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

Related Topics:
Mathematics - Augustin Louis Cauchy - Complex analysis - Holomorphic function

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Suppose 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:

Related Topics:
Open subset - Complex plane - Boundary - Interior

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:f(a) = {1 over 2pi i} oint_C {f(z) over z-a}, dz

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where the integral is to be taken counter-clockwise.

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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

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:f^{(n)}(a) = {n! over 2pi i} oint_C {f(z) over (z-a)^{n+1}}, dz.

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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.

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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.

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

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