INTEGRAL FUNCTIONS DEFINED BY TAYLOR’S SERIES. 
273 
III. The function g p (x; 9)-V( 1 —/3) (— log xy^x' 0 is one-valued near x = 1, and 
iii the vicinity of this point admits the convergent expansion 
2 (fi.fi). 
u=o X 
where £„+i (y3, 9) denotes the (n +1) pie Riemann ^-function of equal parameters unity. 
IV. If 6 be not real, the function 
0)+g p (^; -0je* vlP , 
the negative or positive sign being taken as I ( 6 ) is > or < 0, is one-valued near 
x — 1, and has no singularity at this point. 
V. If 6 be not zero or a positive or negative integer, g p ( x ; 9) admits, when | x | is 
very large, the asymptotic expansion 
- % 1 , [log (-a)f- 1 v ^ ^ (sin? t9) 
»=i x n (9—nY (-x)° nto-n ! r (/3-n) [log (-a;)]"' 
This theorem is true when (3 is a positive integer, in which case the final series is a 
finite series of (3 terms. 
When 9 is a positive integer, a modification of the formula can be deduced. 
When 9=1 and (3 is a positive integer, we obtain Spence’s formula. # 
VI. Similar analysis holds for the more general function defined when | x | < 1 by 
^ x 11 [log (n + 9y\ c 
n=o (n + 9) a 
Part V. 
The Function F s (x ; 9) = % ^x( n + &) 
n ’ nto n\(n + 9y 
§ 30. We proceed now to use the previous asymptotic expansions to obtain similar 
expansions for very general types of integral functions. 
Let y (x) be a function of x which outside a circle of radius l admits the expansion 
b r /x r , so that for values of r greater than an assignable quantity Pt, | h r [ < J,'\ 
where V > l. 
Suppose further that the points 9, 6+1, 9 + 2... all lie outside this circle, and that 
the modulus of the least of them is taken to be > V. 
De Morgan, ‘Differential and Integral Calculus,’ 1842, p. 659. 
2 N 
VOL. CCVI.-A. 
