( 627 ) 
There exist, as Mrrrag-LerLer has shown us, series of polyno- 
mials, which have, whatever F'(#) may be, an almost unlimited 
region of convergence and for which the region of divergence 
shrivels up to a group of right lines, namely the rays of the star 
belonging to /(). Such series of polynomials are not to be con- 
structed by making use of an auxiliary function g(u) everywhere 
uniform. We must choose for g(u) a function holomorphic within 
the circle | u | =k, which shows somewhere on the circumference 
of that circle an algebraical or logarithmical singularity fit for that 
purpose. 
Each singular point 4A; causes a certain part Gs,; of the region 
of divergence Gs to appear, namely the image made in the z-plane 
of the circle unity in the u-plane with the aid of the equation 
x g(u) = Aj. 
We have now to act in such a way that this image can be 
reduced to an exceedingly narrow loop drawn from the point « = oo 
round the ray of the star of MrrraG-LerFLerR passing through A;, 
By this demand we are more limited in our choice of g(u), but 
yet undoubtedly auxiliary functions of a most divergent nature are 
still possible. A very simple auxiliary function g(u) which allows 
of the observation of all important phenomena, even though it does 
not furnish perhaps in all circumstances the most converging series 
of polynomials, is the function 
(lu? — 1—(1— eu 
mea ae we 
u and / being here positive real fractions. The function is 
g(u) = 
1 oe 
uniform within the circle | u | = —> 1 and the conditions g (0) — 0, 
17 
ere 
g(1)=1 are satisfied, if the power (l—wu)* for | u | <— is defined 
in such a way that its argument is naught with real u. The quan- 
. 1 » . 
tity — takes here the place of the quantity 4. 
iv 
The selected function g(u) we substitute into (V) and we obtain 
1 m= 0 h=m 
Ke = Buh ah, 
lr mik Pill 
1 m (— um 1 m 
Jr pm eee EEE —#P \h 
ak m! aen AO Nh “ m! DL (Lt ys 
It is perhaps simpler to write here 
1 1 — "oc x if 
= <— (u mel Me wes es Sok es 4 
—— 2 ie “) h=1 iy Ge (XI) 
x 
Then 
