THE ANALYTIC REPRESENTATION OF COMPLEX 
FUNCTIONS. 
BY 
Frank G. Radelfinger. 
[Read before the Society January 31, 1903 ] 
INTRODUCTION. 
The value of an analytic complex function F (a;) can be 
calculated, for all values of x within a simply connected 
domain C containing no singular point, by Cauchy’s inte- 
gral. 
F(x) = ^—. C Z1& 
2 * i J c — 
F (c) dc 
x 
(i) 
But if instead we employ the corresponding form of Tay¬ 
lor’s theorem— 
F (x) F (a) + F "'( a ) 
x — a + 
(2) 
we are limited to values of x lying within a circle described 
about a and passing through the nearest singular point. In 
order, therefore, to reach all points of the domain G y , we must 
resort to prolongation by means of a series of intersecting 
circles of convergence. That such prolongation is possible 
shows that an analytic function is completely determined 
by its initial value and that of its successive derivatives at 
an analytic point a, and suggests the possibility of obtaining 
a development depending only on the elements 
F (a), F a) (a), . . . F (n) (a) . . . (3) 
and independent constants, which development would be 
( 227 ) 
