# 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

$\nabla^2 = \frac {\partial^{2}}{\partial x^{2}} + \frac {\partial^{2}}{\partial y^{2}}$

$\nabla^2 = \frac {\partial^{2}}{\partial x_1^{2}} + \frac {\partial^{2}}{\partial x_2^{2}} + \cdots + \frac {\partial^{2}}{\partial x_n^{2}}$

$\nabla^2 f = \frac {\partial^{2} f}{\partial x^{2}} + \frac {\partial^{2} f}{\partial y^{2}}$

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:

$\nabla^2 = \nabla \cdot \nabla = \left ( \frac{\partial}{\partial x} \mathbf{i} + \frac{\partial}{\partial x} \mathbf{j} \right ) \cdot \left ( \frac{\partial}{\partial x} \mathbf{i} + \frac{\partial}{\partial x} \mathbf{j} \right ) = \frac {\partial^{2}}{\partial x^{2}} + \frac {\partial^{2}}{\partial y^{2}}$

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 $c$.

## Laplace equation

I.e.

$\nabla^2 f = 0$

## 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.

$\phi \text{ is harmonic} \iff \nabla^2 \phi = 0$

Considering laplacian is a linear operator, we have:

$\phi \text{ and } \psi \text{ harmonic} \Rightarrow (\phi + \psi) \text{ and } c\phi \text{ are harmonic}$

## 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.:
$\phi(x, y) = a(x^2 − y^2) + bxy$

where $a b$ are constants.

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