1921 - 22 .] The Confluent Hypergeometric Function. 
89 
VIII. — The Asymptotic Expansion of the Confluent Hyper- 
geometric Function, and a Fourier-Bessel Expansion. By 
Dr T. M. MacRobert, M.A. 
(MS. received February 16, 1922. Kead May 1, 1922.) 
PART I. 
The Asymptotic Expansion of the Confluent Hyper- 
geometric Function. 
In Whittaker and Watson’s Modern Analysis, chap, xvi, the asymptotic 
expansion of the confluent hypergeometric function Wfc,^( 0 ) is established 
for the region — 7 t< amp0<7r, 2 ; AO. The object of the first part of this 
paper is to show that this expansion is valid in the extended region 
— 87t/ 2< amp2^<37r/2, z^O. 
In the equation 
(1) 
which is valid for R(m — /(j-f- 1)>0, — tt < amp < tt, z^O, assume for the 
moment that amp 2 ^ = 0. Then the path of integration can be deformed 
into a straight line from the origin to infinity, making an angle \[r with 
the ^-axis, where — 7 r/ 2 <'i/A< 7 r/ 2 . Thus 
r( J - k + 7n)Jo 
/*'* / y\k—i+m 
J + ij 
dl, 
( 2 ) 
an equation which is valid for R(m — /c-hl)>0, \f/' — Tr< z <\j/- ir , 0 AO. 
Now 
where* 
~„r\ I + )» — »•) \z/ ' ‘ 
Tl(k-^ + m) 
n{/c— j + jrt - s)\zJ Jo 
k—i+m—s 
dt . 
Hence, employing the formula 
T{z)= 
Jo 
* Of. Prof. G. A. Gibson, Proc. Edin. Math. Soc., vol. xxxviii. 
