Identity theorem - Knowlpedia

Statement

Let $D\subseteq\mathbb C$ be a domain and let $f,g:D\to\mathbb C$ be holomorphic. If the set

\{z\in D:f(z)=g(z)\}

has an accumulation point in $D$ , then $f=g$ on all of $D$ .

Equivalent zero-set form

A nonzero holomorphic function on a domain has isolated zeros. Indeed, at any zero $a$ , its convergent power series factors as

f(z)=(z-a)^m h(z)

with $m\ge1$ and $h(a)\ne0$ . The integer $m$ is the order of the zero.

To see the factorization, expand $f$ in its Taylor series at $a$ . If $f$ is not identically zero, let $m$ be the first index with nonzero coefficient; factoring $(z-a)^m$ leaves a holomorphic $h$ with $h(a)\ne0$ , so the zero is isolated. An accumulation point of zeros forces every Taylor coefficient there to vanish, hence $f-g$ vanishes on a neighborhood. The zero set is then both open and closed in the connected domain, and therefore is all of $D$ .

Hypotheses matter

The accumulation point must lie inside the domain. For example, a sequence of zeros may accumulate at a boundary point without forcing the function to vanish. Connectedness is also essential: agreement on one component says nothing about another.

References

John B. Conway, Functions of One Complex Variable I, 2nd ed., Springer, 1978. Publisher record. Relevant: Chapter III, §7.