theorems on complex function series
Theorem 1. If the complex functions f1,f2,f3,… are continuous 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)=f(n)1(z)+f(n)2(z)+f(n)3(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 series (the so-called Taylor series
)
f(z)=∞∑n=0an(z-z0)n |
This is valid at least in the greatest disk which contains points of 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 |