x) I dx, I |/(a-) I dx, e arbitrarily small and positive, Jo J e exist and if P^s;) be the function defined for all values of z and for f > — 1 by the equation (104), then each of the three series: 00 (2. + 2) r x'^+%v)dx + IlqnT^iK'x), Jf n = \ 160 SUMMABILITY OF FOUEIER SERIES AND ALLIED DEVELOPMENTS n=l in which X„, X„' and X„" represent respectively the nth positive roots of the equations P,(2) = 0, P/(2) = 0, zP/(s) - hPM = 0; h = constant + and in which qn' = p^2 J x'^+'f{x)PX\n'x)dx, will converge provided i; ^ — ^ at any point x (0 < .r < 1) in the arbitrarily small neighborhood of which f{x) has limited total fluctuation, and the sum will be H/(-^--0)+/(.r + 0)]. IVIoreover, the convergence will be uniform to the limit j{x) throughout any interval {a', h') enclosed within a second interval (ai, 6i) such that < Oi < a' < // < 6i < 1 provided /(.r) is continuous throughout {a', h') inclusive of the end points x = a', x = h' and has limited total fluctuation throughout (ai, hi). Also, if f{x) remains finite throughout the interval (0, 1) with the possible exception of a finite number of points and is such that the integrals (140) exist, then each of the three series above {v ^ — ^ )will be summable (r = 1) at any point .T (0 < ar < 1) at which the limits /(.r — 0), j{x + 0) exist and the sum will be i[/(.i--0)+/Gr + 0)]. Moreover, the summability will be uniform to the limit /(.r) throughout any interval (a', h') such that < a' < 6' < 1 provided that at all points within (a', h') inclusive of the end points x = a', x = b' the function /(a-) is continuous. Under the same conditions for f(x) when considered throughout the ivhole interval (0, 1) the three series (i/ ^ — |), when considered for the value x = 1, will converge to the respective limits 0, /(I — 0) and /(I — 0) provided that f(x) is of limited total fluctuation in the neighborhood at the left of the point x= 1. The same series when considered for the value .t = 1 will be summable to the respective limits 0, /(I — 0) and /(I — 0) whenever /(I — 0) exists." 67. If we now introduce Bessel functions into this result through the relation Developments in Bessel Functions 161 Py(z) = z~''J^(z) and then apply the theorem to the function x'^ix) instead of f(x) we obtain the following: Theorem. If f(x) remams finite throughout the interval (0, 1) with the possible exception of a finite numher of points and is such that the integrals (141) I x^\f{x)\dx, I \f{x)\dx; e = arbitrarily small positive constant Jo Je exist and if J^{z) be Bessel' s function of the first kind of order v then each of the three series 2gn/v(Xn.X-), (2^ + 2) r x'^+'f{x)dx + flqn'J.iK'x), 00 Y.qn"JX^n"x), in which \n, X^' a7id X,/' represent respectively the nth positive roots of the equations JM = 0, ^ (S-'J.CZ)) = 2j;(2) - vJ^iz) = 0, zJJ{z) — (h + v)J^{z) = 0, h = constant =f= 0, and in which 2 r^ qn = J ,/^ N2 I xf{x)J^{\nX)dx, qn ^^ ~T /-v /\2 I •V C*^/" vvXn XjuX, qn" = 2\n"' {h{2p-\-h)+Xn"V.0^n') — J^ xfix)J^(Kn"x)d:i will converge provided v > — ^ at any point x {0 < x < 1) in the arbitrarily small neighborhood of which f{x) has limited total fluctuation, and the sum will be |[/(^-0)+/(.r + 0)]. Moreover, the convergence will be uniform to the limit f(x) throughout any interval (a', 6') enclosed within a second interval (ai, bi) such that < ai < a' < b' < bi < I provided f(x) is continuous throughout {a', b') inclusive of the end points x = a', x = b' and has limited total fluctuation throughout (ai, by). Also, if f{x) remains finite throughout the interval (0, 1) loith the possible ex- ception of a finite number of points and is such that the integrals (141) exist, then 12 162 SUMMABILITY OF FoUHIEE SERIES AND AlLIED DEVELOPMENTS each of the three series above (v > — ^) will be summable (r = 1) at any point x (0 < .T < 1) at lohich the limits f{x — 0),f(x + 0) exist and the sum will be H/G-^ -0) +/(.!