470 
MR. E. W. BARNES ON INTEGRAL FUNCTIONS. 
the limits of the summable divergent series as k tends to infinity are taken, the 
arithmetically asymptotic expansion 
log R p ( 2 ) = (on — 1) log 2 — -{(on ■— \) log on — on + \ log 2 77 -} 
q (-)- 1 
+ S' fr-F - v// 
S=-p 
SZ a 
In n-r 1 on°/p 
=-* < & S 
+ 1 
p 
( — '\~ p , 
-f-—— log m — ^ S 
1 S 1 ' y > _ 
2 ' q/y s 
s=—k * >n ' 
on s ’ p 
a (-Y_B I+1 _1 1-J20! * (-V* 
^ £ 0 2^ + 2! m~ t+l 1 P ^ 
m' 
S/p 
' r^ +] 
PA +1 
,x' 
s/p 
*=1 
where the double accent denotes that in the corresponding summation the terms for 
which s = 0 and s = — p are to be omitted. 
As before, the coefficient of B /+1 vanishes identically. 
7c ( _ \s-l # 1/p 
The series S' ; ??h /p is equal to log 
S= -7.: S2 S 
It is then only necessary for us to consider the series 
!■- (—y~ l 
S" 
s=—k SZ S S 
m 
-+ 1 
P 
If we }mt t = log m _ , we may write this series in the form 
f(t) = 
7: ( _\s-l p »t 7c 
■ v." 1 — l —A _ s'." / y-i 
( — y- l e st (- 
«=-*• s + 1 1 S— 7c V s S + P 
Remembering that a summahle divergent series may be differentiated, we find 
P 5=-/j S 
or 
/'(0 + p /(0 = ^ + (-) p - 1 e ^ 
Therefore 
f(t) = Ae~ pi + t — - + ( —) p “Re“ p/ , 
where A is a constant of integration. 
Now when t — 0, 
* (— y-i 
f(t) = S" ( > 
s ~-l 
\ 
S( + 1 
P 
7c /I 
s" (-y-M- 
s= —/■ \S 
S + pj ’ 
