Complex Analysis/Harmonic function

Definition

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

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

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

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

Proof

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

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

since partial derivatives commute. 1. $⟹$ 2.Define the function $g:=\frac{\partial u}{\partial x}-i\frac{\partial u}{\partial y}$. By the Cauchy-Riemann-Differential equation, $g$ is holomorphic.since$U$ is simply connected, there exists a primitive $f:U\to ℂ$ from $g$, assume (by Adding a constant) ,that $u(z_0)=f(z_0)$ for a $z_0\in U$ applies. write $f=u_1+i{v}_1$. it is

$\frac{\partial u_1}{\partial x}-i\frac{\partial u_1}{\partial y}=f^{\prime }=g=\frac{\partial u}{\partial x}-i\frac{\partial u}{\partial y}$

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

de:Kurs:Funktionentheorie/Harmonische Funktion