# Harmonic polynomial

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

A polynomial with $x_1,\dotsc,x_n$ as variables that satisfies the Laplace equation. Any harmonic polynomial may be represented as the sum of homogeneous harmonic polynomials. If $n=2$, there are only two linearly independent homogeneous harmonic polynomials of degree $m$ — for example, the real and the imaginary part of the expression $(x_1+ix_2)^m$. If $n=3$, the number of linearly independent homogeneous polynomials of degree $m$ is $2m+1$. In the general case — $n\geq2$ — the number of linearly independent homogeneous harmonic polynomials of degree $m$ is

$$K_n^m-K_n^{m-2},\quad m\geq2,$$

where

$$K_n^m=\frac{n(n+1)\dotsm(n+m-1)}{m!}$$

is the number of permutations of $n$ objects taken $m$ at a time with $m$ repetitions. The homogeneous harmonic polynomials, $V_m(x)$, are also known as spherical functions (in particular if $n=3$). If $n=3$, one may write, in spherical coordinates

$$V_m(x)=r^mY_m(\theta,\phi),$$

where $r=\sqrt{x_1^2+x_2^2+x_3^2}$ and $Y_m(\theta,\phi)$ is a spherical function of degree $m$.

How to Cite This Entry:
Harmonic polynomial. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Harmonic_polynomial&oldid=44612
This article was adapted from an original article by V.F. Emel'yanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article