DOUBLE GAMMA FUNCTION. 
3(L3 
Integral Formulre. 
§ 70. In the general theory of multiple gamma functions the fundamental integral 
formula expresses the fact that the integral of the ?i-ple gamma function can be 
expressed in terms of {n-\- l)-ple gamma functions of specialised parameters. As we 
have not yet defined the treble gamma function, we cannot prove this theorem for the 
case when n = 2. In the case when n = 1, this proposition reduces to Alexeiewsky’s 
theorem (“Theory of the G Function,” § 13). We proceed first to translate this 
theorem into the notation of the present paper, and then to give an alternative proof 
capable of extension to the ?z-ple gamma functions. 
The G function is substantially the double gamma function with equal parameters, 
the two being connected by the relation 
r.r' {z h, co) = G (^ 
\<j} 
[“ Theory of the G Function,” § 29.] 
Bv differentiatino; Alexeiewsky’s theoi’em we obtain 
a o 
0 = T log 277 - 2 - « + i + (2 + « - 1) £ log r (2 + «) - £ log G (2 + a), 
and therefore, writing 2 for 2 + a, and substituting from the relation just quoted, we 
find 
0 = — - -l-| 4 -(' — log r (+ oj log 1^3 ( 21 oj) +' “ log oj. 
But logr(^) = logr,(t|‘»)-(^-1 
We thus have 21 /;/( 2 1 w) + (2 + w [ w) = S/( 2 1 w).(i.) 
On integration we have 
j log {a I (o) da = a log Fj [a | co) — Sj (n | w) -f w log 
j 0 
To (O) I co) 
We may put 2 + « in place of 2 , so that 
(2 + a) log (2 + « I oj) + Y, ^‘^8’ ^3 (2 + ri -{- oj j w) = S/ (2 + a I (o). 
And now, on integration with respect to 2 between the limits 0 and 2 , we obtain the 
extended formula 
logG(2 + a I (o) dz = (2 + a) log (2 + a I w) + &) log 
J 0 
To H“ (( “h CO I co) 
To (« + w I CO ) 
(2 + a I w) + Si {a\(o) — a log G (a | co). 
3 A 2 
