Definition:Harmonic Polynomial

Definition

Let $\map P z$ be a polynomial over the complex numbers.

Then $\map P z$ is a harmonic polynomial if and only if its Laplacian is $0$.

That is, if and only if it satisfies Laplace's equation:

$\nabla^2 \map P z = 0$

That is, if and only if $\map P z$ is a harmonic function.

Also see

• Results about harmonic polynomials can be found here.

Sources

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $3$: Elementary Analytic Methods: $3.7$ Complex Numbers and Functions: Powers: $3.7.23$