• + 0)]. Moreover, the siimmability will be uniform (§ 45) to the limit f{x) throughout any interval (a', b') such that < a' < 6' < 1 'provided that at all points ivithin (a', b') inclusive of the end points x = a', x = b' the function f{x) is continuous. Under the same conditions for f(x) when considered throughout the whole interval {0, 1), the three series, ichen considered for the value x = 1 will converge to the respective limits 0, /(I — 0) and /(I — 0) provided that f(x) is of limited total fluctuation in the neighborhood at the left of the point x = 1. The same series ichen considered for the value a; = 1 loill be summable (r = 1) to the respective limits 0, /(I — 0) and /(I — 0) wheriever /(I — 0) exists, it being always assumed that the integrals (141) exist?'^ 3. The Developments in Terms of Legendre Functions. 68. We proceed to consider the well known development (142) » 2n-\- 1 r^ fix) = zlqnXnix); qn = ''~^— f{x)Xn{x)dx, in which Xn{x) represents the polynomial of Legendre (Zonal Harmonic) of order n. In the notation of § 60 we here have a development of the form (54) in which H{z, x) = A''z(.t), a = — l,b = 1 and in which equation (53) becomes a^|(i-^-^>l^| + * + i)-^' = °- Moreover, since z is to take only integral values, the equation u{z) = must 22 It may be noted that oiu* results, in so far as they concern convergence at a special point a; (0 < X < 1), are not in entire accord with those of Dini ("Serie di Fourier," pp. 2G6-269). In fact, instead of the existence of the first of the integrals (141) Dini requires that |/(a;)|.r''+a~P, where p is the greater of the two numbers v, \, shall be integrable in the neighborhood at the right of the point x = 0. This discrepancy is due chiefly to a slight error occurring in formula (95), p. 237 of DiNi's work, the last term of which should contain under the integral sign e^'^r^""!"" instead of e~''T''~i~'», as appears from the analysis on p. 237. If this formula (95) be altered as just indicated and resulting changes be made on pp. 242, 243, 265-269, we are led to the above theorem. This same theorem, so far as it concerns convergence, is in accord with the results published in recent years by Hobson {Proc. London Math. Soc, Vol. 7 (1908), pp. 359-388), while, as regards summabihty, the theorem is in accord with the results of C. N. Moore {Trans. Am. Math. Soc, Vol. 10 (1909), p. 428). It may also be remarked at this point that, except in the study of uniform convergence, the results published of late years by Hobson and others respecting the convergence of Fourier series and other developments in terms of special normal functions were originally obtained rigorously for the first time by Dini — a fact apparently not well imderstood. See, however NiELSON, "Handbuch der Theorie der Cylindcrfunktionen," p. 353. Developments in Legendre Functions 163 here be regarded as given in advance and may be taken for example as u(z) = sin TTZ = 0. Furthermore, we have in the present instance K(x) = 1 — x^ so that equations (60) become satisfied identically by taking h' = 0, h = 0. However, since K(zL 1) = it follows that the general formulae of § 61 for the determination of the integral (36) corresponding to the present development cannot be used. It becomes necessary, therefore, in order to ascertain whether this integral satis- fies the conditions of the fundamental Theorem I, §51, to proceed independently of such formulae. Now, the integral (36) here becomes (143) 7r/2) and in which In{t) is defined in one of two ways as follows: (a) li6 <6' ovd^-K- 6', (149) Ut) = l;,£ \ ^n{co^ 7)]r=% A in which c and tt — c represent the two values of ;/' determined by the planes of the two great circles through the point 7=0 tangent to the circle 6=6 and in which 72 (^) and yz{6) represent the two values of 7 pertaining to the points upon the circle 6=6 having the common coordinate yp. (6) li6' <6 K-K- 6', (150) hit) = ^f' Ynicos yz{6))dyp, in which yz{6) represents the value of 7 pertaining to the point upon the circle 6=6 having the coordinate i/'. Upon writing [7„(C0S 7)]^'=i''o^ + [YnicOS 7)];=o - [Fn(C0S 7)]v=v.