DOUBLE GAMMA FUKCTION. 
331 
+ 
(<Z + &)^ + CO.t) oSj^(« + (Wo) 
^ ^m+s 
m = 1 
m 
J 
(^JCWj + ( 7 CW„)'« + ^ 
{jpnw^ + qnw^Y 
* (—)” f'^'n + ^ — 1\ JoS„j (« + cwi + (Wo) + oB,„+| oS„; (ft + (Wj) E qDm+i 
( 7 -^"i) 
m-Ys 
o^ 7 « “ 1 " ^i} "f" 
(( 2 cwo)“+* 
where, if a is positive with respect to the (w’s, t, [s, a \ cwj, cwo) may be expressed ])y 
e-‘‘-{—zy-'^dz 
the integral 
,21IS7ri 
sr(i-.,) 
L (1 — (1 — 6““=^ 
(§ 39 -) 
Make now s = e, wdiere e is very small; ex2:)and the various terms of the 
asymptotic equality in powers of e, neglecting those higher than the first, and we 
obtain, if the real part of a is positive. 
'pit qn 
(p7i + 1) {qn + 1) — e log II 11 [a fi- + m^ojo) 
mi = 0 y/lo = U 
= ^ fj^ (1 _ (1 + ^log(~ ^)} {1 + ye} {1 + 2M€7rt} dz 
+ ' 1 H~ e (y + ^)] [rd {pMi + qwo)“ [l — e log [paypi fi- qoypi)] 
-j(W]^(Wo 
— [npyoy^^ [1 — elog^^y^cwj — (n()'cwo)'[l — elog qntwo]} 
n {poy^ + ^cwo)[l — elog {pno}^ + qno).^)} 
— np)o)Y [1 — elogpncwj] — ^ 
+ (1 + e)[p7i(l — €logj9»(w^) + (/ft(l — el(jg(^7icw^)] 
+ oS/ (u + (W^ + (Wo) 1^1 “■ e log {^qjyKo^ ~|“ (y?icwo)J {pt “h ^i) [^1 — e log^9/?cWjJ 
oS/ (rt + (Wo) [1 — elog qno},^ 
“b (® “k ~k ^ 2 ) (1 “k e) 
IV 
dd= 1 m 
(a + (W]^ + (Wq) + oB,n^.- | A ^l) + 0^*'” + ! oS,,, (ft + 0}^ + 2 b»J + 2 
iP^l + 2^2)“ 
(^CWi)" 
This ecpiality will hold for all values of s, a, cw^, cw., if the integral be replaced by 
C 2 (e, cwo), the logarithms having their principal value with respect to — ("1 + (W2). 
Equate now the absolute terms in this asymptotic equality, and we find, if a is 
positive with respect to the- (w’s. 
( — 2)~1 
IL (1 — (1 — £;-“^q 
But we have seen (§ 17 ) tliat 
dz “k2*^1 ~k ^i“k^2) — 2^/ ~k ^i)—2^L ~k ^2)* 
{p ~k ewj -k cwo) — 2b] ip ~k ^1) — 2^1 (® “b ^2) — —2^1 (^^') ”k b 
2 u 2 
