DOUBLE GAMMA FUNCTION. 
827 
which express as contour integrals the double gamma function, and the third double 
gamma modular form. The first relation only holds when a is positive with respect 
to the w’s. The second is valid for all values of and subject of course to the 
dominant condition that is not real and negative. 
It is worth noticing that the first formulce may also be written 
oSq (a) {m -f- 7n') ‘Itti + [a) {2M7ri + y] 
+ f dz. 
2-77 J L {l—e~ “‘b (1 — 
§ 46. Subject to the condition that the real parts of a and L are positive, we may 
now express our contour-integrals as line-integrals. 
Consider the integral 
r z)-'^ log (- z) , 
277 JL 1 — (1 — 6~“2-) 
, r„(a| WpWo) 
log ^ 
By hypothesis the logarithm has a cross-cut along the axis of L, the initial value 
of its imaginary part lying between 0 and — 27n. Hence if the contour of the 
integral be reduced to a straight line from co to e, where e is a point on the axis of L 
very near the origin, a circle of small radius | e | round the origin, and a straight 
line from e back again to -fi 00 , we shall have 
_i f z)-^ Iog(- , _ _ r e-^%-z)-^dz 
277jL(l - je^ (1 “ ^““^0 
I J_ p”" c-”^^‘^ {loge + 1(6 — 77)} ^^0 
The logarithm in this second integral, which results from the small circular contour 
surrounding the origin, has its })rincipal value. The integral itself is evidently equal 
to (§ 15) 
rd0[ioge-f .(d-77)]rf^, 
e _|_ 2 ^ 1 (^) 
1! 
+ 
= i,S-,(a)log£ - . + 
1 . 
+ terms involvijig positive powers of e] 
-f- terms which vanish with | e |. 
Thus 
i_ r log ( - z) , 
277 jb (1 — (1 — e-“2b 
_ r” z)-\iz 
. e (1 — e““ib (1 — e““2b 
gS'i (a) log e — 
1 , 
,2 d" 
+ terms which vanish with 