(«') ^ We here employ the common notation [f{x)Y'z=a = f{b) — f{a). Developments in Legendre Functions 165 and observing that F„(cos 0) = 2, F„(cos tt) = 0, we thus obtain (151) r ip{:n, a, t)dt = ± i =F^ J Fn(cos 7i(^'))# ± hit). In order to show that relation (I) of § 51 is satisfied it therefore suffices to show that for all values of a and t such that (152) ^ . ^ . (6 > 0) the last two terms of (151) converge (n = oo) uniformly to zero. In doing this we shall make use of the following two fundamental results respecting 7„(cos 7): (A) For values of 7 in any interval such that 0<^^7^7r— ^<7r the expression yn(cos 7) converges (n = co) uniformly to zero. (B) For all values of n we have uniformly hm F„(cos 7) = — i. e., corre- sponding to an arbitrarily small positive quantity ) uniformly to zero. In showing this we shall find it convenient to divide these circles into three classes as follows : (a) < d < 6' - K, (6) 6' + K< e <7r- 6', (c) IT - d' < 6 Kir. Also, we shall assume for the present (as above) that B' < 7r/2 (a > 0). First, for the circles (a) we have In{i) defined by (149) in which 72(^) and yz{d) are such that k ^ 72 ^ 7r/2, /c ^ 73 ^ 20' — k < tt, while c lies between fixed limits dependent only upon e (as again appears after noting the significance of the various letters upon the unit sphere). Wlience, by result (A) we reach the desired result for the circles (a). As regards the circles (6), let us divide these into two sub-classes as follows: (by Tr-e'-v<0 0. Whence, using result (J5), we see as before that if v be any preassigned arbi- trarily small positive quantity, we may take ju so small that for all the circles (c)' we shall have uniformly | In(t) | < v. With fx thus chosen, let us consider the resulting circles (c)". Here again we are to use the form (149), but the values of 72 and 73 which enter lie between assignable limits m, n such that m > 0, n < TT (m = K, n = IT — n). Hence, for the circles (c)" the expression 7„(0 has the desired properties, and in summary we may say that the same is true for all the circles (c). Thus, relation (I) of § 51 becomes satisfied for all values of a within the interval ^ a ^ b' < 1. That it is satisfied also when — l0, it appears that (III) is here satisfied for all values of a and t such that — 1 < a' ^ a ^ b' <1 and -l-a+^^^^-e, e^t^l - a- ^ (^>0). Whether the same is also true (as desired by (III)) when t lies in the intervals — 1 — a ^ t ^ — 1 — a -\- ^ or 1 — a— ^^^^1 — a remains in doubt, thus leading eventually to an application of remark (1) of § 56. Due account of this ex- ceptional character will be taken before the final summary of our results into a theorem. We turn to the consideration of (143) when a = d= 1. First, if a: = 1 we have (156) f cp{n, 1, t)dt = i E (2n + 1) f Xn{l + t)dt Jo n=0 Jo = - I Z) 1 A"„(cos 6) sin ddd n=0 Jo " Cf. Fej£r, I. c, p. 103. sin edd (158) Developments in Legendre Functions 169 and we shall now show that for values of t such that — 2 + e ^ ^ ^ — e; i. e., of 6 such that OK'n'^B'^ir— r}{r] arbitrarily small but > 0) the last member of (156) converges (uniformly) to the value — 1 when n = co, thus satisfying relation (I) 6 of § 55 {Gi = 1) when exception is there made of the value t = — 2 (d = tt). In fact, when a = 1 we have 6' = 0, so that in using (147) we have y = 6 while (p becomes independent of 6. Thus we may write r' r^ d cp{n, 1, t)dt = i -y^[X„(cos d) + Xn+i(cos e)]dd (157) •^0 '^y ^^ = i[^n(C0S 6) + X„+i(C0S e)]l = - 1 + ^rn(C0S d) . The indicated statement thus follows upon noting the properties already men- tioned of F„(cos 6) when 00). Again, if a = — 1 we may write I (p{n, - 1, t)dt = - ^ Z (2w + 1)(- 1)" I A^„(cos 6) sin ddd = -|Z(2n+l) f Z„{cos(7r-0)} = -I f F/{cos (tt- e)]de = - UYnicOS {it - e)]]l^^ = 1 - |7n{cOS (tT - 6)], from which it appears that for values of 6 such that rj ^ 6 '^ tt — r] (rj > 0) i. e., of t such that e ^ t ^ 2 — e, the first member of (158) converges uniformly (n = ) to the limit + 1, thus satisfying relation (!)