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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

Every meromorphic function on a plane domain is a quotient of holomorphic functions Corollary
Every meromorphic function on ℂ is a quotient of entire functions Corollary
Prescribed principal parts on a compact Riemann surface Corollary
Admissible cycles for the residue theorem Definition
Counting, chordal proximity and characteristic Definition
Divisors, principal divisors and canonical divisors on a Riemann surface Definition
Elliptic function for a lattice Definition
Holomorphic line bundles and meromorphic sections on a Riemann surface Definition
Holomorphic maps and meromorphic functions on Riemann surfaces Definition
Level-one modular forms and cusp forms Definition
Meromorphic differentials, orders and residues Definition
Meromorphic functions on the Riemann sphere Definition
The logarithmic derivative of a meromorphic function Definition
The modular discriminant and the j-invariant Definition
Zero and pole counts weighted by multiplicity and winding number Definition
Addition and duplication for ℘ Example
Meromorphic Jensen identity with a zero or pole at the centre Lemma
The point-divisor exact sequence and the Euler-characteristic step Lemma
Weak solutions of a degree-zero divisor and the logarithmic-derivative identity Lemma
Addition formula for ℘ Theorem
Convergence, zeros and quasi-periods of the Weierstrass zeta and sigma functions Theorem
Poisson–Jensen formula for a meromorphic function on a disc Theorem
Poles of a meromorphic function form a closed discrete set and are at most countable Theorem
Residue theorem on a compact Riemann surface Theorem
The field of elliptic functions is generated by ℘ and ℘' Theorem
The level-one valence formula 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)