DOUBLE GAMMA FUNCTION. 
.Sin 
find that ^o(.s, a\o)^, ojo) as so defined wonld involve n. In otlier words, we should 
have made an assumption which could not be justified. 
If we wish to obtain an expansion valid for all values of and we must con¬ 
sider as our starting point the integral 
ir( — s) r e~‘^~ (— zy~'^(h 
2 TT J L (1 — (1 — ^““ 2 ^ 
where (— = gO-i)iog(-o^ iPg logarithm being rendered uniform by a cut along 
an axis L, coinciding with the bisector of the smaller angle between the axes of ^ 
0)1 
and —, and where the integral is taken along a contour having this axis L for axis 
Wo 
(as in the figure), and enclosing the origin, but no other possible pole of the subject 
of integration. That value of log(— z) is to be taken which is such that the imagin¬ 
ary part of the initial value of log (— L) lies between 0 and — 2ttl. 
This integral of course is only valid when a lies between the smaller area bounded 
by the axes of Wj and w.,, or, as we may say, when « is positive with respect to c^i 
and We notice that the line L is uniquely defined, since the ratio wo/wi cannot 
be real and negative. The definition of the integral is not complete when Wj and Wo 
include and are equally inclined to the axis of'— 1 ; in this case we may take L to 
be a line nearly coinciding with this axis. 
We now define the double Ivieniann { function, when the variable a is positive 
with respect to the w’s, and s, ojj, and have any complex values, by means of the 
equality 
tr(l -s) 
tri 1 