„ of § 55 (6^1 = 1) when exception is there made of the value t = 2 (6 = 0). Relations (II)a and (11)6 of § 55 are evidently satisfied as a result of (157) and (158), but relations (Ill)a and (111)6 are not satisfied. For example, we have 0. Since, as already shown, relations (!)& and (II) & are satisfied, it follows (of. (25)) that the first term here appearing on the right approaches the limit /(I + 0) provided only that the integral "" \Kx)\dx exists and that f{x) has limited total fluctuation in the neighborhood at the left of the point x = 1. It remains, therefore, but to impose such further conditions upon /(.r) that the last term of (159) shall approach the limit zero as n = co, and we shall now show that this will be the case whenever f{x) is of limited total fluctuation throughout the whole interval (— 1, 1). First, let us consider the integral (160) r ' /(I + t)cp{n, 1, t)dL «/-2+e Considering that n has any fixed value (positive integral), let us divide the interval (— 2 + e, — e) into a certain number m of parts such that in each the function (p{n, 1, t) does not change sign. Let pi, jp2, •'-, Vm-\ be the corre- sponding points of division. We may then write r' /(I + t)^{n, 1, t)dt = f f + f" + • • • + r ) /(I + t)« represents the fluctu- ation of /(I + t) in the interval ih < t < p^+i, the last equation enables us to write I r~^ / "*"' \ /(I + t)^{7l, 1, t)dt < C7 X + E £>. , I «^-2+e \ s=l / from which the indicated result concerning the last term of (159) becomes evident. Similarly, when x = — I we may obtain the corresponding result so that the discussion of the convergence of the series (142) may now be readily completed, both for the case of a point x such that — 1 < .r < 1 and for the end points X = ±1, except that, following remark (1) of § 56, it remains to consider the integrals f{(x^-t) 0)." 71=00 It will thus appear that although the summability (r = 1) of (142) at an internal point (— 1 < x < 1) cannot be assured under conditions so slightly limitive as those met with in the corresponding studies of Fourier series (§ 46) or the Bessel expansions (§ 67), nor indeed under restrictions upon f{x) which are any less than those stated in Theorem I for convergence (r = 0) at such a point, yet at the end points a; = ± 1 the conditions for summability may be stated in a less restrictive form than the corresponding ones in Theorem I. We begin by noting that, as a result of (144) and (145), the function $(«, n, t) corresponding to the present development is such that where $(n, a, t) = ^^-qj^ [(n, «, = ^ f^ Sn'{y)dcp, where *„'(7) is defined as in (166). Now, when t is such that — € ^ < ^ e (as occurs in relation (II) of the general theorem of § 52) the corresponding values of 7 pertaining to the neighborhood " Cf. Fej6r, I. c, p. 107. Developments in Legendre Functions 175 of the (fixed) point {6', (p') lie in an interval of the form ^ 7 ^ 77 where 77 vanishes with e. Thus, while formula (168) is general, holding for all values of a and t with which we are concerned in applying the theorem of § 52 to the present development, we are unable to determine whether relation (II)' of the same theorem is here satisfied until more than is given by (177) is known of the behavior of Sn'i'^) for large values of n. A critical study of Sn'{y) for 0^7 ^ € is here needed and such study has apparently not yet been made. Again, it cannot be argued from (167) and (168) that relation (III) of § 51 is here satisfied by ^{n, a, t) (cf. remark (2), § 56). This relation, however, is seen to be satisfied if we confine ourselves to the intervals — 1 — a+^^< ^ — 6, e^i^l — a — ^(^>0) instead of — 1 — a^^^ — e, e^i^l — a, but this is nothing