# Harmonic Function / Laplace's equation

Yao Yao on June 5, 2018

harmonic 这个词的意思太多了，比如在 periodic signals 里翻译成 “谐波”。而 Harmonic Function 的翻译是 “调和函数

## Laplace operator

The two-dimensional Laplace operator, or laplacian as it is often called, is denoted by $\nabla^2$ or $lap$, and defined by

where $f(x, y)$ is a twice differentiable functions.

The notation $\nabla^2$ comes from thinking of the operator as a sort of symbolic scalar product:

Notice that the laplacian is a linear operator, that is it satisfies the two rules

• $\nabla^2 (u + v) = \nabla^2 u + \nabla^2 v$
• $\nabla^2 cu = c(\nabla^2 u)$

for any two twice differentiable functions $u(x, y)$ and $v(x, y)$ and any constant $c$.

I.e.

## Harmonic Function

A function $\phi(x, y)$ which has continuous second partial derivatives and satisfies Laplace’s equation is called a harmonic function. I.e.

Considering laplacian is a linear operator, we have:

## Examples of harmonic functions

• Degree $0$: all constants $c$ are harmonic.
• Degree $1$: all linear polynomials $ax + by$ are harmonic.
• Degree $2$: the quadratic polynomials $x^2 − y^2$ and $xy$ are harmonic; all other harmonic homogeneous quadratic polynomials are linear combinations of these, e.g.:

where $a b$ are constants.

• Degree $n$: the real and imaginary parts of the complex polynomial $(x + \mathrm{i} y)^n$ are harmonic.