ANALYTIC REPRESENTATION OF COMPLEX FUNCTIONS. 229 
A representation of the branch FA (x) is obtained in the 
form of an infinite series of polonomials 
FA (x) = Z (x) (5) 
M = 0 
(x) = Z v C ? F v (a) (x — a) v ( 6 ) 
the coefficients C* being independent of the choice of F v (a), 
and may be given a variety of forms all consistent with the 
convergence of the expression (5) within the star A. The 
method employed in deducing the expression (5) consists 
in extending the function backward from x to a along a 
vector by means of a series of n intersecting circles of con¬ 
veyance of equal radii. To reach all parts of the star A, 
limit n—oo. 
A very simple form for the coefficients C\* is obtained by 
Mittag-Leffler which gives a development of the form 
FA (x) —^Z n (x) = g n (x) (7) 
V- = 0 
n 2 n* 
(*) - S 2 . 
= O = 0 
«2n 
2 
F{a) 
Al + Ao . 
It is pointed out that instead of prolonging the functions 
along a vector passing through a, a family of non-intersect¬ 
ing curves may be employed. 
Fixed Curve .—It may be of interest to note in passing that by 
choosing a fixed curve in the x plane surrounding a singular 
point it is possible to pass completely around an isolated sin¬ 
gular point any number of times and obtain a development 
of F(x) valid for all points of the curve and in terms of a 
real parameter.* 
When a circle of radius R with its center at the origin is 
chosen as the fixed curve we obtain the expression 
* Bulletin American Mathematical Society, Vol. 8, page 15. 
33—Bull. Phil. Soc., Wash., Vol. 14. 
