Harmonic polynomial
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
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where
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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
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where and
is a spherical function of degree
.
References
[1] | S.L. Sobolev, "Partial differential equations of mathematical physics" , Pergamon (1964) (Translated from Russian) MR0178220 Zbl 0123.06508 |
[2] | A.N. [A.N. Tikhonov] Tichonoff, A.A. Samarskii, "Differentialgleichungen der mathematischen Physik" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian) MR104888 |
[3] | M. Brélot, "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris (1959) MR0106366 Zbl 0084.30903 |
E.D. Solomentsev
A finite linear combination of harmonics. Real-valued harmonic polynomials can be represented in the form
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for a given natural number , non-negative
, and real
,
,
. Complex-valued harmonic polynomials can be represented in the form
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where and
are natural numbers,
is real and the
,
, are complex. Harmonic polynomials are the simplest almost-periodic functions (cf. Almost-periodic function).
Harmonic polynomial. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Harmonic_polynomial&oldid=32574