# Cauchy integral formula

## The formulas.

Let $D=\{z\in\mathbb{C}:|z-z_{0}| be an open disk in the complex plane, and let $f(z)$ be a holomorphic function defined on some open domain that contains $D$ and its boundary. Then, for every $z\in D$ we have

 $\displaystyle f(z)$ $\displaystyle=$ $\displaystyle\frac{1}{2\pi i}\oint_{C}\frac{f(\zeta)}{\zeta-z}\ d\zeta$ $\displaystyle f^{\prime}(z)$ $\displaystyle=$ $\displaystyle\frac{1}{2\pi i}\oint_{C}\frac{f(\zeta)}{(\zeta-z)^{2}}\ d\zeta$ $\displaystyle\vdots$ $\displaystyle f^{(n)}(z)$ $\displaystyle=$ $\displaystyle\frac{n!}{2\pi i}\oint_{C}\frac{f(\zeta)}{(\zeta-z)^{n+1}}\ d\zeta$

Here $C=\partial D$ is the corresponding circular boundary contour, oriented counterclockwise, with the most obvious parameterization given by

 $\zeta=z_{0}+Re^{it},\quad 0\leq t\leq 2\pi.$

## Discussion.

The first of the above formulas   underscores the “rigidity” of holomorphic functions. Indeed, the values of the holomorphic function inside a disk $D$ are completely specified by its values on the boundary of the disk. The second formula is useful, because it gives the derivative  in terms of an integral, rather than as the outcome of a limit process.

## Generalization.

The following technical generalization  of the formula is needed for the treatment of removable singularities. Let $S$ be a finite subset of $D$, and suppose that $f(z)$ is holomorphic for all $z\notin S$, but also that $f(z)$ is bounded   near all $z\in S$. Then, the above formulas are valid for all $z\in D\setminus S$.

Using the Cauchy residue theorem, one can further generalize the integral formula to the situation where $D$ is any domain and $C$ is any closed rectifiable curve in $D$; in this case, the formula becomes

 $\eta(C,z)f(z)=\frac{1}{2\pi i}\oint_{C}\frac{f(\zeta)}{\zeta-z}\ d\zeta$

where $\eta(C,z)$ denotes the winding number of $C$. It is valid for all points $z\in D\setminus S$ which are not on the curve $C$.

Title Cauchy integral formula CauchyIntegralFormula 2013-03-22 12:04:46 2013-03-22 12:04:46 djao (24) djao (24) 25 djao (24) Theorem msc 30E20