Mathematics. — “Series of analytical functions’. By Prof. J. 
Worrr. (Communicated by Prof. L. E. J. BROUWER). 
(Communicated at the meeting of September 29, 1918). 
Oscoop’s theorem: “Jf the series f, + f,+..., all the terms 
whereof are analytical functions within a region T of the complex 
plane, is convergent at the points of a set (B) everywhere dense in 
T, and if besides, | fi Hf, H.+ fn | SG at every point of T 
(G being a constant), the series converges everywhere in T and there 
represents an analytical function” *) has been again demonstrated 
by ARZELA ®). 
Viraut*) and Porter’) have extended the theorem by proving 
that it is sufficient if only the set (@) of the points where the series 
converges has an internal point of 7 for a limiting point. 
Of the thus extended theorem a simple demonstration shall be 
given in the sequel. 
1. To this end we suppose that the f; are analytical in 7’, that 
in T everywhere |S, |< G for every n, G being a constant, and 
that the series is convergent at the points ò,, 8,,...., having the 
internal point z, of 7’ for limiting point, and we shall prove that 
the series converges uniformly in every region lying with its boun- 
dary within 7’. 
Now describe a circle (f) with centre z, and radius R, lying in 
T. Let 8; be a point of (5) inside a circle (4 R) with centre z,, and 
let f(8;) denote the sum of the series at (8), S,(8;) the sum of n 
terms, then 
1 ('S(@dt Tes S,(t)dt 
Su) 221 t—B; andes (2) Aad tim f t—B; 
(R) (R) 
If 8, denotes another point of (8) inside (3 R), then 
4G 
|Sn(Bi )—S,.(8x) | << 7 |B; Fes Bel ’ 
1) W. F. Oseoop. Functions defined by infinite series. Annals of Mathematics, 
Series 2, Vol. 3, Oct. 1901, p. 26. 
3) V. C. Arzera. Annals of Mathematics, Series 2, Vol. 5, 1904, p. 51. 
3) G. Virau. Sopra le serie di funzioni analitiche. Annali di Matematica, Serie 32, 
tomo 10, 1904, p. 65. 
4) M. B. Porter. Annals of Math. Series 2, Vol. 6, 1904—5, p. 45 and p. 190. 
