Meromorphic function

Definition

A meromorphic function on a domain $D$ is a function holomorphic away from a discrete subset $P\subset D$ such that every point of $P$ is a pole. Equivalently, it is locally a quotient $g/h$ of holomorphic functions with $h$ not identically zero.

Riemann-sphere viewpoint

After assigning the value $\infty$ at each pole, a meromorphic function is the same as a holomorphic map not identically equal to $\infty$ ,

D\longrightarrow\widehat{\mathbb C},

where $\widehat{\mathbb C}$ is the Riemann sphere. This formulation extends unchanged to meromorphic functions on Riemann surfaces.

Algebraic structure

Meromorphic functions on a connected Riemann surface form a field. Zeros and poles carry integer orders, and the reciprocal of a nonzero meromorphic function exchanges them.

Warning

A function with an essential singularity is not meromorphic across that point. In several complex variables and algebraic geometry, “meromorphic” has sheaf-theoretic refinements; this knowl records the one-variable notion.

References

Otto Forster, Lectures on Riemann Surfaces, Springer, 1981. Publisher record. Relevant: Chapter 1, §§8–10.