m lAVvTl-XX* UL 23 1913 HE ASYMPTOTIC DEVELOPMENT FOR A CERTAIN INTEGRAL FUNCTION OF ZERO ORDER Of n{£ UNIVfiLiiS«T> BY . CHARLES W. COBB ASSISTANT PROFESSOR OF MATHEMATICS IN AMHERST COLLEGE PORTION OF A THESIS PRESENTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF MICHIGAN IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY PRINTED AT K\jt Nortoootr ^re^g; NORWOOD, MASS. 1913 THE ASYMPTOTIC DEVELOPMENT FOR A CERTAIN INTEGRAL FUNCTION OF ZERO ORDER BY CHARLES W. COBB ASSISTANT PROFESSOR OF MATHEMATICS IN AMHERST COLLEGE A PORTION OF A THESIS PRESENTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF MICHIGAN IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY PRINTED AT , K\it Nortoociti Press NORWOOD, MASS. 1913 %' BIBLIOGRAPHY Barnes, E. W. A Memoir on Integral Functions. Phil. Trans, {k), vol.199. (1902). The Classification of Integral Functions. Camb. Phil. Trans., vol. 19. The Maclaurin Sum Formula. Proc. Lond. Math. Soc, Series 2, vol. 3. The Asymptotic Expansion of Inteo-ral Functions of Finite Non-zero Order. Proc. Land. Math. Soc, Series 2, vol. 3. Ford, W. B, On the Determination of the Asymptotic Develop- ments of a given Function. Annals of Math., 2d Series, vol. 11 (1910). Hardy, G. H. On the Function Pp(x). Quarterly Journal, vol. 37 (190.5). LiTTLEWOOD, J. E. On the Asymptotic Approximation to Integral Functions of Zero Order. Proc. Lond. Math. Soc, Series 2, vol. 5 (1907). On the Dirichlet Series, and Asymptotic Expansions of Inte- gral Functions of Zero Order. Pi^oc. Lond. Math. Soc, Series 2, vol. 7 (1909). On a Class of Integral Functions. Trans. Camb. Phil. Soc, vol.21 (1910). jNIattson, R. Contributions a la Th^orie des Fonctions entieres. (These) Upsal, 1905. INIellin, H. Om definita integraler etc. Acta Soc. Fenn., t. 20, No. 7. Uber eine Verallgemeimerung der Riemannschen Function I {s). Acta Soc Fenn., t. 24, No. 10. 268473 '2 ASYxMPTOTlC DEVELOPMENT M ELLIN, H. Ein Forinel fiir den Logarithnms transcendenter Funk- tionen von endlichem Geschlecht. Acta Soc. Fenn., t. 29 (1900). Reprinted Acta Math., vol. 28 (1904). Peterson, J. Vorlesungen iiber Functionentheorie. Copenhagen, 1898. PoixcARE, H. Siir les int^grales irregulieres des Equations lin^aires. Acta Math., vol. 8 (1886). 1. Introduction. By means of the Maclaurin Sum Formula, Barnes * and Ford f have obtained asymptotic developments for certain integral functions of non-zero order, but for functions of zero order Barnes' analysis breaks down, while Ford has not treated the case, and Littlewood J holds that these functions should be studied by special methods. The present paper, however, uses the Maclaurin Sam Formula to obtain the asymptotic development for a typical function of zero order, and may therefore be considered as supplementing the work of Barnes and Ford, by showing that a common method of investigation, based on the Maclaurin formula, may be employed in determining asymptotic developments for integral functions of both orders. 2. The Maclaurin Sum Formula. The Maclaurin Sum Formula with remainder will be used in the following form : || \if(w') is analytic throughout a vertical strip of the w complex plane, extending to an infinite distance above *Bakxes, E. W. p. Phil. Trans. {A) vol. 199, pp. 411-500; Proc. Lond. Math. Soc, Series 2, vol. 8, pp. 27.3-295. t Ford, W. B. Annals of Math., second series, vol. 11 (1910). X LiTTLEWooi), J. E. Proc. Lond. Math. Soc, Series 2, vol. 6 (1907). II Ford, VV. B. Lectures on Divergent Series given at Univ. of Michigan, 1910-1911. ASYMPTOTIC DEVELOPMENT 3 and bslow the axis of reals and including the real points w =a^w — h^ and is such that where y is some assignable positive quantity, we may write + ^"P""!