Legendre polynomials: Difference between revisions

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*[http://mathworld.wolfram.com/LegendrePolynomial.html Legendre Polynomial -- from Wolfram MathWorld]
*[http://mathworld.wolfram.com/LegendrePolynomial.html Legendre Polynomial -- from Wolfram MathWorld]
[[category: mathematics]]
[[category: mathematics]]
==References==
# B. P. Demidotwitsch, I.A. Maron, and E.S. Schuwalowa, "Métodos numéricos de
Análisis", (Ed. Paraninfo, Madrid, 1980) (original in Russian)

Revision as of 18:02, 20 June 2008

Legendre polynomials (also known as Legendre functions of the first kind, Legendre coefficients, or zonal harmonics) are solutions of the Legendre differential equation. The Legendre polynomial, can be defined by the contour integral

Legendre polynomials can also be defined (Ref 1) using Rodrigues formula as:

Legendre polynomials form an orthogonal system in the range [-1:1], i.e.:

for

whereas

The first seven Legendre polynomials are:







"shifted" Legendre polynomials (which obey the orthogonality relationship in the range [0:1]):




Powers in terms of Legendre polynomials:






See also

References

  1. B. P. Demidotwitsch, I.A. Maron, and E.S. Schuwalowa, "Métodos numéricos de

Análisis", (Ed. Paraninfo, Madrid, 1980) (original in Russian)