must certainly diverge if | «| has reached the limit | @| , because 
then g() and also z = zg(u) might become infinite. 
The region in which e= g(u) is moving, must never contain the 
singular points of the function /(z). So if we call these singular points 
Aj = rj ej, 
(J — i 2, 9, es Ld . ) 
then, when | u | increases fluently, we may never have 
We suppose that for given z from this equation (II) w is deduced. 
Perhaps that solution is impossible, and not to be satisfied by any 
value of w with a modulus smaller than &. In that case we shall 
decide, that for the point # under consideration the radius of con- 
vergence of the series (I) is equal to k. If there are solutions with a 
modulus smaller than &, we then call #,,; the smallest of the moduli 
of the obtained solutions and #, the smallest of all moduli &,; 
G= 1,2,3,...) For |u| < BR; now z=zg(u) will not be able 
to reach any of the singular points A; , F'(ag(u)) will have a finite 
and definite value and the series (I) will converge if only the nature 
of the function F(z) in z=oo does not limit | u | more closely. 
So the result of the foregoing considerations respecting the radius 
of convergence of the series (1) is the following. For a multiform 
function F(x) the radius of convergence when « is given is equal to 
the smallest of the two quantities PR, or |a |. If there are no 
singular points in the finite part of the plane the radius of conver- 
gence is equal to | @|. On the other hand for a uniform function 
F(x), holomorphic in «=o, the radius of convergence is always 
BR, In some cases, where the limit /, is wanting, the radius of con- 
vergence can increase to k. 
In the series (1) we now substitute w = 1 and we obtain in this 
way formally the development of /() in a series of polynomials. 
We find 
ze leze PO) Do glu)? 
_F@)=F0)+ 5 Sal aA BO) Dalai) 
ill hi m | m=1 
and now the question arises whether this series converges or diverges 
in a given point «x. If it converges it will produce that value of 
Fe) which is deduced out of F(ag(u)) when | u | gradually 
increases from 0 to 1. To judge its nature we must arrange the 
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