Cauchy's integral theorem

Cauchy's integral theorem states that a holomorphic function has zero contour integral around a closed curve when it is holomorphic throughout the enclosed region.

Holomorphic function: A complex function that is complex differentiable at every point of an open domain. Holomorphy throughout the enclosed region is the theorem’s central condition.
Complex differentiability: Differentiability of a complex function with respect to a complex variable. Complex differentiability throughout an open set is equivalent to holomorphy there.
Augustin-Louis Cauchy: A French mathematician whose work established foundational results in analysis, including complex integration. Cauchy formulated the integral theorem as part of his development of complex analysis.
Residue theorem: A theorem that evaluates contour integrals using residues at enclosed isolated singularities. It extends vanishing contour integrals to functions with isolated singularities.
Contour integral: An integral of a function along an oriented curve, defined by parametrizing the curve. The theorem asserts that this integral is zero for eligible closed contours.
Open set: A subset of a topological space in which every point has a neighborhood contained in the subset. Holomorphy is defined on open domains, and the theorem concerns such regions.
Complex analysis: The study of functions of complex variables and their analytic properties. The theorem became a foundational result in this field.
Cauchy's theorem for multiply connected regions: A contour-integral result accounting for multiple boundary components surrounding holes in a domain. It adapts the zero-integral conclusion to regions that are not simply connected.
Simply connected space: A path-connected topological space in which every loop can be continuously contracted to a point. Simple connectivity lets the theorem apply to every closed contour in the domain.
Closed curve: A continuous curve whose starting point and ending point coincide. The theorem evaluates integrals over closed curves that bound suitable regions.

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