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Meromorphic functions on a plane domain

Definition

Let $Ω \subseteq C$ be a nonempty connected open set. A function $f : Ω ∖ P \to C$ , where $P \subseteq Ω$ , is meromorphic on $Ω$ when

$f$ is holomorphic on $Ω ∖ P$ , and
every point of $P$ is a pole of $f$ in the sense of Isolated singularities: removable, poles, and essential singularities.

The set $P$ is the pole set of the meromorphic function.

Remarks

If $P = \emptyset$ , the function is simply holomorphic on $Ω$ .

This definition is deliberately local. The later page on the argument principle adds the quotient and divisor viewpoints, but this page works only with the isolated-pole description.

Depends on

Isolated singularities: removable, poles, and essential singularities

Used by

Poles of a meromorphic function form a closed discrete set and are at most countable Theorem

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Sources

Patrick Brosnan, UMD complex analysis notes, §3.10 Meromorphic functions (standard reference, not scraped)
Jean-Baptiste Campesato, MAT334 course page and notes index (standard reference, not scraped)