Cauchy's integral formula
provides integral formulas for all derivatives of a holomorphic function
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.
