270 
me. E. W. BARNES ON THE ASYMPTOTIC EXPANSION OF 
§ 25. We will now shoiv that , if lx {fi+s) > 0, and H($) > 0, 
f y f3 ~ 1 e~ Bv (1 — e~ y ) s dy, 
J 0 
the integration being taken along the positive half of the real axis. 
e~ ny + UN- 
Then, if ?/> 0, and H(s) > 0, and, as before, a = -s-\, 
S (s) 1 
r(-s) T(p) 
Let (1 -e~"Y = s' ( *)-( ■•< + »■ !) 
?i=0 J- • " • • • ^ 
|K»| = 
co 
S 
II = N 
( s)...( s + n 1) - ny 
ftl 
< 2 
n=N 
n 
n 
(l +-) C"r 
e~ ny 
r -1 
A rj J 
f / 1 
1 \ 1 
exp 
L (r \i + "' + w/J 
Now 
n 
e^U 
[L CT\ _«fl 
( 1 + - e »■ 
r = 1 
A W J 
tends to a definite finite limit as n tends to infinity. 
Let g be its maximum Value when n > N. 
GO 
Then I|£ N | < g t e~ ny \ n a{1+en) |, where e n tends to zero as n tends to infinity. 
Hence, if y > 0, | Tv N | < Ke~ Ny , if Tv (s) > 0, where K is a definite finite quantity 
independent of N and y. 
Hence, if I N = ( y^~ x e~ By T\ N (y) dy, 
Jo 
II N I < K [ y^~ D e -H N -™} dy. 
Jo 
Thus I N tends to zero as N tends to infinity if lv (ft) > 0 and Tv (s) > 0. 
Now 
r (p) J 
f y 13 3 e 6y (1 — e y ) s dy = 
J 0 
1 [ v^~ 1 e~ 6y \ 1 ^_ s )‘"( _ s + u _i) e~ ny + L 
(i8)Jo y «o «1 
li) e -(«+0y f ^ + I N 
rM 
N_1 r ^-. r(n-^ - ( 
w 
r(-.s) r(/3) »=o 
= -h-\ T r f~ s L +1 if &(/8) > °- 
r(-s) »= 0 n! (ti + d)' 3 v 7 
Make now N tend to infinity, and we see that, if Tv (/3) > 0 and Tv (s) > 0, 
s (s>= dy ■ 
But both sides of this equality are continuous and finite except for the poles of 
r(-s), if |£(0)>O and Tl(/3 + s)>0. Therefore the equality holds under this 
limitation. 
