DOUBLE GAMMA FUNCTION. 
329 
Therefore on making a = 0, we have by § 45, 
^*-4? ~ ~~ 022 ( 0 ) 2 , 0)o) 
r?~J 1 , oSi^^h'') , sSit^.'hy) 
[ (1 ~ e “ 1 =) (1 — c ~“20 
+ 
e ' [1 — 
d 
On diiferentiating the formulfe whicli expre.ss the logarithm of the double gamma 
function as line and contour integrals, we obtain 
_yr g-'^--log(-OrL~ 
hi 97 - 1 T n _ “k 2 *^ oM^l 
0),, OJo 
= f>)fl(TV7^ 
U’ 
cl 
27rjL (1 — e “1^(1— c “aO 
y + (M + m 4- 'ni'). TTi 
+ 
(1 — e““2=) (1 — 
Similarly, again diiferentiating, 
. _ I C C-«(— 2 ){log(— 
-1 “ 9„ I I 
e--oS'o (a) + oS'o (o-) [(M + m + my27n]. 
CO] ^ CO.-) \ 
- 2 ^ I(1 _ 
= -L(l)“'^{(T3v4u 
And, if .s be greater than 2, 
+ .SV"(«) 
-A + ^- ^^0-^ ^«) I + («) 
(1_ C-“1=)(1~ c-“2d ' ^2-0 v/jl.O \ / 
y -j- 27n(M A d“ 
(M + m 4- in') . 2771 
yya\co^, coo) = 
I r c “* (— 4 ^ ^ Og (— d' 
(1 — c "rj (1 — c “ 2 =) 
- (2) 
c “4- ~) 
5 — 1 
(1 — c““iO (1 — c “24 
ck 
O) 
Notice that, when we have the more narrow restrictions, the real parts of «, 
2 , and o)o are all positive, the constants ?>g m', and M all vanish, and there is a 
substantial simplification in the formulae. 
§ 47. We may ikjw deduce expressions as line and contour integrals for the first 
and second double gamma modular forms 
y 22 (o) 2 , ojo) and y 2 e(<^i) 
We have seen (§22) that 
CO 
n 
( 02 ) = yoo(o)^, 0 ) 2 ) 4” ) 4- 
rc 
■» CO 
s' s'' 
+ 7 *0 
1 
1 
+ 
+ m-^co-^ + + '‘h^2 (”h“i 4“ ’” 2 *^ 3 )" 
and, therefore, on making a — 0, 
yi’3("n "i) = ~ 
so that, by the last paragraph, 
aOO 
y 22 (w 2 , Wo) = (L) dz- 
I 0)2, OJ 2 ) 4" “ 
a = 0 
(1 — C "14 (^- — ^ “^) 
1 - + 4.8/^(0 
J 
oS'o ( 0 ) 27rt(M -{- ra + in'). 
2 u 
VOL. CXCVI.—A. 
