INTEGRAL FUNCTIONS DEFINED BY TAYLOR’S SERIES. 
251 
Such were considered by, among others, Cauchy, 1 Binet, 2 and Raabe. 3 Other references to the 
history of the subject will be found in the “ Encyklopiidie der Mathematischen Wissenschaften.” 4 
But the behaviour of a function defined by a Weierstrassian product, when considered ordy for real 
values of the variable near infinity, affords little knowledge of the essential singularity. Stieltjes 5 first 
proved the asymptotic expansion for F (x) to be valid for all values of |arg x\ < w. His result was 
subsequently obtained by Mellin. 1 ’ Immediately afterwards the author," from an idea suggested by one 
of Melon’s earlier papers and due originally to Riemann, 8 extended the result to the multiple gamma 
functions. Then, simultaneously, Mellin 9 and the author 10 discovered the asymptotic expansions for 
large classes of integral functions defined by Weierstrassian products. Such investigations have been 
developed by the author in a series of papers. 11 
It is natural to expect that similar results can be obtained for functions defined by Taylor’s series. 
An asymptotic expansion for Bessel’s function J 0 (x) was first given for real values of a; by PoiSSON. 12 
The lesult was extended to other integral values of n, that is to say, to functions J n (a;), where n is an 
integer, by Jacobi. 13 Then, in a noteworthy paper, Hankel 14 extended the result to general complex 
values both of the parameter n and the variable a; ; and though his statement of his results merited the 
criticism of Hurwitz, 15 it deserves recognition as a valuable discovery. The question has since been 
considered, among others, by Weber 10 and Nielsen 17 . Further references will be found in the 
‘ Encyklopiidie ’ ls and in Nielsen’s text book. 17 
In this connection mention may be made of a similar investigation by Hobson 19 in the theory of 
Legendre’s functions. 
Closely allied to Bessel s function are integral functions defined by generalised hypergeometric 
1 Cauchy, ‘Exercices d’Analyse,’ tome 2, p. 386. 
2 Binet, ‘Journal de l’Fcole Polytechnique,’ tome 27, p. 220. 
3 Raabe, ‘Crelle,’ vol. 25, p. 147; vol. 28, p. 10. 
4 Brunel, loc. cit., vol. 2, A, p. 166. 
5 Stieltjes, ‘ Liouville,’ ser. 4, vol. 5, p. 425. 
6 Mellin, ‘Acta Societatis Scientiarum Fennicae,’ tome 24, No. 10. 
7 Barnes, ‘Phil. Trans. Roy. Soc.,’ A, vol. 196, p. 265. 
8 Riemann, ‘CEuvres,’ 1898, p. 165. 
9 Mellin, ‘Acta Societatis Scientiarum Fennicae,’ tome 29, No. 4. 
10 Barnes, ‘Phil. Trans. Roy. Soc.,’ A, vol. 199, pp. 411-500. 
11 Barnes, ‘Cambridge Phil. Trans.,’ vol. 19, pp. 322-355; pp. 426-429; ‘ Proc. Lond. Math. Soc., 
ser. 2, vol. 3, pp. 253-272, and pp. 273-295. 
12 Poisson, ‘Journal de l’Fcole Poly technique,’ tome 19, p. 349. 
13 Jacobi, ‘Gesammelte Werke,’ vol. 7, p. 174. 
14 Hankel, ‘ Mathematische Annalen,’ vol. 1, jop. 467-501. 
[° ne of the referees has pointed out that I had omitted to mention the brilliant investigation of 
Stokes, which remained long unknown to continental mathematicians. Stokes obtained the asymptotic 
expansions of the solutions of Bessel’s equation for complex values of the variable in two papers published 
m 1857 and 1868 respectively (‘Cambridge Philosophical Transactions,’ vol. 10, p. 105; vol. 11, p. 412). 
The reader may also notice Stokes’ ‘Cambridge Philosophical Proceedings,’ vol. 6, p. 362, and ‘Acta 
Mathematica,’ vol. 26, pp. 393-397.] 
15 Hurwitz, ‘Mathematische Annalen,’ vol. 33, p. 246. 
10 Weber, ‘Mathematische Annalen,’ vol. 6, p. 148. 
17 Nielsen, ‘Handbuch der Cylinderfunctionen,’ 1904, pp. 156, &c. 
18 Wangerin, ‘ Encyklopiidie der Mathematischen Wissenschaften,’ Band 2, A, p. 748, 
19 Hobson, ‘Phil. Trans.,’ A, vol. 187, pp. 443-531. 
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