28G 
MR. E. W. P3ARNES OX THE THEORY OF THE 
In an analogous treatment of the double gamma function we may expect that 
similar limit theorems will be i-equired. This, in fact, is the case ; but for our 
present purpose it is sufficient to take particular cases of the asymptotic expansion 
for log r( 2 :). 
To make use of this approximation we need only remember that (“Theory of the 
Gamma Function,” § 39) if 2 and w he any finite complex quantities, and n a positive 
Integer, 
log H {z + nioj) = log [2 + (n + l)w | w] — log Fj (2 1 w). 
m = 0 
We suppose that such values of the logarithms are chosen that additive terms 
involving 'Ittl do not enter. In other words, we shall say that the logarithms have 
their absolute values, the formula just written ])eing merely a convenient way of 
w]'iting the identity 
n 
II (2 ffi wiw) 
a> = 0 
I \[2 + {n + !)&)] 
I>) 
On differentiating this identity with respect to 2 we have 
k 1 
, = 0 + '»;&)) 
= v///^'[2 + (n + l)w i w] — i//f^)(2 1 w) 
with the notation of § 2 of the “ Theory of the Gamma Function.” 
§ 19. The double gamma function of 2 with jmrameters and we write 
ffi( 2 |w^, 6 J 0 ). 
When there is no doubt as to their presence the parameters are omitted. From 
tills function we form the subsidiary system 
1 / 1 . 3 '^’ (2 1 0 )^, C 0 . 2 ) = log To (2 1 013 , W 3 ). 
^^2® (21 "2) = log r2(21 "n "2)- 
and so on. 
As a definition we assume 
I 01 ,, 013 ) = 
_ 2 V-^ 
n!, = 0 i)ij=oG "t wq&l^ + 
This double series is, liy Eisenstein’s Theorem,^ absolutely convergent, provided 
the ratio of oij to 013 is not real and negative. This limitation on the parameters 
holds throughout the whole theory of the double gamma functions. It corresponds 
to the limitation in Weierstkass’ theory of elliptic functions that F must not be a 
real quantity. 
* r. Forsyth, “ Theory of Functions,” § 56. 
