theorems on complex function series


Theorem 1.  If the complex functionsf1,f2,f3,  are continuousMathworldPlanetmath on the path γ and the series

f1(z)+f2(z)+f3(z)+ (1)

converges uniformly on γ to the sum function F, then one has

γF(z)𝑑z=γf1(z)𝑑z+γf2(z)𝑑z+γf3(z)𝑑z+

Theorem 2.  If the functions  f1,f2,f3,  are holomorphic in a domain A and the series (1) converges uniformly in every closed (http://planetmath.org/ClosedSet) disc of A, then also the sum function F of (1) is holomorphic in A and the equality

dnF(z)dzn=F(n)(z)=f1(n)(z)+f2(n)(z)+f3(n)(z)+ (2)

is true for every positive integer n in all points of A.  The series (2) converges uniformly in every compact subdomain of A.

Theorem 3.  If f(z) is holomorphic in a domain A and z0 is a point of A, then one can expand f(z) to a power seriesMathworldPlanetmath (the so-called Taylor seriesMathworldPlanetmath)

f(z)=n=0an(z-z0)nwherean=f(n)(z0)n!(n= 0, 1, 2,).

This is valid at least in the greatest disk  |z-z0|<r()  which contains points of A only.

Title theorems on complex function series
Canonical name TheoremsOnComplexFunctionSeries
Date of creation 2013-03-22 16:47:55
Last modified on 2013-03-22 16:47:55
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 9
Author pahio (2872)
Entry type Theorem
Classification msc 30B99
Classification msc 40A30
Related topic IdentityTheoremOfPowerSeries
Related topic WeierstrassDoubleSeriesTheorem