MR. E. AY. BARNES ON INTEGRAL FUNCTIONS. 
483 
§ 77. Consider now, as an application of the general formula just obtained, the 
asymptotic expansion of IT 
>1 = 1 
1 + ,p 
, where <x and p are real positive quantities 
■jj 
such that % u a ~ p is convergent, and, therefore, such that p > a + 1. 
71=1 
With our former notation 
p, = n", 
rm 
X ( m ) = ] dn 
' !'(t) = t 1/p . 
m 
<r+ 1 
CT + 1 
X ['A (0] = 
a + 1 
t~T~ 
CT + 1 
The constant <j {) arises from the asymptotic equality 
iil — 1 
v 
n =1 
n u = 
x (z&i ^ +1 
,r 0 (2« + 2)! dm~ lPl 
and, therefore, M = — x (g 0 ) = £ (— <r). 
Similarly j p n ft (n) dn = — £ ( — ps + <x). 
The constant y 0 arises from the asymptotic equality 
//i-i 
5 n a 
ii=i 
loo - n dn — 
O 
Vo 
m 
9 
log ?n + 
y (-J9T+1 
tio 2* + 2! dm*** 1 
(m 0- log in). 
ry o 
We may readily show that n* log n dn = £' (— <x). 
For, as has been stated, for all values of 6', 
m-1 1 
£W = s i 
7( = l /t 
— s ?/i' 
y —1 + 
+ t 
t -1 
— s 
2t 
( y -i B< 
(s + 2* 
1) m* +s<_1 
If, then, we put 6- = cr + t, and expand each term in powers of t, we may equate 
coefficients of similar powers in the identity. # 
If we equate coefficients of the first power, we find 
Ho-) 
m ~ 1 loo- n 
V _»_ 
a 
n=l 11 
m 
L(i - o-y 
log VI 
1 — a 
log m -F 
v (~ fWi cI2t+1 Ip g 711 
t -o 2^ + 2! dm- t+1 rri 
* Compare the process carried out in §§ 27 and 30 of the “ Theory of the Gamma Function.” 
3 q 2 
