Laurent series

A Laurent series is a series in integer powers of a complex variable, with only finitely many negative powers. It represents a holomorphic function on an annulus within its domain of convergence.

Annulus of convergence: The ring-shaped region in which a Laurent series converges, bounded by its inner and outer radii. The series converges uniformly on compact subsets inside this ring-shaped region.
Holomorphic function: A complex-valued function that is complex differentiable at every point of an open domain. Laurent series represent holomorphic functions on their annuli of convergence.
Residue theorem: A theorem evaluating contour integrals through the sum of enclosed residues. Laurent coefficients supply the residues that determine these contour integrals.
Taylor series: A power series whose coefficients are derivatives of a function at its center. Taylor series contain no negative powers and describe functions on disks around ordinary points.
Laurent coefficient formula: A contour integral formula that extracts each coefficient of a Laurent expansion. Contour integrals determine the coefficients uniquely on circles within the annulus.
Complex differentiability: Differentiability of a complex function with respect to a complex variable. Complex differentiability throughout an annulus is the regularity condition for representation.
Contour integration: Integration of complex-valued functions along oriented curves in the complex plane. Integrating a Laurent series around a circle isolates its inverse-first-power coefficient.
Laurent polynomial: A finite linear combination of integer powers of a variable, allowing negative exponents. A Laurent series may be infinite in its nonnegative powers, unlike a Laurent polynomial.
Cauchy's integral formula: A theorem expressing a holomorphic function's values through an integral over a surrounding contour. Its contour representation leads to the positive- and negative-power parts of the expansion.
Isolated singularity: A point where a function fails to be holomorphic, while remaining holomorphic throughout a punctured neighborhood. A Laurent expansion around the point classifies its isolated singularity.

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