Show a function is harmonic over some space


Posted by 4 years ago
426.1k points

Showing that a function f is harmonic is a straightforward task. For a function to be harmonic it must satisfy Laplace's equation which can be written in a few ways,

\begin{equation} \nabla^2f=0 \end{equation}

or

\begin{equation} f_{{x_1}{x_1}}+f_{{x_2}{x_2}}+\ldots+f_{{x_n}{x_n}}=0 \end{equation}

or

\begin{equation} \frac{\partial^2 f}{\partial x_1^2} + \frac{\partial^2 f}{\partial x_2^2} + \ldots + \frac{\partial^2 f}{\partial x_n^2} =0 \end{equation}

Example

Suppose we have a function $f : U=\mathbb{R}^2 \backslash {(0,0)} \rightarrow \mathbb{R}$ defined by $f(x,y)=\ln{(x^2+y^2)}$ and we want to show that it is harmonic. The first statement just says that the function f's domain includes the whole 2-dimensional space except the point $(0,0)$ and that its range is all the real numbers. To show it is harmonic just calculate the second partial derivatives with respect to x and y, and show that the sum is zero,

$$ f_{x} = \frac{2x}{x^2+y^2}\; \; \longrightarrow \; \; f_{xx}=\frac{-2x^2+2y^2}{(x^2+y^2)^2} $$

$$ f_{y} = \frac{2y}{x^2+y^2}\; \longrightarrow \; f_{yy}=\frac{2x^2-2y^2}{(x^2+y^2)^2} $$

taking the sum of $f_{xx}$ and $f_{yy}$ give the desired result, $ f_{xx} + f_{yy} = 0 $.

This was a great help to me. - baptiste 4 years ago