262 
MR. E. W. BARNES ON THE ASYMPTOTIC EXPANSION OF 
Then* 
(-zf-'e-^dz = {0+n)- 13 
2n J l 
when the integral is taken along a contour, as in the figure, embracing the axis L, 
and (6+n)~ p = exp [—/3 log (6+n)\ the logarithm being rendered one-valued by a 
cross-cut along the negative direction of the axis of 1/L, and log(d + n) being real 
when 6 + 71 is real and positive. It is a value of ( 6 + n)~ p so defined which we suppose 
to intervene in the fundamental series by which (x ; 6) is defined. [ Vide § 8.] 
We now have 
G^(x;d) =- r 4—->[ (-O' 3-1 t -.e- {d+n) *dz 
2lT J L n=0 71 ! 
= iV ( l -P) \ t-zy-'exyl-zO+e-’xjdz. 
27 T JL 
§ 15. We may deform the original contour L till it has the position of the figure. 
It thus consists of a small curcuit round the origin, the real axis from 0-t- to R 
described both ways and a line P from It to co parallel to the original direction of the 
* ‘ M. G. F.,’ p. 388. [In this manner reference will be made to the paper cited in § 9.] 
