DOUBLE GAMMA FUNCTION. 
373 
exists, and then it will be shown that, the possibility of such an expansion being 
established, its actual form is readily ol)tained Ijy a process of difference-integration. 
Subsequently we shall verify the results by an alternative proof l)y means of contour 
integration, this proof being the natural extension to double gamma functions of the 
one employed for functions of a single parameter (“ Theory of the Gamma Function.” 
Part IV.). 
It should be noticed that the expansion under consideration differs materially from 
the expansions obtained in Part III. In those expansions the limits of the number of 
terms of the series and 2 :»roducts, the quantities pn and <pi, formed the infinite basis 
terms; but in the present case that basis is the variable itself 
§ 79. We first write down the asymptotic expansion for 
log Uo (2 d- (.6 1 OJ, (xi) . 
We have obtained (“Theory of the G Function,” § 15) the asymptotic expansion 
1 nr, \ 1 1 A . s ~ 11 r. , — G" 
log G (2 -p a) = -]bj- — log A +-- log 27r + ' - 
a 
00 ( - 1 CO 
^ ^-S4nll,l)+ S 
- iV j ^Og 2 
+1 
\^1n(PLn + 2>A 
+ 2 - 
=i 2)1 ( 2)1 — 1 )^~ 
2 ( 1-1 ’ 
this ex 23 ansion being valid for all values of a and z such that 121 is large, and the 
principal value of log 2 being taken. There is, in the language jireviously employed, 
a barrier-line along the axis of — 1 . 
Now in § 70 it has been seen that 
logG 
log r3(2l w) + 
T-loa; 2tt 
2o} ° 
w. 
and 
Tlierefore 
log A - iP- 
Po(co) 
Pl(«) 
+ i log 01. 
h' 2 (' I 
Pol") 
(z + a — (of 
00 
+ S 
71 = 1 
r -)~" 
^,n{a 1 01) “b 
2bi((fi(c) I 
-I- 1 I 
? 
where now there is a barrier-line along the axis of — 01. 
But by § 5 
gh (<r I oi) 
m + 1 
— 1 ") + 
ob ni + \ (c) 
VI 1 ’ 
SO that we have finally 
