MR. E. W. BARNES ON INTEGRAL FUNCTIONS. 
471 
§ 6G. We proceed now to investigate the asymptotic expansion for 
30 
F (s) = n 
n= 1 
+ 
pa n P 
00 1 00 1 
where ct n is such a function of n, <f> (n), let us say, that S —^- converges, and 2 —— 
n=i b=i n n 
diverges, however small e may be, p being real, finite, positive, non-integral, and 
greater than unity, while is an integer such that p + 1 > p > p. 
Suppose that the result of reversing the equality r — <$> (n) is to give n — ip (o'). 
Let on be a very large integer, such that on — 1 < xfi (j z |) < rn. 
As previously, we have 
log F (z) = (on — 1) lo 
VI — 1 
_ s' 
n=l 
log <f> (n) 
+ 
+ 
wi-i 
v 
n=i 
'(f) (n) 
+ 
+ 
(-r 1 
VI — 1 
s' 
n- 1 
Z ( — )PzP 
--—— 
cf>(n) p (/>?<»_ 
+ 2 
n=m 
(—)PzP +1 
_(p + l)^ +I (n) 
+ 
( — )/ ,+1 ^P +3 
p+2<f)p +2 (n) 
Substitute now the arithmetically asymptotic approximations given by the 
Maclaurin sum formula, and we have 
log F ( 2 ) = (on - 1) log 
"Ml 
log (f) (n) dn + \ log <j) (on) 
yo 
+ 
/.■ 
8=-k 
(-)- 1 
in k 
<f)’ (n) dn — \ 5/ 
y, S=-k 
(~y 1 a ( m ) 
sz 4 
+ 
CO 
t=0 
(_-y a +1 
(2 £ + 2 )! 
log <i> (»o + 
S2" 
cP t + l 
dm~ t+l 
ft M 
In this expansion y_ s is infinite, and there is no corresponding Maclaurin constant 
if, and only if, s > p. 
Use indefinite integrals and transform by integrating by parts in the same way 
and under the same restrictions as in § 54, and we get 
log F ( 2 ) = — i log 2 + 
(yo 
log (f> (n) dn + S' 
5= ~P 
(~) s 
sz* 
f ft ft 1 ) 
dn 
ifr (t) dt 
t 
