330 
MR. E. M". BARNES ON THE THEORY OF THE 
The additive term will of course vanish when the real parts of and cog are positive. 
Similarly, we have 
72i("i> " 2 ) = 
a = 0 
= Lt 
a = 0 
_ jl f c ~)+ 7 } , 1 
IttJ L (1 — c “!"")(1 — 
— .2Sq~’ {a)27n(M + 7/1 + ni), 
so that tlie first double gamma modular form is expressed as a line integral by the 
formula 
721 ("n " 2 ) - (1- c-^b) ~ ^ 
{o)'27tl(}>1 + m + m'). 
It will be noted that for the modular forms 
P-zi^v 72i("]»" 2)/ 722("n " 2 ). 
we have, liy making a vanish, obtained line integrals wliich are in general finite, 
although in our fundamental formulae the restriction was made that the real part of 
a should be positive. 
This restriction was necessary to ensure that the contour integral should be finite 
at infinity. It is clear from the mode of generation of the line integrals, that the 
process which has been carried out is pei'fectly valid, since liy the introduction of the 
terms loii’ n,-— allowance has been made for the manner in which the contour 
integral tends to an infinite value as o. tends to zero. 
§ 48. At the beginning of § 43 we entered on the investigation which has just been 
given by integrating with respect to a under the sign of contour integration, and in 
this way we deduced the contour integral for log ro(a) from that for (n). 
We now proceed to show how the contour integral for log ro(«) may be obtained 
without the employment of this process. 
For this purpose we take the fundamental asymptotic ecpiality of § 38, valid for 
all values of s, a, and Wo, the many-valued functions with s as index having their 
principal values with resj^ect to the axis of — (wj -f- Wo). 
7 m qn 
V s* 
1 
„, = 0 m„ = o(« -I- 
— ^2 1 <^1) <^ 2 ) “k 
(s — 1) (s — 
1 1 
ipncoy ~ (qncoy~~ 
2ci -f- a)^ 4“ coj r 1 
2 (s — 1) [QjTicOj + qno)y~^ 
—2_LI 
