278 
MR. E. W. BARNES ON THE ASYMPTOTIC EXPANSION OF 
taken round a contour L 0 embracing the positive half of the real axis and enclosing 
5 = 0, 1 , 2, ... oo, but no other poles of the subject of integration. If l< \9+n\, 
v — 0, 1, 2, ... oo, the whole circle of convergence of x( s + ^)> a °i rc ^ e °f radius l 
and centre — 9, lies outside the contour L 0 . If the inequality does not hold, the 
contour L 0 has to be indented to include s = 0, 1, 2, ..., but no singularities of y(.s + d). 
Tins is always possible if x(^ + n ) not infinite f° r an y positive integral (including 
zero) value of n. 
Suppose now that | arg (— x) | < 7t/ 2. Then the integral will vanish along any 
part of an infinite contour for which I\ (.s) is greater than a finite negative quanity 
h (say). 
Hence, if | arg (—a?) j < ^ , 
taken alone: a contour L 9 which consists of the infinite line s = —h, and a loop from 
a point on this line which includes the singularities of y(.s + 0), but none of those 
of r(-s). 
Hence if we neglect terms of order l/\x\ k , we see that, when \x\ is large and 
| arg(-x) | < \tt, we have 
f(- 2 ')" f x(- 2 /)«' ,, ° s< '* >r (y+«)rf 2 /. • • • (1) 
taken round a contour which encloses within its bulb the singularities of x(~V)> but 
none of the points -9, -9-1, -9-2,..., and which embraces the positive half 
of the real axis. In this integral the principal value of log (-x), which is real when 
x is real and negative, must be taken. It is by evaluating this integral for assigned 
values of x(~?/) that we can obtain the asymptotic expansion of F^(x;9) when 
ft {x) <0. 
§ 36. Consider the case when /3 is an integer, positive or negative. In this case 
the subject of integration is one-valued. The integral can therefore be taken along 
a finite contour which encloses the singularities of x(2/)> n °t the poles of T (y + 9). 
The residues at the latter points will give rise to a finite number of algebraical terms, 
i.e., terms which involve algebraical powers of x, il there are any such points within 
the circle of convergence of y (y)- Making due allowance for such terms, we may 
take the integral round a circle just larger than this circle of convergence. This 
integral when \x\ is large will be at most of order I x l ~ 6 I . The same is true oj the 
more general integral (1) ivhen /3 is not an integer. We thus get a superior limit to 
the asymptotic value of ( x ; 9) when Tv (a?) < 0 and /3 is or is not integral. 
Further progress must be made by a detailed examination of the singularities of 
X ( y) within its circle of convergence. 