^" [/'^"'-"(S) -/'^'"-"ra)] + E„, (^ m)l where ^--(2^;OJ^.Jo ^ ^ ^ ^ ^ = 1 when m = 0, < 6* < 1 whenm = l, 2, 3.... 3. Problem. We shall apply this theorem to the dis- cussion of a single type-function of zero order, though the m.ethod employed is evidently capable of broader application. The problem is as follows : Given the integral function of zero order, F(^z^ = TT M j = exp. ^(2) ; z real or complex, it is proposed to consider the existence and determina- tion of an asymptotic development for H(z^ in the ^w=/(0+K^) 4 ASYMPTOTIC DEVELOPMENT precise sense of "asymptotic" as originally formulated by Poincare,* viz., a development of the form lime(2)=0; w = 0, 1, 2 .... This problem has been considered by other methods by Mellin,f Barnes, J Hardy,§ Mattson,|| and Littlewood.^ So far as results are concerned, all agree as to the lead- ing terms (terms that go to infinity as z becomes infinite), but no two results are exactly alike for the rest of the development. Such differences as present themselves are doubtless of form only and hence of minor importance, but the present treatment by means of the Maclaurin Sum Formula tends to unify the problem of determining asymp- totic developments for integral functions, by reducing cases of both non-zero and zero order to a common method of treatment. 4. From our definition, § 3, (1) iT (2!) = lim r V log (e^ -z)-VJ and by the logarithm of a complex number Z, say, we shall mean : log Z — log R + i^ ; Z = B(gos -{- 1 sin cp) ; — 2 7r<<0. In order to evaluate ff(z) for large z, we first regard z as fixed and place —= 6 ; p = greatest integer in z * Acta Math., vol. 8 (1886), pp. 295-344. t Acta Soc. Fennicce, t. XXIX. § Quarterly Journal of Math. (1905). I Phil Trans. {A) vol. 199. || These, Upsal (1905). IT Proc. Lond. Math. Soc, Series 2, vol. 5 (1907). ASYMPTOTIC DEVELOPMENT log \z\; -< I ^ I < 1. Inasmuch as the following analysis holds only when -< \d\<\,z may not lie on any circle whose center is at the origin and whose radius is e'' (p = 1, 2, 3 •••) but z may be chosen anywhere else in the complex plane. b-h To evaluate V /(a:), the theorem of § 2 requires that the corresponding f {vf) shall be analytic throughout a vertical strip of the w-plane, including the points w — a^ w = h. So we write :r-l p-\ (2) ^ log {e- - 2) = ^ log (g- - 2) -f log (6^ - z) x=l a-=l and observe that the function f(w') = log (e** — z') is an- alytic in the vertical strip containing the points ?^ = 1, w =p^ and also in the strip containing the points w —p + 1, w = X. We have then, putting w = 0, ^ = 1, V log (e^ -z)= P log (g^ - 2)t7a: - | [log (e^ - z) -iog(.-^)] + xi,(p)-n,(i) where fi, (a:) = - * j^* [log (e^-^^-^ - 2) - log (.-'^^ - O] . ^^. Also, y log (e^ - 2) = f log (e^ -z)dx-\ [log (g^ - 3) - log (e^+^ - 2)] + fl,(:r) -n,(p-\' 1). ar=30+l 6 ASYMPTOTIC DEVELOPMENT Taking the terms of the right-hand members in the above order, r log (e^-z) dx = [2: log {e^-z)-^- f-^ =:p\og(e^-^)-^-\og(e-z)-^l- T-^ Now, CP xz , C^ xz n — I dx = I dx •^1 e^ — z *^i z — e-^ dx dx. X"' z \z The power series in — is uniformly convergent for all x^ 1 -^ X ^ p^ since in this interval | g* | < | 2 [. Hence, integrating term by term, -- I dx ^1 e'^ — z = g + J(._l)H.1gJ(2.