[ 249 ] 
V. The Asymptotic Expansion of Integral Functions defined by Taylor’s 
By E. W. Barnes, M.A., Fellow of Trinity College, Cambridge. 
Communicated by Professor A. R. Forsyth, Sc.D., LL.D., F.Ii.S. 
Received May 9, —Read June 8, 1905, —Received in condensed form March 2 
Series. 
2 
1906. 
Contents. 
§ 1. Introduction. 
§ 2. Object of the paper. 
§§ 3-4. Preliminary definitions. 
§ 5. A general theorem. 
Page 
250 
252 
253 
254 
Part I. 
* gfl 
The function G (x; 0) = 2 —-—--— x 
256 
Part II. 
03 x n • 
The function g(x\0) = 2 -when I a; I < 1.257 
Part III. 
“ x n 
The function G« (x; 6) = 2 — - f- - 
' n=o T(n+l)(n + d)P 
258 
Part IY. 
x n 
The function g B (x: 6) = 2 -— -, when I x I < 1 
PK ' n=o (n + Of 1 1 
272 
Part V. 
The function FR#;#) = 
' vhen x f is regular at the ori « in ' ■ 
x n exp 
The function 2 
n + 6 
«=o r («- + !) 
273 
279 
VOL. CCVI.-A 406. 
2 K 
29.6.06 
