272 
MR. E. W. BARNES ON THE ASYMPTOTIC EXPANSION OF 
Suppose now that N tends to infinity. Each side of the equality tends to a 
definite finite limit, and we have the given theorem. 
§28. We will now shoiv that, if l he a positive quantity such that \n + 0\> l, 
(n = 0, 1, 2, ... co), and if R he any finite quantity, however large, such that 
E(/3) > R + e, where e > 0, e-*l p G p (x ; 0) = J (x)/x R , where | J {x) | can he made as 
small as we please, for all values o/E (/3) however large, hy sufficiently increasing |a|. 
For we have i r ® r (n — si ft 
e-PG, (x ; ») = - J- 2 
' 27uJcj*=o n \ (n + up 
We take the straight contour to cut the real axis in the point -R-e', where 
0 < e' < e. 
Putting s = — R— e f + LV, the integral becomes 
ar K " £ ' f t r ( 7? + ^ + e - w)! dv = x~ ll J (x) (say). 
27rJc 1 «=o n\(n + 6y 
From this integral we can show that | J (x) | can be made as small as we please by 
taking | x | sufficiently large. And as | (3 | increases indefinitely | J (x) I can be made 
as small as we please. 
We thus have the theorem enunciated. 
It is evident that by studying the singularities of the function S (s) we could 
obtain anew the asymptotic expansion of Gfixffi) when Hi (x) > 0. Ibis problem 
I reserve for a future occasion. 
Part IV. 
The Singularities of g f , (x ; 0). 
8 29. The function qAx\0) defined when |jc|< 1 by series t x n /(n + 6y can be 
v 7i=0 
studied by the methods previously developed. 
We assume that 6 is not zero or a negative integer, and that f3 is not equal to zero 
or a positive integer. When /3 = 1, the function becomes g(x;0) previously considered 
in Part II. When (3 is a positive integer, the function can be derived from the case 
ft — 1 by differentiation with regard to 6. 
On account of the length of this paper I give some theorems relating to this 
function, leaving the development of the theory for publication elsewhere. ' ,v 
I. The function g ? (x ; 6) has a single singularity in the finite part of the plane. This 
singularity occurs at x = 1 and is not an essential singularity. Near x = 1, g$ (x ; 6) 
is many-valued. 
II. The function g ft {x ; 9)-g fi (x ; l) aR 9 has no singularities in the finite part of the 
plane, except the singularity due to x x ~ e at the origin, and if [ log as | < 2 tt, it admits 
the expansion 
rr-o v 
n) 
x- i nr 
71 = 0 Vb ! 
where £ (s, a) is the simple Riemann ^-function of parameter unity. 
* See a forthcoming paper in the ‘ Proceedings of the London Mathematical Society. 
