# Harmonic polynomial

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A polynomial with as variables that satisfies the Laplace equation. Any harmonic polynomial may be represented as the sum of homogeneous harmonic polynomials. If , there are only two linearly independent homogeneous harmonic polynomials of degree — for example, the real and the imaginary part of the expression . If , the number of linearly independent homogeneous polynomials of degree is . In the general case — — the number of linearly independent homogeneous harmonic polynomials of degree is where is the number of permutations of objects taken at a time with repetitions. The homogeneous harmonic polynomials, , are also known as spherical functions (in particular if ). If , one may write, in spherical coordinates where and is a spherical function of degree .

How to Cite This Entry:
Harmonic polynomial. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Harmonic_polynomial&oldid=16082
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