Cauchy's integral formula

Not to be confused with Cauchy's integral theorem.

In mathematics, Cauchy's integral formula is a central statement in complex analysis. The statement is named after Augustin-Louis Cauchy. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk. The statement also provides integral formulas for all derivatives of a holomorphic function. Cauchy's formula shows that, in complex analysis, "differentiation is equivalent to integration": complex differentiation, like integration, behaves well under uniform limits – a result denied in real analysis.

Cauchy's Integral Formula Media

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  Surface of the real part of the function g(z) = z2/z2 + 2z + 2 and its singularities, with the contours described in the text.