DOUBLE GAMMA FUNCTION. 
337 
pH -f" 1 j pH “f 1 
co^ 
+ 
pH + 1 
0)., 
<pl + 1 
log (/>W| + qoj^) -f- ^ " (log (/oj, + 
CO.-I 
CO, 
(log (JCO., 4” 
where log (^^^i -j- geo.,) lias its jirincijial value with respect to the axis of —(ai^ + <^ 2 )- 
§ 52. Let us next put 5 = 2 in the final equality of § 50. 
We shall obtain, in the limit when n is infinite, 
irii q'li 
V V 
1 
- ^ , -; = Lf. 
m,= 0 m 2 = 0 {cH + IHjWj + }ll^CO.,y s=2 
Co(s, a I oj^, Wo) — ^ ^ 
+ (1 + log n)oS/-'’(o) 
— Fo{oS;"'(w) log^^w], 
the logarithms having their principal values with respect to the axis of — ("i + Wo). 
.oS/3)(0) 1 
Thus 
Lt 
i^is, a W^, Wo) 
oS 
s 
a = 0 
qyii 
— V 
cgil 
y^/ 
1 
J 
0 
0 (hIjWj -t 'IH^CO^y 
cP 
+ 
WiWo 
[log n — log ( 2 ) 0)1 + qwo) + log pw^ + log gwj. 
Now the left-hand side of this relation is independent of p and cj, and therefore we 
see, by § 22, on putting = g' = 1, that 
Lt 
s = 2 
a = 0 
'r(‘ n\ \ 
a\co^, CO.,) - 
s 2 
cP 
+ — = — 7;’i(a>i, Wo) — 2(7?i + m')7r(,oS['^^(o). 
Wj^Wo ~ 
This formula again agrees with results which can be deduced from the integral 
formulse. Incidentally yo;(wj, Wo) has been expressed as the more general limit 
I p"' 111- I 
yn("n - -log ^-v- 
0 ) 3^0)0 
[log (pco^ + (/Wo) - log pwi - log qco.^ 
where log (y^w^ +'Uywo) has its principal value with respect to tlie axis of — (w^ -b Wo). 
§ 53. We can now finish the investigation indicated at the beginning of § 50, and 
oljtain the expression for hju — ^ , without integrating under the sign of contour 
^ ^ po(Wi, CO.p tab fc, 
integration. We have 
— log To (a 1 w^. Wo) — y “b "b lo 
where n = nqwj^ + m.,co.,. 
VOL. UXOVi.—A. 
iga 
P'll qil 
+ Lt t 'Z' 
n=<x) 0 0 
}}(i ht2 
2 X 
h.g (a + O) - log + 2^ 
