Holomorphic function

Description

A complex function is complex differentiable or holomorphic if it has a complex derivative in every point. This is analogous to the fact that infinitesimal squares are preserved under the map. This is the complex analog to a real Differentiable function.

Definition

Let $U\subseteq \mathbb{C}$ open and $f:U\to \mathbb{C}$. $f$ is complex differentiable in $a\in \mathbb{C}$ if the following limit exists: ${\lim }_{z\to a}\frac{f(z)-f(a)}{z-a}$ The limit is called the derivative of $f$ in $a$, denoted as $f^{\prime }(a)$

Characterisaton by real differentiability

Let $a\in U\subseteq {\mathbb{R}}^2$ and $f=(f_1,f_2):U\to {\mathbb{R}}^2$ be some real map. We can identify ${\mathbb{R}}^2$ with $\mathbb{C}$. $f$ is complex differentiable in $a$ if and only if $f$ is differentiable in Partial derivative and the Cauchy-Riemann Partial derivative hold: ${\partial }_x{f}_1(a)={\partial }_y{f}_2(a)\text{ and }{\partial }_x{f}_2(a)=-{\partial }_y{f}_1(a)$

Characterisation by power series

Let $U\subseteq \mathbb{C}$. A map $f:U\to \mathbb{C}$ is holomorphic i.f.f. in every point $a\in U$ there is a neighbouhood around which $f$ can be written as a power series. For a given holomorphic map, the characterising power series can be determined by

1. Taylorentwicklung: $a_n=\frac{1}{n!}{f}^{(n)}(a)$
2. Cauchy integral formula: $a_n=\frac{1}{2\pi i}{\int }_{\partial B_r(a)}\frac{f(w)}{(w-a{)}^{n+1}}dw$

This means that in the complex plane holomorphic functions are exactly the Analytische Funktion.

Properties

Description by Laurent series

Is a holomorphic function $f$ defined on a ring-like domain around a point, then there is a Laurent series describing the function on that domain. The coefficients are given by a generalised Cauchy integral formula, written down in Laurent series.

Examples

Function which is holomorphic only in one point

The function $f(z)=|z{|}^2$ is holomorphic only in $0$. This can be shown by taking the real part of $f$ and proving that it is harmonic only in $0$.