276 
MR. E. W. BARNES ON THE ASYMPTOTIC EXPANSION OF 
By the previous investigation this equality is seen to hold good for all values of r, 
however large. 
Let F K (;r) denote the sum of the first (R+l) terms of the series (1). Then, 
asymptotically, 
F„ (*■) = (-*)-’{log(-*)}"-I & 
r=0 
{log 
(-x)y 
N 
•s' 
()" r w(ff) 
+ 
> J N (fl?) 
t =oT(p+r-n)T(n+l) (log(-rr)f |log(-x)} N 
It is at once evident that any attempt to make R tend positively to infinity will 
introduce a series in ascending powers of log (—a?). This series cannot be asymptotic : 
it may be convergent, or it may be divergent but summable. In order to investigate 
its nature, we shall limit ourselves to the case where /3 is a positive integer. In this 
case the series proceeding in descending powers of log (~x) will be finite, and if we 
take N = /3 + r-l, | r J N (x) | is less than 1 / 1 x \ \ when | x | is large, 1 lowever large l may be. 
We have then 
F„(») = (—*)-'{l°g (—*)}* 
R 
-1 ^ 
'P + r —1 
V 
(-)" r (B) (9) b r 
r —0 L n =0 
P (i3+r-n ) T (n+ 1) {log (-x)} n ~ 
+ b T ,.J ( x ) 
The double series may be written 
v s' y y 
Z* Zt 
s= — /3 +1 ?‘=0 s=0 r—s 
'( — y s r (r s) (0) b r {log (—.t)} s 
T(/3 + s) E (r-s+1) 
Thus 
R 
+ t 
{log (-»)}■ V (-)’-fe,.,r M (0 
+ 2 b r ,J 
r =0 
s=o r (/3+s) )■=0 r(r+i) 
CO 
§ 34. Suppose now that R tends to infinity. The series | S b r ,.J (x) | can, even 
w hen multiplied by any finite positive power of | x [, be made as small as we please 
by taking | x | sufficiently large. 
The series % (~~) b r T - (0) - g absolutely convergent. For if y be the minimum 
r =0 r (r + s+ 1) 
value of \(n + 6)\, so that y is the distance of 6 from the nearest of the points 
0, — 1, -2, ..., we have, if \x \ <y, 
Therefore 
v(x + e) = 2 
m =0 r (yn +1) 
Lt 
1 
t—j 
_1 
1/m 
Lr (m + 1 )_ 
1 
y 
Thus, since L/ | b r Vr \ = l , the series is absolutely convergent, since y>l. 
