Cauchy's integral formula

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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 celebrated formula shows that, in complex analysis, "differentiation is equivalent to integration": complex differentation, like integration, behaves well under uniform limits - a result denied in real analysis.

Contents

Theorem

Suppose U is an open subset of the complex plane C, f : UC is a holomorphic function and the closed disk D = { z : | zz0| ≤ r} is completely contained in U. Let γ 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 similarly 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), it follows that holomorphic functions are analytic. In particular f is actually infinitely differentiable, with

This formula is sometimes referred to as Cauchy's differentiation formula.

The circle γ can be replaced by any closed rectifiable curve in U which has winding number one about a. Moreover, as for the Cauchy integral theorem, it is sufficient to require that f be holomorphic in the open region enclosed by the path and continuous on its closure.

Proof sketch

By using the Cauchy integral theorem, one can show that the integral over C (or the closed rectifiable curve) is equal to the same integral taken over an arbitrarily small circle around a. Since f(z) is continuous, we can choose a circle small enough on which f(z) is arbitrarily close to f(a). On the other hand, the integral

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