DOUBLE GAMMA FUNCTION. 
355 
ni - 1 »t - 1 
IT n r3(,-i + 
p z= ° ° _ ^ g(-/ir) log + 2 („i+„(Vc,S|'(o) 
Po''‘"~ V«l> «A 
We now proceed to obtain this result at once from the expression of log 
as a contour integral. 
We have seen (§45) that, when a is positive with respect to the w’s, 
pJrOi, Wo) 
1 1.0 («(-) 
log- —;- 
Pii^V ^i) 
— .ohjj (tt) (j\'l -)“ d“ ) ^TTt -(~ (o) 2]\'l77t 
I [• e SOog(-G A 7} , 
^27rjL 
(1 — C -“ 1 =) (1 - C ■ “-0 
and, therefore, m being a positive integer, 
rwj + •'Jw.o! 
lit — 1 //6 — 1 , 
1\ ( C + 
log n n 
m 
. =0 .5=0 - 
p.2 (Wp O).,) 
;/l — 1 ~ 1 
~ r = (M + m + m) ‘Zm 1 £ ,-,80 ( a + 
• = 0 5 = 0 ' 
Til 
1 /rt-i 
+ .WS/(o)2M7r. + 
,)-l| log ( _ , ) + V V 
7* = 0 5 = 0 
(1 — ('““>=) (1 — e- 
But 
and (§ 14) 
, _,"i , , _ (m~ l) lu, 
I — c e ' m 
1 — e““i- 
, _“i: 
3 - A 
III — 1 ill — 1 
I' = 0 8 = 0 \ >1' I ' 
^ i\ ^ TO / \1 "D/^I ^2 \ 
-, “ ) — m ni)^ (wj, Wo) + , , 
III 111 / ~ 1 -/ ~ i \ 
— .3&0 I ^ 1 ) ^3) “h (1 — ?n') oBj (w|, w.i). 
Therefore when a is positive with respect to tlie w’s, 
logll n I — 2 ---!f-L j :z= (M + 7n + ?u')27rt 2 SQ('?na j Wj, wo) + 2 M 7 n 38 /( 0 ) 
'(•=0 5=0 
p/wi, Wo) 
-j~ (n2 -[“ ill )27rt(l — ^U') .-iB^ W]^, Wo) -/ 1 
•*' -- ’'TT 
i r e "A —‘d Hlog(— ') + 7 }d~ 
27rjL (1 • 
8 ince m is a positive integer, the axis L defined with reference to the parameters 
Wj, Wo is the same as that defined with reference to the parameters mco^, ntco., for the 
lines representing these two sets of parameters are coincident. If then we change 
z into mz, the integral last written becomes one which (§ 45) is equal to 
log -j- .o8/(7Ha){log'm — 2 Al7Ttj — o8o(7»«)(/?; + m')' 27 n, 
Po(Wj, Wo) 
the arithmetic value of log m being taken. 
o O 
z 2 
