DOUBLE GAMMA FUNCTION. 
321 
unless the axes of L and-- embrace the neo-ative half of the real axis. In this latter 
n ^ 
case, since the imaginary part of the initial value of log (— L) lies between 0 and 
— 2771, as also does that of log ^ — we are giving a different prescription to the 
many-valued function which occurs in the subject of integration. 
We therefore have 
dXl - s) 
2it 
zy~\lz = 
tr(i — .<!) 
i'TT 
J /I 
dz 
where [jl has the values which have been assigned to it. 
Now fU (1 — s) 
ZTT 
ye-“^-{—zy-^dz = 
the logarithm having its principal value (“ Theory of the Gamma Function,” p. 107). 
Hence ~ g }| g-nc^_ zy~^Jz = ^-suog n + 2 ix'7ti] 
CTT 
where /x' = 0, unless n and ^ embrace the axis of — 1, in which case fx' = :^ I, the 
upper or lower sign being taken as ) is positive or negative. 
Finally then 
1 
where the latter function = e where log n has a cross-cut along the axis of 
— y, and is real when n is real and positive. In other words, log n has its principal 
JLi 
value with respect to the axis of — p 
§ 41. If now we apply to the integral 
dui —s)r —zy-hh 
277 J L 
(1 — (1 — e~ 
the procedure of § 38, we shall, for all values of s, a, Wj, and Wo, such that a is positive 
with respect to the w’s, obtain the asymjjtotic e(|uality 
pn qn 
dffl - s) 
z)’-hh 
ui = o 111.2 = 0 {y + 4- 
277 
r ( — .c 
J L (1 — I 
+ 
-)(1 — 6"“^ 
_1_ J 1 1 _ 1 
(s — l)(s — ‘2) -1- qw. 2 )'~^ 
1 
1 
2 cc -j- (Uj -|- CO 3 
2(s — 1) [(27/^60^ + 
[over] 
2 T 
VOL. CXCVI.—A. 
