258 
MR. E. W. BARNES ON THE ASYMPTOTIC EXPANSION OF 
By considering this equation we may again deduce the expansion, valid when 
^ ^ 2 ’ g{x;0) + x~ e log{l-x) = {i//(l)-t//(d)}ar 
„-e 
+ 
(1 —x) n {0—1) ... ( 0—n) /1 
- + ... + 
1 nj 
„=1 X " ' x n\ 
Y. We may equally show that, when | l-x\ is sufficiently small, 
co 
x e g{x-,0)+log{l-x) = xp(l)-xp{0)+logx- S S„_! {0) (log x) n /n !, 
where S n _! (^) is the (n-l) th simple Bernoullian function of 0 of parameter unity. 
VI. When 6 is not a positive or negative integer or zero, we have, if \x \ >1, and 
I arg ( x) | < 7r, 
9 {x ; 6) = -t 
■-{-x)~ e 
7r 
n =ix n {0—n) v w/ sin 7 tO 
This formula gives the asymptotic value of g {x ; 6) when | x | is very large. 
Pabt III. 
The Function G /3 (x ; 6) 
= 1 
x n 
n =on \ {n + 0y 
§ 8. This function is the generalisation of that considered in Part I. "W e assume 
that 9 is not zero or a negative integer, and further that 
(n + $Y = exp {/3 log {n + 6 )}, 
wherein the absolute value of the imaginary part of log (n + 0) is less than tt. 
If 0 be real and negative, this convention fails to define {n + 6f for those terms 
for which (n + 0) < 0. In these cases we may assume that the imaginary part of 
log {n + 0) is equal to +th. 
§ 9. Suppose in the first place that (x) > 0. Then we have the following 
lemma*':— 
If I arg x I < 7 r/ 2 , the integral J- f xV ^ j) ds vanishes when taken along any part 
1 & 1 1 ° 2ttl J (s + 0) p 
of the great circle at infinity for which 3ft (s)> — l, where l is any finite positive 
quantity, provided the circle pass between the points s = n, n being a positive 
integer. 
The same integral is finite when taken along any parallel to the nnagmaiy axis in 
the finite part of the plane, which does not pass through finite singularities of the 
subject of integration. 
* With this theorem the reader may compare the method used by the author to obtain the asymptotic 
expansion of the multiple gamma function. ‘Cambridge Philosophical Transactions, vol. 1J, §§ 55-57. 
