290 
ME, e. w. baenes on the asymptotic expansion of 
We therefore have 
E a (-x ; 0, P) = - 
(-) n x- n _ J_ f x ,J e (-y) p irdy 
2m J V (l-a9-ay) sil 
xy) sin tt (0+y) ’ 
the latter integral embracing the positive half of the real axis and including the 
origin, but no other singularity of the subject of integration. Since the zeros of 
Y (l — ad—ay) sin tt (9 + y) lie outside the contour, we may employ the summable- 
divergent series ® 7 „ 
77 -/{r (l—ad— ay) sin n {0 + y)} = 2 ( — ) d n y 
n= 0 
under the sign of integration. The integral will then be represented asymptotically by 
x 
2m n =o 
oo * 
% (-y)-P +n d n e~' Jlosx dy = - x~ e 2 
d. 
.to vjfi — n) (log x )’ 1 ^ 1 ' 
If then | arg x\ < tt( 1-«/2), we have the asymptotic expansion 
00 ^ dr, 
E a (-x;9,P)=- 2 
- + £ 9 (log xf 1 2 
n ti V (1 -an) {9—nf (-x) n 7 »=o T (& — n) (log x) n 
Therefore when 0 < a < 2, and 2ir-£«ir > arg x > we have 
e.(., ; e,p ) (A) - 
In this formula the principal value of log (-x), whose imaginary part lies between 
± 7 T, is to be taken, and ( O—nf is defined with reference to the cross-cut previously 
taken 
§ 49. To obtain the asymptotic expansion of E a (x; 9, /3) for other values of , aig x j 
when 0 < a < 2, and for all values of arg x when a >2, we consider the contour 
integral 
_Lfr(-as) ds 
2m j ' 7 sm 7rs (0 + s)^ 
round a contour which embraces the positive half of the real axis and includes the 
poles of r(—as) and the points s = 0, 1, 2,... », but no other singularities of the 
subject of integration. 
In the subject of integration q is an odd positive integer equal to 2p+l (say), and 
x s and (d + sf are defined as in the previous section. The integral is evidently 
equal to 
sin 7 Tginja. _^ 
1 ^ x 
,m/a 
x m 
a m =o ml sin 7rm/a (i 9-Vmfaf m =o Y (1 + am) (9 + m) p 
The first series is equal to 
= —a' 3-1 2 G p {x l,a e 2n ^ ,a ] 0.9}, 
^m/ag2irtmp./a 
1 P 
_± £ £ __ 
a fx=—p m= o ml (9+m/a) 
