428 
where J/ denotes the maximum of «, and y, ,m the minimum for 
k 
1 | 
k=1,2,...k,—1,) and such that at the same time N eN, 
4 1 
and a <i y, |< N, for &=1, 2,...4,—1. The points P,, Pan 
eN if 
then are contained in //y, + //1y,, and we have 
| NDE: 
| Py, (es Yn nn 
1 
Since now at every point of the plane lim y, =O, at each point 
of V one of the two indices N, and MN, satisfies the relation 
LW, —lhimw, | <e, whieh proves the quasi-uniform convergence. 
n= 00 
Besides at 0 the derivative of the limiting function is zero, and 
lim P’ (0) = Jo"). 
n= 00 
It follows from § 8 that the function Py (x, y) cannot be quasi- 
uniformly convergent at the points of every set V. For V we choose: 
«=0, O<y<a. Since the continuous function P* (#, y) for y 0 
converges to zero and for y=0O to + oo, the convergence is not 
quasi-uniform. 
14. If in a region G a series of analytical functions converges to 
an analytical function and if, besides, the series of derived functions 
converges quasi-uniformly at the points of every closed set contained 
in G, then it is permitted to differentiate the series termaise every- 
where within G. 
In the first place we observe that every region within G contains 
another region where the convergence is uniform’), so that the 
termwise differentiation is permitted in this last region. 
Since / (z) converges to a function which is continuous within G 
and coincides on an everywhere dense set of points with /'(z), we 
have everywhere /'(z)= lm / (2). 
n= 0 
1) For any », indeed, 
|P’ (0)—9/ (0) =- 
An? +2 2n* +1 
n 9 4 Ta 
2 3 
1. f= (t) ah (t); dé | < 
le t | 
I 
7 
ni—1 
Cr 
Since Pp, (0) =n’, we have ae OT 
*) P. Monrer. Thése, p. 83. 
