286 
ME. E. W. BAENES ON THE ASYMPTOTIC EXPANSION OF 
§ 44. We consider next the asymptotic expansion of E a (x) for other values of 
arg x. 
For this purpose we investigate the contour integral 
2771 j Sin 7 TS 
which is taken along a contour which embraces the positive half of the real axis and 
encloses the poles of T( — as) and the points s = 0, 1 , 2, ... go, but no other poles of 
the subject of integration. 
It is equal to 
GO 
v 
7)1 =0 
/1 —a 
. . . ™ sin 77 - 
(~) wl ~U- V a 
am! • m 
sin 77 — 
a 
m 
I 
^ (— )T (— an) sin {77 ( 1 — a) n)x n 
■*w--—— ' 
»=0 77 
1 
-exp 
a 
[a«]+E 0 (x). 
Now when s = u+iv and || is very large, the subject of integration tends 
exponentially to zero if 
— t^CCTT + 77 | 1—a] —77+ | arg X | <0. 
If a < I, this condition gives jarg x \ < f 77a ; and if a > 1, we must have 
| arg x | < 277 — hair. 
If these conditions are satisfied, the integral vanishes round that part of the circle 
at infinity for which n> —k, where k is a finite positive quantity. 
We may deform the contour in the usual way, and we see that the integral is 
equal to 
4 ( —)”T (an) sin 77(1 —a) n ( T 
where J* denotes the integral along a line parallel to the imaginary axis cutting 
the real axis between s — —k and s = — (&+1). Thus, when \x\ is large, | J*| is of 
order less than I /1 x | k . 
. Under the assigned conditions, we therefore have the asymptotic equality 
(x) 
1 i 
- exp [a; a ] — — X 
6C ?L= 
«=i x n r (1 — an) 
• • (B), 
in which we have at most neglected terms of order lower than any algebraical power 
1/1 x |, when | x \ is large. 
The conditions show that, when 2>a> 1, the expansion is valid over the whole 
plane at infinity. When 1 > a > 0 we see that the expansion is valid over all the 
area at infinity not covered by the condition 277 — 3 ^x 77 > arg x > ^a77 of the previous 
paragraph as well as over part of that area. 
In the investigation of the present paragraph we may have 4>a>2. If a = 2 + &, 
