Encyclopedia > Cauchy's integral formula

  Article Content

Cauchy's integral formula

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 : UC 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:

<math>f(a) = {1 \over 2\pi i} \int_C {f(z) \over (z-a)} dz </math>

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

<math>f^{(n)}(a) = {n! \over 2\pi i} \int_C {f(z) \over (z-a)^{n+1}} dz</math>

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

All Wikipedia text is available under the terms of the GNU Free Documentation License

  Search Encyclopedia

Search over one million articles, find something about almost anything!
  Featured Article
Sanskrit language

... words for mother, matr, and father, pitr. The similarities between Latin, Greek, and Sanskrit led to the discovery of this language family by Sir William Jones, and thus ...

This page was created in 35 ms