meromorphically

Meromorphically is an adverb used in complex analysis to describe properties, extensions, or mappings that are defined by or involve meromorphic functions. A meromorphic function on a domain is holomorphic except at isolated poles, where the function may blow up to infinity but has a well-defined Laurent expansion with a finite principal part near each pole. Meromorphic maps generalize this notion to maps between Riemann surfaces by requiring that, in local charts, the coordinate expressions are meromorphic functions.

In practical terms, a function f is meromorphically extendable to a larger domain if there exists a meromorphic function on that domain that agrees with f wherever f is defined. The local behavior near a pole z0 is described by a Laurent expansion with a finite principal part, and the order of the pole is the degree of that principal part. Meromorphicity is preserved under basic operations: the sum, product, and ratio of meromorphic functions are again meromorphic.

On compact Riemann surfaces, a nonconstant meromorphic function has only a finite number of poles and zeros and induces a holomorphic map to the Riemann sphere. Meromorphic functions thus play a central role in the study of algebraic curves and complex manifolds, linking analytic, geometric, and algebraic perspectives. In practice, the term meromorphically often appears when describing extensions, equivalence under meromorphic maps, or the manner in which a function behaves near singularities. An understanding of meromorphic behavior is foundational in topics such as analytic continuation, divisor theory, and the classification of Riemann surfaces.