322 
1 
s — 1 
MR. E, W. BARNES ON THE THEORY OF THE 
1 
1 
1 co^iqno)^)^ ^ 
+ 
+ 0)^ + COj) 2&' ^l) 2^ "F ^ 2 ) 
{pnaiy + 
(q)lCL).2y 
+ s 
(—)“ /'>>1 + S — I 2^»i (® + + f<^2^ “F 2^^"‘ + l o^mia + Wj) + 2^^'" + l 
I 'll 
III — 1 
111 
{'po)2 + q^i) 
m + s 
YmY F (O.i) + 2^"‘+l l > 
wherein all the many-valued functions with s as index have their principal values 
with res^^ect to the axis of — It proves convenient to consider these functions as 
1j 
having their principal value with respect to the axis of — (wj^ + co.^). In order that 
this may be the case, we must multiply the integral by 
^•2Mif7ri 
where, as in § 39, /x = 0, unless -f 7 includes the axis of — 1, in which 
case M = di 1, as I(wi -+- wo) is negative or positive. 
Remembering the definition of « j W], wo) given at the end of § 39, we see that 
we obtain for our fundamental asymptotic equality an expression which in form is 
identical with (A) § 38, but in which the many-valued functions with s as index have 
their principal values with respect to the axis of — (*^1 + It IS evident that 
the equality will hold for all values of a, and will thus serve to define {o('S, a ] oj^) 
for all values of s, a, and oj^. 
§ 42. We proceed now to take such particular cases of the general asymptotic 
equality which has just been obtained as lead to expressions for the logarithm of the 
double gamma function and its derivatives. 
Sujjpose that s is a positive integer greater than 2 ; then, making i/ infinite in the 
general asynqitotic equality, we see that 
where 
(— y 
Ci(s, a 1 ojy, OJ,) = "'(«1 "n 
^ 2 '^ (a 1 wi, 0 J. 2 ) = log lh(a 1 Wi, oj^). 
This relation is true for all values of a, oj, and o)., ; it is the first of a series 
connecting the double zeta and double gamma functions. 
Let us next put s + e for s, where e is a small real quantity and 6' is, as before, a 
positive integer greater than 2. Then, provided a is positive with respect to wj 
and Wg, 
Hhl - 
e) 
2MTri(5+ e) 
r e~Y — z} 
JL (1 — 
‘Itt 
,S ~1 + I 
dz 
) (1 — e “ 2 ^) 
the integral being taken along the L-contour, and M being the integer defined at the 
end of S 39. 
