DOUBLE GAM^fA FUNCTION. 
339 
Hence if — m, when in is a positive integer greater than or ef[nal to zero, 
l,{s, a 1 oji, a>,) = r (1 + m) 
Thus, when 5 is a negative integer, 
(<5, tt I Ct)j, Wo) - 
(ft) 
1 - S 
But by § 12 corollary. 
oh K+1 
n + 1 
— + l {^l ’ nS,, («) , 
and therefore finally, when s is a negative integer, 
^0 {s, a I Wj, Wo) = (c^) “h (*^i > ^.o)- 
When a is not positive with respect to the w’s this formula continues to hold, as is 
immediately evident liy the theory of the function ^o(- 9 , o j oj,, w.,), to whicli we shall 
shortly proceed. 
§ 55. We proceed next to find the values of ^o(.'?, «| Wo) for positive integral 
(Including zero) values of s. 
We have seen, in § 48, that when a is positive witli respect to tlie w’s, 
— z)~^dz 
^ r /T \ ^ _ Q/ / \ 
27r) (1- (1- “ 2^ 1 
(1 — c “1') (1— e "2^) 
so that ^.o(o, a|w^, w.j) — oS'j(«). 
Differentiate with regard to a, and we find 
c~^d 
(L) 
27rJ ^ (1 — 6“"*') (1 — c~“D 
Now liy § 39, when e is a small quantity, 
Cd 1 — e, a Wi, = Uo~‘ dz 
’ 1 n 2 tt JL ( 1 — 
I r c~"-{1 — e loo' ( — z) + }{l“7e+ • • - HI" 2Me7r6 . . .} 
= -f 
d -rr ! 
^ttJL e(l — H~ 
so that, neglecting powers of e above the first. 
dz, 
efo(l — e, a|a)^, Wo) = oS/'‘d«){^ — ye ~~ 2 Me7rt} 
log ( — z)c~''dh 
— * e 
27r jL (1-■ 
But we immediately deduce from § 53 that 
d 
-*,logra«) = 
i r c log ( — z) dz 
27rjL(l—c f 
2x2 
— 127n(M + w + in') + 7 } oS/-d«)- 
