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. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... Augustin Louis Cauchy Augustin Louis Cauchy (August 21, 1789 – May 23, 1857) was a French mathematician. ... Complex analysis is the branch of mathematics investigating holomorphic functions, i. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ...

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: In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ... In topology, the boundary of a subset S of a topological space X is the sets closure minus its interior. ... In mathematics, the interior of a set S consists of all points which are intuitively not on the edge of S. A point which is in the interior of S is an interior point of S. The notion of interior is in many ways dual to the notion of closure. ...

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 In mathematics, the Cauchy integral theorem in complex analysis, named after Augustin Louis Cauchy, is an important statement about path integrals for holomorphic functions in the complex plane. ...

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. In complex analysis, a complex-valued function f of a complex variable is holomorphic at a point a iff it is differentiable at every point within some open disk centered at a, and is analytic at a if in some open disk centered at a it can be expanded as...

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. In mathematics, the concept of a curve tries to capture our intuitive idea of a geometrical one-dimensional and continuous object. ...

These formulas can be used to prove the residue theorem, which is a far-reaching generalization. The residue theorem in complex analysis is a powerful tool to evaluate path integrals of meromorphic functions over closed curves and can often be used to compute real integrals as well. ...

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

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.

Example usage

Consider the function

and the contour described by |z|=2, call it C.

To find out the integral of f(z) around the contour, we need to know the singularities of f(z). Observe that we can rewrite f as follows:

Clearly the poles become evident, their moduli are less than 2 and thus lie inside the contour and are subject to consideration by the formula. By the Cauchy-Goursat theorem, we can express the integral around the contour as the sum of the integral around z₁ and z₂ where the contour is a small circle around each pole. Call these contours C₁ around z₁ and C₂ around z₂. The Cauchy integral theorem in complex analysis is an important statement about path integrals for holomorphic functions in the complex plane. ...

Now, around C₁, f is analytic (since the contour does not contain the other singularity), and this allows us to write f in the form we require, viz:

and now

Doing likewise for the other contour:

The integral around the original contour C then is the sum of these two integrals: