Cauchy's integral formula

Cauchy's integral formula expresses the value of a holomorphic function inside a suitable closed contour as a contour integral over that contour. Its derivative form gives every higher derivative by a related integral.

Cauchy integral theorem: A theorem stating that the contour integral of a holomorphic function around a contractible closed curve is zero. Its vanishing-integral result underlies the deformation arguments used to establish the formula.
Jordan curve theorem: The theorem that a simple closed plane curve separates the plane into an inside and an outside. It gives a precise meaning to the interior enclosed by the formula’s simple contour.
Taylor series: A power series whose coefficients encode a function’s derivatives at a chosen point. The derivative formula identifies Taylor coefficients with weighted integrals of boundary values.
Augustin-Louis Cauchy: A French mathematician whose work established major foundations of complex analysis and mathematical analysis. Cauchy developed the integral methods and theorems associated with this formula.
Contour integral: An integral that accumulates a function along an oriented curve in the complex plane. The formula’s boundary data are integrated along an oriented contour.
Simply connected space: A path-connected space in which every loop can be continuously contracted to a point. Simple connectivity is a common domain condition behind standard versions of the formula.
Analytic function: A function that agrees locally with a convergent power series at every point in its domain. The formula implies that every holomorphic function is locally represented by a Taylor series.
Complex analysis: The branch of mathematics studying functions of complex variables and their distinctive analytic properties. The formula is a foundational result in the theory of one complex variable.
Residue theorem: A theorem evaluating contour integrals of meromorphic functions through the residues of enclosed singularities. A simple-pole residue calculation gives the formula’s kernel integral.
Winding number: An integer measuring how many times a closed curve wraps around a point, with orientation counted. For general contours, the formula’s value is multiplied by the contour’s winding number.

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