identity theorem

Identity theoremFernando Sanz Gamiz

Lemma 1.

Let $f$ be analytic on $\mathrm{\Omega }\mathrm{\subseteq }\mathrm{C}$ and let $L$ be the set ofaccumulation points (also called limit points or cluster points) of$\mathrm{\{}z\mathrm{\in }\mathrm{\Omega }\mathrm{:}f\mathit{}\mathrm{(}z\mathrm{)}\mathrm{=}\mathrm{0}\mathrm{\}}$ in $\mathrm{\Omega }$. Then $L$ is both openand closed in $\mathrm{\Omega }$.

Proof.

By definition of accumulation point, $L$ is closed. To see that itis also open, let $z_0\in L$, choose an open ball $B(z_0,r)\subseteq \mathrm{\Omega }$ and write $f(z)={\sum }_{n=0}^{\mathrm{\infty }}{a}_n{(z-z_0)}^n,z\in B(z_0,r)$. Now $f(z_0)=0$, and hence either $f$ has a zeroof order $m$ at $z_0$ (for some $m$), or else $a_n=0$ for all $n$.In the former case, there is a function g analytic on $\mathrm{\Omega }$ suchthat $f(z)={(z-z_0)}^{m}g(z),z\in \mathrm{\Omega }$, with $g(z_0)\ne 0$. Bycontinuity of $g$, $g(z)\ne 0$ for all $z$ sufficiently close to$z_0$, and consequently $z_0$ is an isolated point of$\{z\in \mathrm{\Omega }:f(z)=0\}$ . But then $z_0\notin L$, contradicting ourassumption. Thus, it must be the case that $a_n=0$ for all n, sothat $f\equiv 0$ on $B(z_0,r)$. Consequently, $B(z_0,r)\in L$,proving that $L$ is open in $\mathrm{\Omega }$.∎

Theorem 1 (Identity theorem).

Let $\mathrm{\Omega }$ be a open connected subset of $\mathrm{C}$ (i.e., a domain). If $f$ and$g$ are analytic on $\mathrm{\Omega }$ and $\mathrm{\{}z\mathrm{\in }\mathrm{\Omega }\mathrm{:}f\mathit{}\mathrm{(}z\mathrm{)}\mathrm{=}g\mathit{}\mathrm{(}z\mathrm{)}\mathrm{\}}$has an accumulation point in $\mathrm{\Omega }$, then $f\mathrm{\equiv }g$ on$\mathrm{\Omega }$.

Proof.

We have that $\{z\in \mathrm{\Omega }:f(z)-g(z)=0\}$ has anaccumulation point, hence, according to the previous lemma, it isopen and closed (also called ”clopen”). But, as $\mathrm{\Omega }$ isconnected, the only closed and open subset at once is $\mathrm{\Omega }$itself, therefore $\{z\in \mathrm{\Omega }:f(z)-g(z)=0\}=\mathrm{\Omega }$,i.e., $f\equiv g$ on $\mathrm{\Omega }$.∎

Remark 1.

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