254 
MR. E. W. BARNES ON THE ASYMPTOTIC EXPANSION OF 
Then for any assigned value of n it is possible to find a value X for | x | such that, 
whenever \x\ > X, 
[ e~ x x n Y\, n \ < e, 
where e is an arbitrarily assigned positive quantity. 
It is evidently possible that an asymptotic expansion may hold for some values of 
arg x and not for others. 
§ 4. The following definitions give precision to subsequent statements. 
When we say of a quantity J (x, k) that, for any assigned value or range of values 
of arg x, it is of order less than 1 /\x\ k when | x is large, we mean that for any 
assigned value of k it is possible to find a value X of x such that, when | x | > X, 
| J (. x , k) x k | < e, 
e being defined as before. 
c5 
When we say that J ( x , k ) tends exponentially to zero with 1/ x , we mean that 
it is such that, when | x | > X and <|£t(x)> 0, 
| J (x, k) e px | < e, 
p being a definite finite positive quantity. 
§ 5. Our fundamental procedure is based upon the following theorem. 
Suppose that, when 
x 
is large, we wish to find an 
asymptotic expansion for the integral 
1 = ^~\ e~* z f{z)(-zy-'dz. 
The integral is taken round 
a gamma-function contour C 
which encloses the origin and embraces an axis P from the 
origin to infinity along which 2ft (xz) is positive. 
In the subject of integration f(z) is a function which, for 
values of | z | < l, admits the convergent expansion 
f(z)= X 
Further, f(z ) is such that the integral I is convergent. This condition, of course, 
limits the behaviour of f(z) at infinity along the axis P. Suppose that the plane of 
the complex variable 2 is dissected by lines passing away from the poles of f(z) to 
infinity in a direction away from the origin. We assume that the contour C does not 
contain or cut any of these lines. 
Then the integral I admits the asymptotic expansion 
cc 
2 
»=0 
— ( e~ xz (—zY +n ~ 1 dz 
2n - c 
^ _fn_ 
,Zo T(l-/3-n)x p+n ' 
Divide up the contour C into two parts L and M. 
L lies wholly within the circle 
