Complex Analysis/Harmonic function

Definition

Let $U\subseteq \mathbb {C}$ be an open set. A function $u\colon U\to \mathbb {R}$ is called harmonic if it is twice differentiable and satisfies

\Delta u:={\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}=0

have. The real part of a holomorphic function is harmonic, as follows from the Cauchy-Riemann-Differential equation. Interestingly, the converse also holds: every harmonic function is the real part of a holomorphic function.

Connection to Holomorphic Functions

Let $U\subseteq \mathbb {C}$ be simply connected. For $u\in C^{2}(U,\mathbb {R} )$ , the following are equivalent:

$\Delta u=0$
There exists $v\in C^{2}(U,\mathbb {R} )$ such that $u+iv$ is holomorphic.

Proof

(2). $\implies$ (1).By the Cauchy-Riemann-Differential equation, we have:

{\begin{array}{rl}{\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}&={\frac {\partial }{\partial x}}\left({\frac {\partial u}{\partial x}}\right)+{\frac {\partial }{\partial y}}\left({\frac {\partial u}{\partial y}}\right)\\&={\frac {\partial }{\partial x}}\left({\frac {\partial v}{\partial y}}\right)+{\frac {\partial }{\partial y}}\left(-{\frac {\partial v}{\partial x}}\right)\\&={\frac {\partial ^{2}v}{\partial x\partial y}}-{\frac {\partial ^{2}v}{\partial y\partial x}}\\&=0.\end{array}}

since partial derivatives commute. 1. $\implies$ 2.Define the function $\textstyle g:={\frac {\partial u}{\partial x}}-i{\frac {\partial u}{\partial y}}$ . By the Cauchy-Riemann-Differential equation, $g$ is holomorphic.since $U$ is simply connected, there exists a primitive $f\colon U\to \mathbb {C}$ from $g$ , assume (by Adding a constant) ,that $u(z_{0})=f(z_{0})$ for a $z_{0}\in U$ applies. write $f=u_{1}+iv_{1}$ . it is

{\frac {\partial u_{1}}{\partial x}}-i{\frac {\partial u_{1}}{\partial y}}=f'=g={\frac {\partial u}{\partial x}}-i{\frac {\partial u}{\partial y}}

so is $u_{1}-u$ constant. because $u_{1}(z_{0})=f(z_{0})=u(z_{0})$ ist $u_{1}=u$ und $v:=v_{1}$ does what is desired.

Translation and Version Control

This page was translated based on the following Wikiversity source page and uses the concept of Translation and Version Control for a transparent language fork in a Wikiversity:

Source: Harmonische Funktion - URL:

https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Harmonische_Funktion

Date: 01/08/2024

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