Cauchy's integral formula

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Cauchy's integral formula 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. It can also be used to formulate integral formulas for all derivatives of a holomorphic function.

Supppose U is an open subset of the complex plane C, and f : U → C is a holomorphic function, and the disk D = { z : | z − z₀| ≤ r} is completely contained in U. Let C be the circle forming the boundary of D. Then we have for every a in the interior of D:

where the integral is to be taken counter-clockwise.

The proof of this statement uses the Cauchy integral theorem and, just like that theorem, only needs that f is complex differentiable. One can then deduce from the formula that f must actually be infinitely often continuously differentiable, with

Some call this identity Cauchy's differentiation formula. A proof of this last identity is a by-product of the proof that holomorphic functions are analytic.

One may replace the circle C with any closed rectifiable curve in U which doesn't have any self-intersections and which is oriented counter-clockwise. The formulas remain valid for any point a from the region enclosed by this path. Moreover, just as in the case of the Cauchy integral theorem, it is sufficient to require that f be holomorphic in the open region enclosed by the path and continuous on that region's closure.

These formulas can be used to prove the residue theorem, which is a far-reaching generalization.

Sketch of the proof of Cauchy's integral formula

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 a tiny circle around a. Since f(z) is continuous, we can choose a circle small enough on which f(z) is almost constant and equal to f(a). We then need to evaluate the integral

∫ 1/(z-a) dz

over this small circle. We may do it by choosing the parametrization (variable substitution)

where and . It turns out that the value of this integral is independent of the circle's radius: it is equal to 2πi.