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

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Meromorphic functions on a connected plane domain form a field

Statement

Let $Ω \subseteq C$ be a connected plane domain. Then the meromorphic functions on $Ω$ form a field under pointwise addition and multiplication.

Facts & Assumptions

Given: A connected plane domain $Ω$ .

[L1]

Every meromorphic function on $Ω$ is a quotient $g / h$ of holomorphic functions with $h \equiv̸ 0$ (Every meromorphic function on a plane domain is a quotient of holomorphic functions).

Proof

technique · direct

1.1given

Sums and products of meromorphic functions are meromorphic by the pointwise formulas on the common holomorphic locus.

2.1L1step 1.1algebra

Let $f$ be a nonzero meromorphic function. By [L1], write $f = g / h$ with $g, h$ holomorphic and $h \equiv̸ 0$ . Since $Ω$ is connected and $f \equiv̸ 0$ , one also has $g \equiv̸ 0$ . On the set where $g \neq 0$ , $1 / f = h / g$ , which is meromorphic; at a zero of $g$ this quotient has at worst a pole. Hence $1 / f$ is meromorphic on $Ω$ .

3.1step 1.1step 2.1algebra∎

Therefore every nonzero meromorphic function has a multiplicative inverse, and together with step 1.1 this makes the meromorphic functions a field.

Depends on

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

Used by

Elliptic function for a lattice Definition
Addition and duplication for ℘ Example
Moving the boundary of a fundamental parallelogram Example
Addition formula for ℘ Theorem
Divisor and residue laws for elliptic functions Theorem
The field of elliptic functions is generated by ℘ and ℘' Theorem

Dependency tree · two levels

4 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 (standard reference, not scraped)