282 
ME. E. W. BAENES ON THE ASYMPTOTIC EXPANSION OF 
The investigation just concluded has emphasised the fact that the asymptotic 
expansion of the more general function considered at the beginning of this Part of 
the memoir demands, when lu (x) <C 0, a knowledge of the nature of the finite 
singularities of y (x). 
Part VI. 
The Function f p (x;0) = t > when I x I < L 
J PK ' n =o {n+0f 
°° x n v (n + 0) 
§41. The function f,(x;0) is defined when | sc | < 1 by the series ( n + df ; 
where outside a circle of radius l where /x is the least of the quantities |r+$|, 
oo 
n = 0, 1, 2, ... oo, x {x) admits the absolutely convergent expansion Zb r /x r . 
The following propositions may be established :— 
I. When \x\ <1, f p (x ; 6) can be written in the form 
co 
t b r g p+r (x ; 6). 
II. The function f p (x; 6) has a single singularity in the finite part of the plane. 
This singularity occurs at x — 1, and is a multiform point. 
III. Near x = 1, f p (x ; 6) behaves like 
7 T 
sin 7r/3 
(-log a') 
3-1 x ~e v. 
b r (logx) r 
,=o T (/3 + r) 
if (3 lie not a positive or negative integer 
between f p (x ; 6) and this expression can, 
absolutely convergent series 
(zero included). In fact, the difference 
when (x) > 1/2, be expanded in the 
l h i 
-< u r ^ n +1 
r=0 ?i = 0 ‘V 
Cn+l + b)- 
This theorem I first prove for the case when $£(#)> by means of contoui 
integrals similar to those employed throughout this memoir, and then extend to all 
values of 6 such that g > l, by means of the difference formulae for multiple Eiemann 
£ functions. 
IV. Whether (3 be integral or not, in fact, for all values of (3 of finite modulus, 
provided the points 0±n, n = 0, 1, 2, ... oo all lie outside the circle of convergence 
of x(*)> , x 7 
pp. : x(0-»O 1 [ ni-x)-^x(-y)Jy 
Zi x n (0-nY 2m 3 c> sii 
( 0—nf 
sin t T(0 + y)(-yY 
