Cauchy's integral formula

In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk, and it 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.

We begin with a theorem that is less general than what can actually be said. Suppose U is an open subset of the complex plane C, f : U → C is a holomorphic function and the closed disk D = { z : | z − z₀| ≤ r} is completely contained in U. Let ${\displaystyle \gamma }$ be the circle forming the boundary of D. Then for every a in the interior of D:

where the contour integral is taken counter-clockwise.

The proof of this statement uses the Cauchy integral theorem and like that theorem it only requires f to be complex differentiable. Since the reciprocal of the denominator of the integrand in Cauchy's integral formula can be expanded as a power series in the variable (a − z₀) (namely, when z₀=0, ${\displaystyle [1+a/z+(a/z)^{2}+...]/z}$), it follows that holomorphic functions are analytic. In particular f is actually infinitely differentiable, with

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