DOUBLE GAMMA FUNCTION. 
323 
Hence if log (— 2 ) is real when 2 is negative, and is rendered uniform by a cut 
along the positive direction of the axis L, 
{ 0 ( 6 ' -f- e, «IW], Wj) 
. (-)-i 1 
- ch 
_ ^^5—l^SMsTTt 
‘Iir (s — 1) ! e J L (1 — c (1 — 
X 11 — e(T + • • • + — 
for 
1.s - 1 
X "11 -j- 2]\!l7rte -{- . . . 1, 
r(e-s+ 1) = 
1 + ye + . . . 
1 + e log ( ~ 2 ) + . . 
r(e) 
(e — s + 1) (e — s + 2) 
( - 
1 ~ ye -f . . . 
(s - 1) ! e 
Now when s is an integer greater than 2, 
zy~'^d: 
1 + ^( Y^" 
(e-1) 
+ • • • 
r e-‘‘H-zy-^dz _ ^ 
J L (1 — (1 — ’ 
(1 —e e “ 2 ^) 
for the integral may be reduced to two line integrals which destroy one another, 
and an intesfral round a small circle enclosincj the origin whose value is zero. 
o o 
We have then, on making e = 0, 
C(5, a\o}^, wo) = 
^ (-yy^ f e-«(_ 2 ;y-i|_ log(_ z) + 2M7ri + ^ + . . . + - yj 
27r (s - 1) ! 
{1 — e “'b (1 ~ e “ 2 ^) 
dz. 
But, when s is an integer greater than 2, 
d- 2y-i|2M7rt + Y + • • • + J - 7 
(1 — c““‘b (1 — 
dz 
vanishes for the reason just assigned. 
We see, then, that when a is positive with respect to and Wo, and .s is a positive 
integer greater than 2, 
al«i, (o.2)={ — y 
2 )''“^ log (— 2 ) 
27r (s — 1) ! J L (1 — {1 — e 
and therefore under the same conditions, 
e““(— 2 )*“^ log ( — z) 
dz, 
1 ^ 2 ’ ^ 2 ) — 9 
dz. 
27rjL(l —e "1^(1“"^ 
43. Put now in this result 5 = 3; then with the assigned limitations 
— 2 )- log (— 2 ) 
" 2 )-Yrfp (i_6- 
(1 — e *^1*) (1 — 
2 T 2 
dz. 
