CorollaryStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Every meromorphic function on a plane domain is a quotient of holomorphic functions

Statement

Every meromorphic function on a plane domain is a quotient of holomorphic functions.

Facts & Assumptions

Given: A meromorphic function $f$ on a plane domain $Ω$ .

[L1]

A meromorphic function is holomorphic away from a discrete pole set (Meromorphic functions on a plane domain).

[L2]

Every discrete effective divisor on a plane domain is the zero divisor of a holomorphic function (Every discrete effective divisor on a plane domain is the zero divisor of a holomorphic function).

[L3]

A locally bounded punctured singularity is removable (Characterizations of removable singularities).

Proof

technique · direct

1.1

Let $P$ be the pole set of $f$ , with multiplicities equal to pole orders. By [L2], choose a holomorphic function $h$ on $Ω$ whose zero divisor is exactly $P$ .

L1L2givenconstruct

2.1

On $Ω ∖ P$ , define $g : = f h$ . Near a pole $a \in P$ , the zero of $h$ has exactly the same order as the pole of $f$ , so $g$ is locally bounded on a punctured neighbourhood of $a$ . By [L3], $g$ extends holomorphically across every point of $P$ .

L1L3step 1.1algebra

3.1

Away from $P$ , one has $f = g / h$ . Since both sides are meromorphic and agree on the dense open set $Ω ∖ P$ , this quotient represents $f$ on all of $Ω$ .

step 1.1step 2.1algebra∎

Depends on

Used by

Meromorphic functions on a connected plane domain form a field Corollary

Dependency tree · two levels

13 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, §9.4 (standard reference, not scraped)
M. Weber, Complex Analysis, §3.3 and §4.4 (standard reference, not scraped)