-l):,lgJ(3.-l)+...] where Collecting terms, (3) log (^p - 2) + ^ log (gx _ 2j) = (^ + 1) log (eP - 2) .r=l - log Ce-z) + s\ + .sy + njp) - n,(i). ASYMPTOTIC DEVELOPMENT Consider next the terms arising from x-l We have x-l X x=p+l log (e=^ — z)dx = \ X k)g (^^ - ^) — -;t — I — *^^ ^^'i-;hi'- dx\ Now \^\<\f\ •- X=IJ + \ if we neglect terms that go to zero exponentially when x is infinite. Calling the power series in may write >p+i aS'o, we x-l X X=p+1 x'^ (4) ]^log(e^-2) = a;log(g^-^)--2 -(i?H-l)log(6!^+i-2!) ^ (P + '^y _hogCe^-z) + hog(eP^-^-z) + S,^-hn,(x-) x-l Noting that V a: = — - — and that (2:--jlog (e^-2)-| ^2 a^ — X 2 = 0, we have (5) II(z) = (p + 1) log (e^ -z)- I log (e - z^ + ^1 + s^ + aS'^ + n,(2;) - n, (i? + 1) + n,(jt>)-n,(i). 8 ASYMPTOTIC DEVELOPMENT To evaluate jS^ and xS'g, note that since each series is ab- solutely convergent, we may write, setting for, loga-'-) ^_i_l-_rL^_... r 2 3 \eej AedJ 9\edj ' ' P>oni the definition oi 6, p = log z + log ^ and hence (as will be shown presently) for large z, (6) H(z) = I (log 2)2 + log z [- 1 + log(- 1)] + log(l-.?) + <0)+{^+... + !^-hl(£)J ; lhr.e(.) = where c(^) is a constant depending only on 6.* Since 6 is by definition a periodic function of z, provided z moves along a straight line passing through the origin, the same is true for c{6}. It remains to establish equation (G) and to evaluate c(6) and the as. * Cf. LiTTLEwoop, loc. cit., 397 (13). ASYMPTOTIC DEVELOPMENT 9 We have from (5), after the substitution p = log z -\- log ^, (7) H(z^ = (log ^ 4- log ^ + 1) [log 2^ + log (^ - 1 )] _ llog Qe - 2)- (log 2 + log(9 + J)[log z 4- log (60 - 1 j] + (logg + logl^ + iy + (log z + log 6) [log (ee - 1) - 1 - log ^ - log (1 - 6')] +.S'/ -f n,(2:) - oxi? + 1) + n,(j9) - n, (i). Consider now We may write z e^ y - 1 ^os[(i-^^y-^)]-iog[(i-^i (-^) logfl gl+. 2; 1 Puttinof = — -, and proceeding^ as above, we have ^a(,.i,=-i-[i(.!^;-.) \(] V^l e2 + 1 1\ 1/1 Vn gS^i IN 1 22-1 27 3WV2e3_i 3; J' + '2.\e6. 1 . and the power series in -— is convergent since eu <1. 12 ASYMPTOTIC DEVELOPMENT In the same way o/^ l_^rzfle-{-l ,\ , Ifzyfl e'+l 1 ^8VWV2e3-l yy^ J and liin fl^ (x) = ^^g. Certain further reductions may be made, viz., -a.(i) + .;.,og^i-^J.i:^;^;(^). Also, and - n. (^ + 1) + fl. (X) + r^i-'^n-r)^^ 1 e" 4- 1 n Y'. •^ 2n e« - 1 Vg^y We have, then, finally, (8) If(z) = Klog zy + log 2 [- I + log (- 1)] + where ^ + + ^^ + ^- + "^^^ 1; l'me(2) = <^) = - iG^s ^y + i log r^ - 1) + i log (.^ - 1) + logn-l+log(-l)]-^-S:7 1 e" + l 1 2 71 e" — 1 6>^ ASYMPTOTIC DEVELOPMENT 13 and ''^ = i-.4±lf-«Y;A=l,2,...»-l. 2* 2 k e^-1 \zj ^ 1 e" 4- 1 feV' . As the power series 2y ;" — ^ ( -J is convergent, evi- dently lim € (z) = 0, and the problem of § 3 is completely solved. UNIVERSITY OF CALIFORNIA LIBRARY, BERKELEY THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW Books not returned on time are subject to a fine of 50c per volume after the third day overdue, increasing to $1.00 per volume after the sixth day. Books not in demand may be renewed if application is made before expiration of loan period. UNIVERSITY OF CAIylFORNIA IvIBRARY