places of holomorphic function

If c is a complex constant and f a holomorphic functionMathworldPlanetmath in a domain D of , then f has in every compactPlanetmathPlanetmath (closed ( and boundedPlanetmathPlanetmathPlanetmath ( subdomain of D at most a finite setMathworldPlanetmath of, i.e. the points z where  f(z)=c,  except when  f(z)c  in the whole D.

Proof.  Let A be a subdomain of D.  Suppose that there is an infiniteMathworldPlanetmath amount of c-places of f in A.  By–Weierstrass theorem, these c-places have an accumulation pointMathworldPlanetmath z0, which belongs to the closed setPlanetmathPlanetmath A.  Define the constant function g such that


for all z in D.  Then g is holomorphic in the domain D and  g(z)=c  in an infinite subset of D with the accumulation point z0.  Thus in the c-places of f we have


Consequently, the identity theorem of holomorphic functions implies that


in the whole D.  Q.E.D.

Title places of holomorphic function
Canonical name PlacesOfHolomorphicFunction
Date of creation 2013-03-22 18:54:18
Last modified on 2013-03-22 18:54:18
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Corollary
Classification msc 30A99
Related topic ZerosAndPolesOfRationalFunction
Related topic IdentityTheorem
Related topic TopologyOfTheComplexPlane