INTEGRAL FUNCTIONS DEFINED BY TAYLOR’S SERIES. 
261 
If m G* be the maximum value of 
sin 7 t/3 
77 
r(y+^) 
y r(r) (^)y r l | 
■=o r\ J | 
on this line, we shall have 
| la | < A\x- d -'\ -r)\. 
Thus for all finite values of k and m, however large, we can take | x | so large that 
x e+v ~% |, where e> 0 and as small as we please, tends to zero as nearly as we please. 
Finally the integral I, 
= —- x 
277 
-e 
M 
(- y)' 
~^x~ y 
Y 
r = m +1 
r (r) (0) r 
— r^y 
T ! 
By the substitution r\ = y log x, we see exactly as in § 5 that | 1+xk (log x) _,3+1+ ” t 
be made as small as we please by taking | x | sufficiently large. 
Hence 
x 
(log x) 
—/3 + 1 + m I 
G f}(—x ; 9) —x 6 % 
(-Y r (r) (6) 
,-=u r ! F (/3—r) (log x) r P+1 
can 
can for any finite value of m be made as small as we please by taking | x | sufficiently 
large. 
Therefore, provided kv(x )<0 and 6 is not real and negative, we have the asymptotic 
expansion 
G „(x;0) = (-x) e [log(-x )] 8 1 
00 
r —0 
(_) r r (r) (d) 
r(c+l)r(/3-r) {log (-*)}-’ 
the principal value of log (—a:), which is real when x is real and negative, being 
taken. 
§ 14. We proceed now to obtain the asymptotic expansion of 
G ? (x ; 9) = S x n ln ! (n + 9) p 
)i = 0 
in the case in which H (x) > 0 . 
We will assume that 9 is not real and negative. In this case the points 9, 9+L, 
9 + 2, ... all lie within an angle, vertex the origin, which is less than t t. 
Let the bisector of this angle be the line 1 /L, and let L be the image of this line in 
the real axis. The figure is drawn for the case in which the imaginary part of 9 is 
positive. 
Suppose now that (—z ) 3-1 = exp {(/ 3 — 1 ) log (— s)} when the logarithm is rendered 
one-valued by a cross-cut along the axis L, and log (—z) is such that it is real when z 
is real and negative. 
