DOUBLE GAMMA FUNCTION. 
315 
subject of integi'ation), and the positive half of the real axis ; and extending from 
+ 00 to + 00 as in the figure. 
Under the limitations specified the integral is, in general, finite. Moreover, by a 
theorem previously obtained,we have under such limitations 
tUd - s) 
(Ct A -f- COoWoV 
the latter expression having its principal value. 
And therefore 
pa qa. 
S' X 
'Itt 
!, = 0 1)12 = 0 (c + + (^- 2 ^ 2 ) 
s)rl — + 
J 1 — (’“"1= 
tr (1 — S) r 1 — e- ^ 1 — ‘ \, 7 
. —-- e ‘ U —z}-' "dz 
iPn — s) 
U is, a\o),, OJa) -+-^ -;-—;-^- ( 
jP ^ /■ [<l+i' (il + llu>i + q(a + I) lu,]; 
+ 
(1 — (1 ~ 
J (1 — ^ 
^y-‘dz 
all the integrals being taken along the fundamental contour. 
When u is a large positive integer, we proceed to throw the two integrals last 
written into the form of asymptotic series. 
For this puipose consider the expaiision obtained in § 16, which may be written 
(1 — 
_ ^ / I \ 1 2 ^ 1 + "ih, I 
— ; — ob o(« + Wj) d-— 2 + 
I (— )'‘ + C^l) I 
^ -r • • • 
We showed that this expansion is valid provided jzj was less than the smaller of 
• • ‘ wTrt- ^TTt 
the two quantities ^— , — . 
&)l 0)0 
Outside this circle the series diverges. But within the region bounded by lines 
going to infinity from the })oles.of 
-g —(«+W, )2 
1 
(1 — C-“U (1 — t'-“U 
* “ Theory of the Gamma Function,” g§ 22, 33 and 34. 
9 Q •> 
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