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Difference between revisions of "Harmonic polynomial"

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====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.L. Sobolev,  "Partial differential equations of mathematical physics" , Pergamon  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.N. [A.N. Tikhonov] Tichonoff,  A.A. Samarskii,  "Differentialgleichungen der mathematischen Physik" , Deutsch. Verlag Wissenschaft.  (1959)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M. Brélot,  "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris  (1959)</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.L. Sobolev,  "Partial differential equations of mathematical physics" , Pergamon  (1964)  (Translated from Russian) {{MR|0178220}} {{ZBL|0123.06508}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.N. [A.N. Tikhonov] Tichonoff,  A.A. Samarskii,  "Differentialgleichungen der mathematischen Physik" , Deutsch. Verlag Wissenschaft.  (1959)  (Translated from Russian) {{MR|104888}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M. Brélot,  "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris  (1959) {{MR|0106366}} {{ZBL|0084.30903}} </TD></TR></table>
  
 
''E.D. Solomentsev''
 
''E.D. Solomentsev''

Revision as of 11:59, 27 September 2012

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 .

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

for a given natural number , non-negative , and real , , . Complex-valued harmonic polynomials can be represented in the form

where and are natural numbers, is real and the , , are complex. Harmonic polynomials are the simplest almost-periodic functions (cf. Almost-periodic function).

How to Cite This Entry:
Harmonic polynomial. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Harmonic_polynomial&oldid=28212
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