# Cauchy integral formula

## The formulas.

Let $$ be an open disk in the
complex plane, and let $f(z)$ be a holomorphic^{1}^{1}It is
necessary to draw a distinction between holomorphic functions (those having
a complex derivative^{}) and analytic functions^{} (those representable by
power series^{}). The two concepts are, in fact, equivalent^{}, but
the standard proof of this fact uses the Cauchy Integral Formula^{}
with the (apparently) weaker holomorphicity hypothesis^{}. function
defined on some open domain that contains $D$ and its boundary. Then,
for every $z\in D$ we have

$f(z)$ | $=$ | $\frac{1}{2\pi i}}{\displaystyle {\oint}_{C}}{\displaystyle \frac{f(\zeta )}{\zeta -z}}\mathit{d}\zeta $ | ||

${f}^{\prime}(z)$ | $=$ | $\frac{1}{2\pi i}}{\displaystyle {\oint}_{C}}{\displaystyle \frac{f(\zeta )}{{(\zeta -z)}^{2}}}\mathit{d}\zeta $ | ||

$\mathrm{\vdots}$ | ||||

${f}^{(n)}(z)$ | $=$ | $\frac{n!}{2\pi i}}{\displaystyle {\oint}_{C}}{\displaystyle \frac{f(\zeta )}{{(\zeta -z)}^{n+1}}}\mathit{d}\zeta $ |

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

$$\zeta ={z}_{0}+R{e}^{it},0\le t\le 2\pi .$$ |

## 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

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

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 |
---|---|

Canonical name | CauchyIntegralFormula |

Date of creation | 2013-03-22 12:04:46 |

Last modified on | 2013-03-22 12:04:46 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 25 |

Author | djao (24) |

Entry type | Theorem |

Classification | msc 30E20 |