zeroes of analytic functions are isolated

The zeroes of a non-constant analytic functionMathworldPlanetmath on are isolated. Let f be an analytic function defined in some domain D and let f(z0)=0 for some z0D. Because f is analytic, there is a Taylor seriesMathworldPlanetmath expansion for f around z0 which converges on an open disk |z-z0|<R. Write it as f(z)=Σn=kan(z-z0)n, with ak0 and k>0 (ak is the first non-zero term). One can factor the series so that f(z)=(z-z0)kΣn=0an+k(z-z0)n and define g(z)=Σn=0an+k(z-z0)n so that f(z)=(z-z0)kg(z). Observe that g(z) is analytic on |z-z0|<R.

To show that z0 is an isolated zero of f, we must find ϵ>0 so that f is non-zero on 0<|z-z0|<ϵ. It is enough to find ϵ>0 so that g is non-zero on |z-z0|<ϵ by the relation f(z)=(z-z0)kg(z). Because g(z) is analytic, it is continuous at z0. Notice that g(z0)=ak0, so there exists an ϵ>0 so that for all z with |z-z0|<ϵ it follows that |g(z)-ak|<|ak|2. This implies that g(z) is non-zero in this set.

Title zeroes of analytic functions are isolated
Canonical name ZeroesOfAnalyticFunctionsAreIsolated
Date of creation 2013-03-22 13:38:10
Last modified on 2013-03-22 13:38:10
Owner brianbirgen (2180)
Last modified by brianbirgen (2180)
Numerical id 8
Author brianbirgen (2180)
Entry type Result
Classification msc 30C15
Synonym zeros of analytic functions are isolated
Related topic Complex
Related topic LeastAndGreatestZero
Related topic IdentityTheorem
Related topic WhenAllSingularitiesArePoles