MR. E. W. BARNES ON INTEGRAL FUNCTIONS. 
475 
The term which we seek is then the value of 
- M> f M(-)' 
log 2 = ( — )P +1 b 1 pzP\og: 
This is, of course, the term independent of 9 in the expansion of 
- 71 - P \ 
zf +e in 
sin 7r 
(p + 9) 
ascending powers of 6. 
In exactly the same manner, if p — ne s (say) is an integer, the corresponding term 
of our asymptotic expansion must undergo the same process of evaluation and will 
give rise to a logarithmic term. If one of the e’s, say e*, is equal to p, we obtain in 
the asymptotic expansion a corresponding logarithmic term 
— pb t log 2 . 
Simple Integral Functions of Finite Integral Order. 
§70. We proceed now to consider the standard function 
R, (*) = n 
n=1 
1 + Ti p ) e " llp 
n 
+ ... + 
(- zU 
pn 
P P 
, where p is an integer > 1. 
Let 2 = re 1 - 6 , where r is very large, and let rn he a large integer such that 
m — 1 < r p < m. 
Then, employing the same process and argument as before, 
log R„(z) = S {y + . . . + +... 
^ m —1 m—l 
+ (m - I) log 2 — - t log n + t 1 i /p 
P 11=1 71 — 1 I 
+ ...+ 
_ V 
n=m 
(-*) p+1 _i_ (~zy +2 
] « n 4- 9 ”1 * * 
p +1 * p + 2 
Ip + 1 )n~p~ (p + 2 )n~ 
. (~*) P 
pil 
P/P 
Now, when = 1 
P 
"N 1 m‘ /p _ 
s ym* ,p + m ,/p log m — . -fi 
re=l 71 ' p S 
(-Y 
S \ Wk 
p 2(1 \2 1 + 2 
p m~ 
and in accordance with the definition of § 49 we put y = F (— 1). 
If, then, we suitably modify the analysis formerly employed we shall obtain, when 
3 P 2 
