MR. E. W. BARNES ON INTEGRAL FUNCTIONS. 
4G9 
Take m an integer such that m — 1 < Il p m. 
Then 
m — 1 
Q,(*) = n (l + 
?i=l I 
n 
\ « (-)V 
~ Tr +... + -- • 
.1 /p/e n'lP pnPlP 
x n i + 
,i/p 
(-) 
VP 
e n 1/p 
so that 
Qp (*) 
m—1 
n 
p=i 
n 1 
m — 1 
1 + A 
x n 
n = 1 
e ii 
i-+ + <=^ 
i/p • • • pn viP 
x n 
« = 111 
(-) p z p+1 (-) P+1 z P+2 
P+1 + P+2 + 
, (/) + ])-,! P (p+2) il P 
ancl hence 
^ 9/1 — 1 
log Q p (?) = (m — 1) log z — 2 log n 
P n = 1 
HI — 1 
+ 2 
71=1 
'/l'P ( — )* 1 il*' P ] 
—— ~r • • ■ “r 
sz" 
m— 1 
+ S 
41 = 1 
_ , l p H“ ■ • • 4" 
(~W 
pn 
?>.p 
+ l (_)/>+!2P+2 
+ + 
| p+ i i p+ 
n=lll \_{p + 1)#' 0» + 2)»p 
Now, when m is a very large integer, 
m ~ 1 1 m * ,p 
v __ ___ 
7>.=i / " s/p 1 _«_ 2 
+ . . ■ + 
(-y 
— -2t- 
v2 1 + 2 
- R/ -k -fF C—- 
+ i \ p 
And, when s is positive and greater than p, 
n=7 » n slp 
wi 1_,,p 
1- 
m~* p 
•+...+ 
(-y 
— s 
n + 1 
- s — 1 V P J m2 ‘-p +1 
p \2/ + 2/ 
+ 
We use these Maclaurin approximations and rearrange the double series which 
results as the arithmetically asymptotic approximation for log Q p ( z ). We obtain, 
in the limit, when the limits of the summable divergent series are taken for h infinite, 
the asymptotic expansion 
