264 
MR. E. W. BARNES ON THE ASYMPTOTIC EXPANSION OF 
the principal value of (—yY \ which is real when y is real and negative and has 
a cross-cut along the line joining 0 and 1, being taken. 
Evidently c 0 = 1. For the evaluation of the other coefficients of the expansion the 
reader may refer to a paper by Fee and.* 
Now 
where G is a gamma-function contour enclosing the origin and embracing the positive 
half of the real axis and the second line integral is taken along the real axis. 
Therefore 
ir(i-£) 
277 
f Cn(-y) n 
Jo. 
+ ' 3 x e ^ dy — 
c,f(l-j8) 
T (l—/3—n) x p+n 
-J„ 
where ] J„ | < K'„ | exp {e" R x} | < K'„ | exp cos (arg a;) |, and K' n is a finite positive 
quantity for all finite values of n. 
Hence 
k -1 
i=o r (l—/3— n) x n+p 
= e 
tr(i-ft) 
277 
2 Cn(-y) 
n=k 
(3 + n—1 
*-l 
e~ xy dy— t J n . 
71 — 0 
But as in § 5 we see that 
X p+k ~ x f 
i Cni-yY**' 1 
e X,J dy 
Jq 
_n— k _ 
can be made as small as we please for any assigned value of k by taking \x\ sufficiently 
large. 
Hence, when (x) > 0, e~% admits the asymptotic expansion 
J c n r(i-/3) 
nio F(l— /3— n)x fi+n ’ 
Reverting to the value which we obtained as the quantity greater than 11 2 [, we 
see that e' x G p (x ; 6) admits, when ft (x) > 0, the same asymptotic expansion. 
^ 17. In the foregoing investigation 6 may have any finite value while not real and 
negative (zero included). 
But the expansion is valid even if 6 be real and negative, provided it be not a 
negative integer. 
* Ferand, “Bordeaux Proces Verbaux,” 1896-97, pp. 93-9 1 ; quoted in the ‘Fortschritte dor 
Mathematik,’ vol. 29, p. 375. 
