( 534 j 
(VI) become infinitely small and the convergence of the series becomes 
infinitely slow. 
It will be noticed that only for | «| > 1 the origin will be inside 
G,. If thus we deduce in the indicated way a development of poly- 
nomials for an arbitrary function F(x) out of the series (VI), we 
have no certainty, if |@| is assumed < 2, that there exists for this 
development a region of convergence G}. And very surely if |a | < 1 
the series deduced for /'(x) would diverge everywhere if this function 
were not holomorphic in z =o or not uniform in the entire z-plane. 
In the second place the question rises: what is for a given point x 
the value of a which causes in « the mark of the series to fall most ? 
It is evidently always most advisable to bring the centre of G in 
the point z, that is by taking 
In that case the mark of the series will be 0. One may wish a 
to be real in the series (VI). To obtain the lowest mark with this 
limitation we must bring a circle through P and A with the centre 
of the axis of the real quantities. For the centre of the circle Gy 
we must take the second point of intersection N of that circle and 
of that axis (fig. 1). In the point x the mark of the series will be 
equal to sin A PAN. 
We will now deduce out of the series of polynomials the devel- 
opment of another function, in the first place of log (1—a). In 
the finite part of the plane there is only the singular point «= 1, 
for the rest the function is multiform and not holomorphic in x =o. 
Therefore we must keep the pole a of g(«) outside the circle 
| «| = 1. In that supposition the region of convergence of the 
new series is identical with that of the series (VI). The series itself 
will be 
m=0 | h=m 
log (lr) = Dn = (m—1)i-a > — (alt= = 
moo a 1 1 m 1 
= r (a— azul ie 
m=1 ma™ | 94 Vé yy | 
Again in the point @ the mark of the series is generally deter- 
mined by the equation 
