MR. E. W. BARNES ON INTEGRAL FUNCTIONS. 
4 77 
and, by putting 6 — 0 in the Fourier’s series, which we considered in the preceding 
1 /_y>—1 
paragraph, we see that this is equal to-+ 1— • 
Therefore 
/(<) - (-r 1 ^ 
- +1 
L P 
+ t-~ 
P 
so 
that 
2" (uuT 1 m* 
^ i + 1 
= (-r 
■i 
m 
T . , m ]p ' 
L P 
+ log 
m 1 '* 1 
+ lo « T - P 
Revert now to the asymptotic expansion for log R p ( 2 ). 
We find on substitution that 
log Pt p ( 2 ) = (m — 1) log z — - {{m — i) log m — m + \ log 2n } 
P 
» ( _ y-i 
_p v' 1 
*=~P 
sz- 
F (: +(-)'-'= 
m 1/p . 1 
* P 
m l/p h L (-«y, n W ,p 
+ m (log — J + v -— log m — log 
And thus, when p is an integer, 1 2 1 very large, and — tt < arg 2 < 77 , 
log R p ( 2 ) 
- |log 2 + £' ( -y^-F l^j - ^ log 2 tt + (- 2 ) p log 2 + (-) p 1 7 
.S=-p 
2 P 
As formerly, this expansion is, in form, independent of the argument of 2 . 
§ 71. We may easily deduce this theorem independently as the limit of our former 
results. 
Take the asymptotic equality 
n ; (i + ^le 
n = l 1 
■ + ... + 
<-*>*- 1 
P~ l 
2 >—111 p 
* 
® 1 
= (2Tr)~ty 2 _i e smnp 
+ 
UU / X O - J 
2 ' k±_ p/q 
S=-p+l \P/’ 
where p lies between — 1 and 
Put now p = p — e ; then R i; ( 2 ) is the limit, when e vanishes, of 
n 
n = 1 
1 + e 
n 1 
1 
,p~ e 
-FA 
U-G n p-* 
X n e* n 
rt. = 1 
p 
p-« 
