Mathematics. — ‘Some applications of the quasi-uniform convergence 
on sequences of real and of holomorphic functions”. By Prof. 
J. Worrr. (Communicated by Prof. L. B. J. Brouwer). 
(Communicated in the meeting of September 27, 1919). 
We consider a region G of the complex plane, and a sequence 
of functions fÁ,f,.... which are all analytical within G. If the 
sequence converges uniformly in any closed region within G, then 
the limiting function f is analytical within G. This theorem however, 
enunciated by Weinrsrrasz, has of late been considerably extended. 
If it is known that the sequence converges at the points of a set 
having a point of the interior of G for a limiting point, and 
besides, that the sequence is uniformly limited in every closed 
region internal to G, then this is already sufficient to conclude to 
the uniform convergence of the sequence in any closed region 
within G, which involves that the limiting function fis analytical *). 
This same conclusion may be drawn, G being a circle, if only the 
sequence of functions is supposed to be uniformly limited within 
oo 
G and convergent at the points z,,z,,..., such that JI (zs—z,) = 0, 
i 
where z, is the centre of G and zj, 4 z, 7); also if it is only supposed 
that there exist two definite numbers a and 6 such that nowhere 
in the interior of G one of these two values is assumed by a 
function of the sequence, and besides that the sequence converges 
at the points of a set having a point internal to G for a limiting 
point ®). None of these theorems lead to other sequences of functions 
than those which are embraced already by Weterstrasz’ theorem. 
The question now could be raised whether it is possible that 
a sequence of functions, each of which is analytical within a 
region G, may converge to a function, analytical within G, without 
this convergence being uniform in every closed region contained in 
G. That such sequences actually exist is shown by an example, 
1) G. Vrrarr. Annali di Matematica, Serie 32, tomo 10 (1904), p. 65. 
For a simple proof vide a.o. Verh. der Kon. Ac. v. W., dl. 27 (1918), p. 319. 
4) W. BrLASCHKE. Leipz. Berichte, Band 67 (1915), p. 194. 
8) C. CARATHÉODORY and E. LANDAU. Sitz. Ber. Ak. v. W. Berlin, Band 32 
(1911), p. 587. 
