. MR, E. W. BARNES ON INTEGRAL FUNCTIONS. 
479 
And now the asymptotic expansion of T (z | p) when \z\ is very large and 
■ 77 < arg 2 < 77 is given by 
(2tt) 2 p z 
_L (-)P- 1 ,-p-A 2 ~r F (—) + 
2p 7 ' 2 p s = l sz s ' p ' P 
When p — 1 , this formula is exactly the asymptotic expansion of T (z) for complex 
values of z, which, as stated in § 3, was first obtained by Stieltjes. 
For, when s is an even positive integer, F (.s) = 0. 
When s is an odd positive integer = 2t 4- 1, let us say, F (s) = * yp 1 . t > 0. 
And F (— 1) = y. 
® 7_y / s 
Thus, when p = 1, T (z ' p) becomes T ( z ), and the series 2 — ~ F ( - 
S = 1 SZ ° \ P 
becomes 
(-y-Bi +1 
t Zo 2t + 1 . 2t + 2 ' z 2t + 1 
, which accords with the usual result. 
§ 73. It is obvious that we can now at once write down the asymptotic expansion 
for G (z) = II 
7, . z \ -A+... + W' 
I 1 + — e a “ “” p 
\ a„ 1 
, where a„ = n? 
1 -L - 1 - + ^ + . 
1 01 € \ ' 01 e 2 * * * 
and 
p is an integer, from the corresponding expansion for the function in which 
1 + + • • • 
and p is not integral. The e’s, of course, are assumed to be 
positive and in ascending order of magnitude. 
The result is 
log IT 
1 + 
+ . . . + 
{-)*zV 
O-nV 
= (~) v zJ log z + (-) 
ZP 
V 
log 
2 p lo §' 2?r + 
00 
s= —p 
(-) 1 
sz s 
i / \v — 1 - p?-*\v 
' sin 7T|?e 1 
. / P(P + 1 ~ , a_ZL 
"I" \ ~~ ) 9 °1 oin 9 m 
sm Z7rpe 1 
_1_ ( __ )P - 7 p-e 2 p I I Z' (_ 1 7 I v g. e V e 2- ■ 
+ ( > sin irpe* * + ‘ + ,=- P ** \ P ’ ’ K h .. 
Zip, 0 ; 
provided 7 y, ^... be not integral (ft = 1, 2, . . . co ). 
Thus, e, not being integral, the first term of the asymptotic expansion of the 
quotient 
is 
( -- z v Z (p, p ; 
We note that Z 
6 1> e 2> • • A _ J 
ij, . . .) n = 1 
' 1 
aJ 
T 
71 
