Harmonic Function

A harmonic function of two variables is a real function with continuous second partial derivatives that satisfies Laplace's equation

(1)

Harmonic functions are called potential functions in physics and engineering. Potential functions are extremely useful, for example, in electromagnetism, where they reduce the study of a 3-component vector field to a 1-component scalar function. A scalar harmonic function is called a scalar potential, and a vector harmonic function is called a vector potential.

To find a class of such functions in the plane, write the Laplace's equation in polar coordinates

(2)

and consider only radial solutions

(3)

This is integrable by quadrature, so define ,

(4)

(5)

(6)

(7)

(8)

(9)

so the solution is

(10)

Ignoring the trivial additive and multiplicative constants, the general pure radial solution then becomes

			(11)
			(12)

Other solutions may be obtained by differentiation, such as

			(13)
			(14)

			(15)
			(16)

and

(17)

Harmonic functions containing azimuthal dependence include

			(18)
			(19)

The Poisson kernel

(20)

is another harmonic function.

A discrete analog on a finite unweighted graph is a discrete harmonic function. At each specified graph vertex of positive vertex degree, its value is the arithmetic mean of its values at adjacent vertices.

A polynomial that is a harmonic function is called a harmonic polynomial.

Boundary values and harmonic measure connect harmonic functions with Brownian motion. For a bounded domain, harmonic measure is the exit distribution of Brownian motion. It represents harmonic extensions of continuous boundary data when those extensions exist.