more than can be at once inferred from the properties already pointed out in § 68 regarding the present function (n, — 1, t) and ^{n, 1, t) (regarded as functions of the type (p there indicated) except that doubt exists in the case of (I)a and (1)6 when t belongs to the respective intervals 2 — ^ ^ t ^ 2, — 2^/^ — 2 + ^(^>0). In other words, nothing more can be said of ^(n, — 1, t) and ^{n, 1, t) than was said of (p{n, — 1, t) and (p{n, 1, t) in § 68. This, however, is not the case in dealing with relations (Ill)a and (III)^. Thus, in (III) 6 we have to consider the expression $(n, 1, t) = ^^qj^ [ =2 Cf. § 44. 33 Cf. Chapman, Quart. Journ. Math., Vol. 43 (1911), p. 51. For summability (;• =1) Chapman places no restrictions upon/(x) at the extremities of the interval (— 1 < x < 1) other than those for the whole interval. 176 SUMMABILITY OF FOURIER SeEIES AND ALLIED DEVELOPMENTS SO that upon introducing (167) we see that (III)^ is here satisfied for all values of t in the interval — 2+^^t^ — e. For the remaining values of t with which (111)6 is concerned, i. e., — 2 ^ t ^ — 2 + ^, doubt exists. Likewise, relation (Ill)a is seen to be satisfied by $(n, — 1, t) except possibly for values of t in the interval 2 — ^ ^ t ^ 2. From the general theorem of § 52 together with the remarks in § 56 and the investigations already made in § 68 of the last two of the integrals (162) we reach the following Theorem IL If the function f{x) of the real variable x satisfies conditions (a), (b) and (c) of the Theorem I (§ 68) then the series (164) ^vhen considered for the values a* = =b 1 will be summable (r = 1) to the respective limits /(I — 0), /(- 1 + 0). 70. The difficulties which present themselves in the study of the summability, r = 1, of the series (164) disappear in large measure when we consider the same problem with r = 2. This fact was first pointed out by Fejer^^ who confined himself, however, to functions f(x) having somewhat greater limitations than we shall here find necessary in view of the general theorems of § 52. In what follows we shall make use without further remark of the following two preliminary results which may be found established on pages 81-87 of Fejer's original memoir. " Having defined Sniy) and Sniy) as in (165) and (166), if we place^^ (170) sn'\7) = :;^W^y) + ^i'(t) + • • • + ^/(t)] then "(1) Whatever the values of n and 7 (0 < 7 < tt), Sn"{y) is never negative. "(2) For values of 7 such that € ^ 7 ^ tt, € being arbitrarily small but > 0, the expression Sn"{y) converges {n = co) uniformly to zero." These results being premised, we shall now endeavor to apply the general theorem of § 52 to the present development. Just as we found the formula (168) for the function $(w, a, t) arising in the study of the summability, r = 1, so it appears that if we represent by xl/{n, a, t) the corresponding function which arises when r = 2, we shall have (171) Hn,a,t) =^f Sn"iy)d'(2«+l)7r k »/(2.s+l)7r k •^ 25+1 ,r L A; fc J In the last integral here appearing the factor sin i is negative (or zero) for all the values of t be- tween (2s + \)ir and (2s + 2)ir and since, for the same values of t, we have A- ^ k = 2 ' the factor appearing in square brackets in the last integral is positive when (2s + l)7r<<^(2s -1-2)7r. In like manner it appears that the least of the minimum values of (6) is J*2T/*sin_fc< , sin t and that this value is positive together with all values of the integral (6) when < i ^ 27r/A;. Thus, for all values of t such that < i ^ 7r/2 the integral (6) is positive and in summary we may say that the greatest absolute value of (6) when ^ i ^ 7r/2 is given by (7). But 2 Cf. DiNi, 1. c, § 18. Appendix 181 Bin kh kti J*"'/*sinfc; ,, sin kit r^l^,, kh r. ^ . ^ tt —. — 7 dt = —. — 7- / at = T —. — - ; < 0, then, corresponding to any e > such that t < 7r/2, e < 7r/a, we shall have for all values of n sufficiently large (8) I C \ t sin kt — I 2 n«^-€ n=o sin t dt