proof of Casorati-Weierstrass theorem

Assume that a is an essential singularityMathworldPlanetmath of f. Let VU be a punctured neighborhoodMathworldPlanetmath of a, and let λ. We have to show that λ is a limit point of f(V). Suppose it is not, then there is an ϵ>0 such that |f(z)-λ|>ϵ for all zV, and the function


is bounded, since |g(z)|=1|f(z)-λ|<ϵ-1 for all zV. According to Riemann’s removable singularityMathworldPlanetmath theorem, this implies that a is a removable singularity of g, so that g can be extended to a holomorphic functionMathworldPlanetmath g¯:V{a}. Now


for za, and a is either a removable singularity of f (if g¯(z)0) or a pole of order n (if g¯ has a zero of order n at a). This contradicts our assumption that a is an essential singularity, which means that λ must be a limit point of f(V). The argument holds for all λ, so f(V) is dense in for any punctured neighborhood V of a.

To prove the converse, assume that f(V) is dense in for any punctured neighborhood V of a. If a is a removable singularity, then f is bounded near a, and if a is a pole, f(z) as za. Either of these possibilities contradicts the assumption that the image of any punctured neighborhood of a under f is dense in , so a must be an essential singularity of f.

Title proof of Casorati-Weierstrass theorem
Canonical name ProofOfCasoratiWeierstrassTheorem
Date of creation 2013-03-22 13:32:40
Last modified on 2013-03-22 13:32:40
Owner pbruin (1001)
Last modified by pbruin (1001)
Numerical id 4
Author pbruin (1001)
Entry type Proof
Classification msc 30D30
Related topic PicardsTheorem