DOUBLE GAMMA FUNCTIOX. 
383 
Let now the perpendicular contour be distorted into a contour which encloses the 
points 2, 1, 0, . . . — n, and which after the point cr = — 7i again goes off to 
infinity perpendicularly to the real axis. 
So far as the value of the integral is concerned this contour will differ from the 
second contour only hy two strips at infinity of length less than (?i -j- 3) parallel to 
the real axis : and by the previous paragraph the integral along these strips will 
vanish. 
By Cauchy’s theorem the value of the integral along this third contour will be 
equal to minus the sum of its residues at the points 2, 1, 0, . . . — n, together with 
the integral . along the perpendicular line cutting the axis of cr between the points 
— 71 and — (n + 1). 
Now, when s = 1 or 2, the residue of the integral is equal to the coefficient 
of 1/e in 
(-f 
e 
T —'>«+' /I 1 \ 
^ ^ («) + (+•••+ 7-1) («) 
eU- 1)! = 
1 . . . 
s 
+ (g + 1^; 2(m + 111) 7TL 2S/*+'>(o) ^ {1 + elog 2 + 
] .yS 
and is therefore equal to 
by § 53 , 
sd 
f + • • . + y - log 2 
(«) — 2{'ni + 7-ii) m (o,)]. 
s : 
where tlie logarithm has its principal value with respect to the axes of — (a)| -f 
AVhen e = 0, the residue is the coefficient of 1/e in 
1 + e ]oo' z . 
S'\ (a) + e log + 
by § GO. 
and is therefore equal to 
, 8 ', (a) log j +log 
When s = — 711, the residue is 
We therefore liave 
(-)”^+Cfr»+i(<0 
m (//; + 1) 
1 haC + ^0 ^ , ni f \ 
loo; -- - {a) 
r 
Loffi) 1! 
log 
: { uyL <’ '"■'“’=‘""'>"1 - 2 S '.(«)[ iog ^ - 2 (.» + «>.] 
[p2 "of J 
2 .vS 
— s — 
ffiis! 
+ («) jlog 2 — 2{7n + Jh') TTt — ^ . . . 
(-)lS/n + l(« ) 
+ “ 
ra(m + ^)z^ 
ffi (2, a 
0),, Wo) 
