DOUBLE GAMMA FUNCTION. 
375 
of the paragraphs leading up to § 49, we evidently obtain for 
Jll 71.2 
log r 3 (a) -f log IT n (a + + wigWo) , 
7J^^ = 0 7Jl2=0 
an expansion in powers of ^ and y, every term of which involves a algebraically, and 
of which the non-ultimately-vanishing terms are typified by (1, n)^ log n -f- (1, ??)“. 
In the second place, consider 
1 T. / ^ 1 ft Tihf + TOi&), l^a) 
loo- To (a) — loa; II 
We have seen, in § 30, that 
To (a) = e- y-1 - “ (a I Oo) 
X IT 
1 = 1 L hp/MjCudwo) 
the product being absolutely convergent. 
The typical term may be written 
rp a 4- g-ai|(,(‘)(miWi I W..) - i|<i(2)(mioji | u,..) 
Exp. 
I 41 l"^) + • • • 
Therefore (“ Genesis of the Double Gamma Function,” §§ 4 and 5) 
«i 
log r 3 («) — log TT [a + I ojj) 
vii — 0 
admits an asymptotic expansion of the form 
A. 
(l,np)log 7ii + (l,72j)2-f N -1, 
r=l 
each term of which involves a algebraically. 
Combining these results we see that 
log r 3 [a + 71.^0)^ + 
admits an asymptotic expansion in powers of — and —, each term of which involves 
a algebraically, and of which the terms which do not ultimately vanish are typified 
by (1, n)‘Gog n + (1, 
But log r 3 (rt Wjoj, -fi is a function of a -|- -}- n.,o}.^. It must then be 
capable of an asymptotic expansion in powers of - - -, each term of which 
“f" 7VctO)2 
involves a algebraically, and of which the terms which do not ultimately vanish are 
typified by 
