MR, E. W. BARNES ON INTEGRAL FUNCTIONS. 
457 
or, 
log P p ( z) — (m — l) log z — P { m ~ 2 ) log m + P m — P l°g \/ 27t 
ps 1 *- (— ) s m ps , 4 (-r 1 
k r ( _ y-i 
+ m t K 
s=—k 
m 
_ i _L x ni _i_ v 
sz* ps + l] s -_ k 2 sz 5 S=1 sz? 
+ 
v (~) f 1 ]B >- [ - P , v/ (~) s_1 /> S \ «l' 
r= 1 
2r-1 
-£-L V' _V_ L 
12r. 2r — 1 ^ s =_, ps - 2r 
ps] 
+ 1 \2r/ ss* J 
This expansion is arithmetically asymptotic with regard to m, and the coefficients of 
various powers of ~ are ultimately to be summable divergent series. 
1 
Let now r = \z\ be large, and such that vp lies between m and m — 1. Then the 
m p . 
modulus of is a quantity which is very nearly equal to unity. We proceed to 
“ sum ” the series 2' —~, 2' v „ - , and 
£ / \ 5 1 pS 
%' / ( ) ^ 
$ (-)- 1 
m ps 4' (~y ™ p> 
= -k s • {P s + 1) 
ZS ’ j/ljfc 2s 3 s 
I 
r * 
2?-! dm~ r ~ i 
i — p log ™ + s' 
l s=-k 
Write t = log — , then the first series becomes 
o r 
Thus 
k ( _ 1 
f(t) = 2' —~ — — e s(( “ ,0) 
s=—k 
S . (ps + 1) 
and 
8/(0 *, (- r'p 
P 3£ «=_i- ps + 1 
So that 
3/(0 v *, (- 
/w = - p » +Ar 
t} 
s(t—iO) 
g.? (t—iB) 
If now we “ sum ” the last series we obtain 
3/(0 
y(£) = — p -f £ — ; and therefore 
f (t) = Ae p + t — i6 — p, where A is independent of t. 
When t = 0, 
Hence 
/( 0 ) 
k ( _y-i 
f{t) = 2' e~* ie . 
J v ’ 3= _i s.ffis + 1) 
t /—y-i * 0_y 
= 2' e~ sle + p 2' 1 \ e~ 
o N 
VOL. CXC1X.—A. 
