Cauchy integral formula

Description

The Cauchy integral formula says that a holomorphic function defined on a disc is completely determined by its values the boundary of the disc.

Definition

Sei

• $U\subseteq \mathbb{C}$ der Definitionsbereich von $f$ und Wertebereich von $\gamma$
• $f:U\to \mathbb{C}$ a Holomorphic function.
• $\gamma :[\alpha ,\beta ]\to U$ eine Komplexe Stückweise Differenzierbare Schleife, die in $U$ nullhomotop ist.

Dann gilt für jedes $z\in U\backslash Spur(\gamma )$: $n(\gamma ,z)f(z)=\frac{1}{2\pi i}\underset{\gamma }{\int }\frac{f(\xi )}{\xi -z}d\xi$ Note: The rotation number $n(\gamma ,z)$ is necessary here because we didn’t ask for a simple closed curve.

Proof: Examine the function $g(\xi )=\frac{f(z)-f(\xi )}{z-\xi }$. By composition, this is a analytic function with a Hebbare Singularität at $z$. Cauchy integral theorem tells us, that the integral around $\gamma$ must be zero. Assuming $\gamma$ is a simple closed curve, we follow: $\underset{\gamma }{\int }\frac{f(\xi )}{\xi -z}d\xi =\underset{\gamma }{\int }\frac{f(z)}{\xi -z}d\xi =f(z)\underset{\gamma }{\int }\frac{1}{\xi -z}d\xi =f(z)2\pi i$ giving us the theorem. $\square$

Taking the derivative in $z$ of the above formula (the derivative is exchangable with integrals), we obtain the formula below.

Definition (Integralformel für Ableitungen)

Für jedes $z\in U\backslash Spur(\gamma )$: $n(\gamma ,z)f^{(k)}(z)=\frac{k!}{2\pi i}\underset{\gamma }{\int }\frac{f(\xi )}{(\xi -z{)}^{(k+1)}}d\xi$ Eine Nullhomologe Schleife ist nullhomotop, daher kann $\gamma$ auch nullhomolog sein.