identity theorem

Identity theoremFernando Sanz Gamiz

Lemma 1.

Let f be analyticPlanetmathPlanetmath on ΩC and let L be the set of accumulation pointsMathworldPlanetmathPlanetmath (also called limit pointsMathworldPlanetmath or cluster points) of {zΩ:f(z)=0} in Ω. Then L is both open and closed in Ω.


By definition of accumulation point, L is closed. To see that it is also open, let z0L, choose an open ballPlanetmathPlanetmath B(z0,r)Ω and write f(z)=n=0an(z-z0)n,zB(z0,r). Now f(z0)=0, and hence either f has a zero of order m at z0 (for some m), or else an=0 for all n. In the former case, there is a function g analytic on Ω such that f(z)=(z-z0)mg(z),zΩ, with g(z0)0. By continuity of g, g(z)0 for all z sufficiently close to z0, and consequently z0 is an isolated point of{zΩ:f(z)=0} . But then z0L, contradicting our assumptionPlanetmathPlanetmath. Thus, it must be the case that an=0 for all n, so that f0 on B(z0,r). Consequently, B(z0,r)L, proving that L is open in Ω. ∎

Theorem 1 (Identity theorem).

Let Ω be a open connected subset of C (i.e., a domain). If f and g are analytic on Ω and {zΩ:f(z)=g(z)} has an accumulation point in Ω, then fg on Ω.


We have that {zΩ:f(z)-g(z)=0} has an accumulation point, hence, according to the previous lemma, it is open and closed (also called ”clopen”). But, as Ω is connected, the only closed and open subset at once is Ω itself, therefore {zΩ:f(z)-g(z)=0}=Ω, i.e., fg on Ω. ∎

Remark 1.

This theorem provides a very powerful and useful tool to test whether two analytic functions, whose values coincide in some points, are indeed the same function. Namely, unless the points in which they are equal are isolated, they are the same function.

Title identity theorem
Canonical name IdentityTheorem
Date of creation 2013-03-22 17:10:38
Last modified on 2013-03-22 17:10:38
Owner fernsanz (8869)
Last modified by fernsanz (8869)
Numerical id 8
Author fernsanz (8869)
Entry type Theorem
Classification msc 30E99
Related topic Complex
Related topic ZeroesOfAnalyticFunctionsAreIsolated
Related topic TopologyOfTheComplexPlane
Related topic ClopenSubset
Related topic IdentityTheoremOfHolomorphicFunctions
Related topic PlacesOfHolomorphicFunction