DOUBLE GAMMA FUNCTION. 
3G7 
we see that 
CO] a>2 Wj + <0-2 
I" + I' + j ' - i f ’ log hi (2 i ojo)d 
L^o Jo JO -'oj 
"1 1 /'"n \ I "2 1 T-. /"’I \ I "1 ■+ "2 1 /"i + 
= log hi ( M 1 "2 + T Ti h 1 + —.0“ ^ log Ti — 
a)i + (O.y I 
O)., 
<Wo I 
— -T'log ri(aJi|w 2 ) +4Si(coi|wo) — Sh Si( S^( ^ 'Iw 
2 ' "hi 
"2 
[r„(a,, + "1 Iwo)ro(wo + 
1 w 1 r (w + "1 + "21 
".d| 
1 Tjfw^ + Wo 
\<^i) ppis^i) 1 
and this expression in turn, by utilising the inultiplicati(')n formulae when m = 2 for 
the simple and double gamma functions and the simple Bernoullian function, is 
equal to 
- 2C0, + log ^ 7 co,|a,„ co,) 
+ Wo I ^ 
log 2 
+ |S/(oiw,) - S, + ^log 
■/ 
„ 1 i 
, Wo 1 I \ A / I , -JWo 1 , . 
+ "y log <1 -^-—-r-^-r '(■ d-log OoloJo, Wo). 
2 ® I ,0 / w. + Wo. \ I - a r ~\ -> -/ 
—":"2jrihTf."2 
L 
J 
Now, on making wj = Wj, we obtain from the multiplication formula 
loo’ 
o 
— 3 log potato, Wo) -|- [oSx^(o 1 ^o)— 1] log 2 . 
Therefore the expression which we have just found reduces to 
- 2w, + ^) log p,(w,) + j f f f ,Sh(w, + w, I w„ w,) 
■h ^ 2^1 0^ 1 ^2’ ^ 2 ) I log 2 
+ I ^2' (0 1 <^2) “ ("y I ^2) + 3 wo log p.2 (wj, w.o) 
And therefore 
f w, 
log ^ 2 ( 2 1 wj, oj.)dz 
0 
— 3w^ log ^ 2 ) ^(^2 log Pzi^Jy log Pi (^ 2 ) 
— 3w^(rn + ni')27n 2 S](o) + (1 + 2inm) 
~ ^1(17 I ^2)+ l^ 2''(0 I 
