MR. E. W. BARNES ON INTEGRAL FUNCTIONS. 
491 
And we shall have 
pf 0 ) l °g z ■ 1 
mzP 
CM” 
C 1 
pq ’ e p « m 2 s =- 7 . L s \ z J J 1 *=-£• V.P + s 
_ 1f-' 
s = 0 
the double dash denoting that the terms for which ^ are to be omitted. 
We therefore have 
pff) ] °g 
tfqm 
- -_ m + 
q pq z n “ l pq \ c?i " 
i 
t z \P 
Therefore 
+ q l0 S \ffn] 
f(z) = —■- z p log z — { ^~z p . 
J \ ' pq & p-q 
\ 1 
P 
- lo 
O' 
We thus have the asymptotic expansion 
n 
>i=l 
e pqn 
1 + -) e* m 
ci' 1 j 
= z ( 1_c n ) l e 
(,-z) P \ogz (~z) p 
M 
P<1 
pH ? 
(i-uA 
o + ' 
c-W 1 ** 
ip 
X <? S =- /J + 1 s (l-f2(2J + «)),« 
We have now given examples of the asymptotic expansions of repeated simple 
functions with transcendental index in the cases when the order is or is not integral. 
And it is evident that such examples might be multiplied indefinitely. In the 
more complex cases the difficulties of the analysis will, no doubt, be very great; but 
such difficulties in no way invalidate the theory which has been developed. 
Part Y. 
Applications of the Previous Asymptotic Expansions. 
§ 85. We proceed now to consider some applications of the previous theorems to 
such questions concerning integral functions as have been raised in the Introduction 
to the present paper. 
In the first place, a knowledge of the asymptotic expansion of a function serves to 
determine the number of roots which it possesses inside a circle of given large radius. 
Let us consider the simple example of the Gamma function, for which we have the 
asymptotic equality 
1 
PW 
= (2-)-^ 
1 ex P \r 
3 R 
+ 
(-)‘B 
V+1 
1 
,=o 2* + 1.2* + 2 2 : 
> -27+1 
