Harmonic function

Description

A function is harmonic if it is a real part of a Holomorphic function (depending on the space). We can use the Cauchy-Riemann-equations to obtain a classical criterion for harmonic functions.

Definition

Let $u:{\mathbb{R}}^2\to \mathbb{R}$ be a real function. $u$ is harmonic if it fulfills the Laplace differential equation: $\Delta u(x,y)={\partial }_x^{2}u(x,y)+{\partial }_y^{2}u(x,y)=0$

Connection to holomorphic functions

1. Let $U\subseteq \mathbb{C}$ open and $f:U\to \mathbb{C}$ be holomorphic. Then $Re(f)$ and $Im(f)$ are harmonic
2. On the other hand, let $U\subseteq {\mathbb{R}}^2$ be simply connected and $u:{\mathbb{R}}^2\to \mathbb{R}$ a harmonic function. Then after identification of ${\mathbb{R}}^2$ with $\mathbb{C}$ there exists a holomorphic function $f:U\to {\mathbb{C}}^2$ where $Ref(x+iy)=u(x,y)$

Properties

Tip

Examples

Function which is harmonic but not a real part of a holomorphic function

The function $u(z)=\log |z|$ is harmonic but it does not the real part of a holomorphic function.

Idea: The idea here is to study the Cauchy-Riemann equation on $U$. If $U$ is simply connected then the complex function can be reconstructed by integration and furthermore all path integrals give the same result.