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Zeros of a nonzero holomorphic function are isolated

Statement

A holomorphic function on a complex domain that is not identically zero has only isolated zeros.

That is, if $f : Ω \to C$ is holomorphic on a complex domain and $f$ is not the zero function, then every $a \in Ω$ with $f (a) = 0$ has a neighbourhood in which $a$ is the only zero of $f$ .

Facts & Assumptions

Given: A complex domain $Ω$ , a holomorphic function $f : Ω \to C$ that is not identically zero, and an arbitrary zero $a \in Ω$ .

[L1]

If two holomorphic functions on a complex domain agree on a set with an accumulation point in the domain, then they agree everywhere on the domain (Identity theorem for holomorphic functions).

[L2]

A holomorphic function has finite order $m$ at $a$ exactly when it factors near $a$ as $(z - a)^{m} g (z)$ with $g (a) \neq 0$ ; its order is $+ \infty$ exactly when it vanishes on a neighbourhood of $a$ (The order of a zero is the exponent in its local holomorphic factorization).

[L3]

A complex differentiable function is continuous at the point of complex differentiability (Complex differentiability at a point implies continuity there).

Proof

technique · direct

1.1

The function $f$ cannot vanish on any neighbourhood of $a$ , for otherwise its zero set would have the interior point $a$ as an accumulation point and [L1], applied to $f$ and the zero function, would make $f$ identically zero on $Ω$ .

L1given

2.1

By step 1.1 and [L2], the order of $f$ at $a$ is finite, so $f (z) = (z - a)^{m} g (z)$ near $a$ with $g (a) \neq 0$ . By [L3], after shrinking the neighbourhood, $g$ is nowhere zero there; since $f (a) = 0$ , the finite order $m$ is positive, and $a$ is the only zero of $f$ in that neighbourhood.

step 1.1L2L3

3.1

The zero $a$ was arbitrary, so every zero of $f$ is isolated; if $f$ has no zeros, the conclusion is vacuous.

step 2.1∎

Depends on

Used by

Local degree of a nonconstant holomorphic map Definition

Dependency tree · two levels

15 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. Lebl, Guide to Cultivating Complex Analysis, Theorem 2.4.7 (standard reference, not scraped)
B. V. Shabat, Introduction to Complex Analysis, Theorem 2.27 (standard reference, not scraped)