Weierstrass double series theorem

If the complex functionsf0,f1,f2,  are holomorphic in the disc  |z-z0|<r  and thus

fn(z)=ν=0anν(z-z0)ν,anν=fn(ν)(z0)ν!n,ν (1)

in this disc, and if the function series

n=0fn=f0+f1+f2+ (2)

converges uniformly to the function F in each disc  |z-z0|ϱ  where  0<ϱ<r,  then also all the series

n=0anν=a0ν+a1ν+a2ν+(ν=0, 1, 2,) (3)

converge, and in the disc  |z-z0|<r  one has

F(z)=ν=0Aν(z-z0)ν (4)

where the Aνs are the sums of the series (3).

Proof.  Apparently, the series (2) converges uniformly also in every closed sub-disc of the open disc   |z-z0|<r.  Therefore the theorem 2 in the entry “theorems on complex function series (http://planetmath.org/TheoremsOnComplexFunctionSeries)” says that the sum F(z) is holomorphic in  |z-z0|<r  and

F(ν)(z)=f0(ν)(z0)+f1(ν)(z0)+f2(ν)(z0)+(ν=0, 1, 2,).

Theorem 3 in the same entry thus guarantees that F(z) has the Taylor expansionMathworldPlanetmath of the form (4) wherein

Aν=1ν!F(ν)(z0)(ν=0, 1, 2,).

According to theorem 2 in the same entry the series (2) may be differentiated termwise,



Note.  In Weierstrass double series theoremMathworldPlanetmath it’s a question of changing the summing :

F(z)=f0(z)+f1(z)++fn(z)+==[a00+a01(z-z0)+a02(z-z0)2++a0ν(z-z0)ν+]+[a10+a11(z-z0)+a12(z-z0)2++a1ν(z-z0)ν+]+[a20+a21(z-z0)+a22(z-z0)2++a2ν(z-z0)ν+]+[an0+an1(z-z0)+an2(z-z0)2++anν(z-z0)ν+]                    ¯=A0+A1(z-z0)+A2(z-z0)2++Aν(z-z0)ν+
Title Weierstrass double series theorem
Canonical name WeierstrassDoubleSeriesTheorem
Date of creation 2013-03-22 16:48:15
Last modified on 2013-03-22 16:48:15
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type Theorem
Classification msc 30B10
Classification msc 40A05
Classification msc 30D30
Related topic TheoremsOnComplexFunctionSeries