Meromorphic function

A complex function holomorphic except at isolated poles, where its magnitude diverges. Locally, each pole is a finite-order singularity.

Laurent series: A series expansion in integer powers that may include finitely many negative powers. Its negative-power terms encode a meromorphic function's behavior near each pole.
Holomorphic function: A complex function that is complex differentiable throughout an open domain. A meromorphic function satisfies this condition everywhere except at its poles.
Contour integration: Integration of complex functions along oriented curves in the complex plane. Poles inside a contour often let integrals be evaluated through residues.
Meromorphic function on a Riemann surface: A function on a Riemann surface that is holomorphic except at isolated poles. It extends the plane-based definition to curved complex spaces.
Residue: The coefficient of the inverse-first-power term in a function's Laurent series around an isolated singularity. At a pole, this coefficient captures the leading contribution used in contour integration.
Complex analysis: The mathematical study of functions of complex variables and their derivatives, integrals, and singularities. Meromorphic functions are a central class within this broader theory.
Argument principle: A theorem relating the winding of a function's values around zero to its zeros and poles inside a contour. It counts zeros and poles of meromorphic functions using boundary values.
Divisor (algebraic geometry): A formal integer-weighted sum of codimension-one subvarieties, encoding zeros and poles in algebraic geometry. Zeros and poles of meromorphic functions define divisors that track their multiplicities.
Order of a pole: The positive integer giving the highest negative power in a function's Laurent expansion at a pole. It distinguishes simple poles from poles where the function diverges more rapidly.
Isolated singularity: A point where a function fails to be holomorphic although it is holomorphic in a punctured neighborhood. Meromorphic functions allow only the pole type of isolated singularity.

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