A treatise on the theory of Bessel functions, by G. N. Watson.

196

THEORY OF BESSEL FUNCTIONS

[CHAP. VII

The general character of the formula for Y, (z) had been indicated by Lommel, Studien
iiber die Bessel'sc/en Functionen (Leipzig, 1868), just before the publication of Hankel's
memoir; and the researches of Weber, Math. Ann. vi. (1873), pp. 146-149 must also be
mentioned.
The asymptotic expansion of Kv (z) was investigated (and proved to be
asymptotic) at an early date by Kummer; this result was reproduced, with
the addition of the corresponding formula for I, (z), by Kirchhofft; and a littleknown paper by Malmsten+ also contains an investigation of the asymptotic
expansion of K, (z).
A close study of the remainders in the asymptotic expansions of Jo (x), Y0 (x), 1o (x)
and K0 (x) has been made by Stieltjes, Ann. Sci. de l'Fcole norm. sup. (3) III. (1886),
pp. 233-252, and parts of his analysis have been extended by Callandreau, Bull. des Sci.
Math. (2) xiv. (1890), pp. 110 —114, to include functions of any integral order; while
results concerning the remainders when the variables are complex have been obtained by
Weber, Mfath. Ann. xxxvII. (1890), pp. 404-416.
The expansions have also been investigated by Adamoff~, Petersburg A nn. Inst. polyt.
1906, pp. 239 —265, and by Valewinkll in a Haarlem dissertation, 1905.
Investigations concerning asymptotic expansions of Jv (z) and YI (z), when
i z is large while v is fixed, seem to be most simply carried out with the aid
of integrals of Poisson's type. But Schlafli1~ has shewn that a large number of
results are obtainable by a peculiar treatment of integrals of Bessel's type,
while, more recently, Barnes** has discussed the asymptotic expansions by
means of the Pincherle-Mellin integrals, involving gamma-functions, which
were examined in ~~ 6'5, 6'51.
7'2. Asymptotic expansions of Hv(1) (z) and It,(2) (z) after Hankel.
We shall now obtain the asymptotic expansions of the functions of the
third kind, valid for large values of z J; the analysis, apart from some slight
modifications, will follow that given by Hankeltt.
Take the formula ~ 6'12 (3), namely
I 2 \ a 1 (27r-o1) 7 ooexp / iu \v-2
HVh Z- (V (+2< j) e-ut (+2z/ du,
valid when - -rw < / < -7r and -,r + / <argz< 7r +/3, provided that
R (v + 2) > 0.
The expansion of the factor (1 + iuz/z)v- in descending powers of z is
(V - ~) iU (,7 - 2) (V -). (iU)2
z + 27 2";
2z 2. 4.z2
* Journalfilr Math. xvii. (1837), pp. 228-242. t Ibid. xLVIII. (1854), pp. 348 —376.
+ K. Svenska V. Akad. Handl. LxII. (1841), pp. 65-74.
~ See the Jahrbuch iiber die Fortschritte der Math. 1907, p. 492.
j1 Ibid. 1905, p. 328. T Ann. di Mat. (2) vi. (1875), pp. 1-20.
** Trans. Camb. Phil. Soc. xx. (1908), pp. 270-279.
tt lMath. Ann. i. (1869), pp. 491-495.