MR. E. W. BARNES ON THE THEORY OF THE 
Now, as has been seen in the “ Theory of the Gamma Function,” § 28 cor., when 
(L) is positive, 
|(L)< 
dz 
= — log e — 7 + terms which vanish with | e |. 
And evidently 
” dz 1 
(L) =-. 
^ 
The integral under consideration is thus equal to 
I (L)'! 
(1 — (1 — 0)^0) 
1 + 
+ terms which vanish with I e I 
If now we make e coincide with the origin, the integral last written remains finite 
and we have 
I ( e "(- 2 ) Gog( — 2 )d.~ 
2?; JL(1 — (1 — 
pQO 
= J(L) 
J 0 
dz 
\ (1 ~ (1 — 
This equality may equally be written 
..2 + 
2Sf“)(c) 
— e 'oS\(a)l— 72 S\(a). 
logEyAnq) _ ^ + 2S'i(o) 2 Mfi 
P 2 ( <^1; <Wo) 
I f ” c «'( — 2 ) G lug (— + 7 r 
^ _L (■ Cl 
27r J L (1 
= f’(L) 
dz 
(1 — C~ “2*) 
dz r e~‘“ 
o I _ 
z [(1 — c -“‘ q (1 
— e 'oS';L(a)| 
under the assigned conditions that a is positive with respect to the w’s, and that the 
real part of L is positive. We thus express the logarithm of the double gamma 
function as a line-integral. 
In order to obtain a line integral for log p.2 (wj, cd.,), we notice that we have 
loi 
>■« = I {e 
Jo 
and therefore 
q_r 
2-7^) E 
e“(-2) ‘ {log(--r 7}d~ 
= (L) 
:7r j L (1 — e “iq (1 — c~ "“h 
(Jz r 
(1 — 
— e 
+ log a 
2Sh«(a) 2Sd->(a) , 0/ / \G 
- 7r H- ^ \ 
