( 526 ) 
We now consider /(xg(w)) first as a function of w, then as a 
function of «g{u). So we arrive at the two developments 
M= ym mM ; 
F (ag(u)) = F(0) + net a Do F(xg(u)) 
n= — 
and 
Fi) EFO Zen FO), 
il 
from which last equation ensues 
Mm l=m FF (h\(0 
D Dag) = > ah SE WETEN 
u—= = . 
So we have 
nr l=n _FW(0 en h 
F(rg(u)) = F(O) + FE um S ok — ( ) 0 gle) Naar area |.) 
i i h} m! 
or 
I (ag(u)) = F(0) 3 Si um Tj (+). 
ml 
The coefficients Z'n(7) of this power series in w are polynomials 
in z and the coefficients of the polynomial 7'„(e) contain of the ori- 
ginally given power series only the m first coefficients 
FQ(0), F@(0),.... FCO). 
The power series (1) has when w is given a definite radius of 
convergence, which may not, as is easy to understand, exceed a 
limit independent of «, i.e. | «| may never become greater than 4, 
because for | ul >k, g(u) and with it /(ry(u)) cannot have a fixed 
value. | 
For the rest with each 2 the series is convergent for very small 
values of w approaching nought, and its sum is that value of 2’(xg(w)) 
which transforms itself for u 0 into the given constant /(0). With 
a steady increase of |u| the series remains convergent and its sum 
is to be stated at F(rg(u)) if only the argument <= zg(u) continues 
to indicate a pomt within a region, extending around z=0, in 
which #(2) is holomorphic. Such a region is for instance the star 
of Mirrac-LEFFLER although the rays of this star need not be right 
lines. If #(z) is uniform in the whole plane and holomorphic in 
«=o, this region can even surround the point z=. In every 
other case «=o must remain outside this region and the series (1) 
