INTEGRAL FUNCTIONS DEFINED BY TAYLOR’S SERIES. 
255 
of convergence of f(z) and, on L, \z\ -cl', where l = l — e and e is a positive quantity 
as small as we please. M forms the remainder of the contour. 
We have 
I_ £ 0 r(l-/3-n) x? +n = 2^\ c e J 0 C ^~ 2 ) n } ( -zf^dz 
= l! + I 2 (say), where Ij is the integral taken along the contour L and I 2 the sum of 
the integrals along the two parts of the contour M. 
In the hist integral I] put xz = £ and let L/ be the transformed contour. The 
integral becomes 
x p+k 2Tr\u 
-?Y i J-& 
c 
n=k 
X 
For any assigned finite value of k, however large, 
I od-D" 
i 
n=k 
X 
l 
©■- 
C k~ C/c+1-tC A . + 2 - 
X 
< |c*| + | Ck+\ 1 1 ' + | Cf. +2 1 1'“ + ... 
This series is absolutely convergent and independent of x or £. We may 
say that , , w „ 
Y*,/_£Y 
therefore 
n=k 
X 
< R*, 
where R* is independent of x or £, and is finite when k is finite. 
Hence 
|Ii|< 
x 
,p+k 
2t r 
• L 
Thus | x^ +k 1 I 1 j can be made as small as we please by taking \x\ sufficiently 
Consider in the next place the integral I 2 . 
If the original contour cut none of the lines of 
dissection of the plane, it may be closed up as in the 
figure. lor, as we pass over no poles of the subject of 
integration, by Cauchy’s theorem we do not alter 
its value. The contour integral I 2 can therefore be 
replaced by 
large. 
sin 7r/3 
e~ xz z^~ 1 
77 
/(«)- % c n (-z) n 
n =0 
dz, 
a line integral taken from the point a along the axis 
P to infinity. 
If we put z = a + ^jx, we get 
t Sill 77/8 [ 00 r A-— 1 
= e "d« + W /(“ + {/*)- 2 c.(-a-i/xf 
u J o L n=o 
and the integral is taken along a line for which I& (£) is positive. 
d£jx, 
