Legendre polynomials: Difference between revisions

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'''Legendre polynomials''' (aka. Legendre functions of the first kind, Legendre coefficients, or zonal harmonics)
'''Legendre polynomials''' (also known as Legendre functions of the first kind, Legendre coefficients, or zonal harmonics)
are solutions of the [[Legendre differential equation]].
are solutions of the [[Legendre differential equation]].
The Legendre polynomial, <math>P_n (z)</math> can be defined by the contour integral
The Legendre polynomial, <math>P_n (z)</math> can be defined by the contour integral


:<math>P_n (z) = \frac{1}{2 \pi i} \oint ( 1-2tz + t^2)^{1/2}~t^{-n-1} {\rm d}t</math>
:<math>P_n (z) = \frac{1}{2 \pi i} \oint ( 1-2tz + t^2)^{1/2}~t^{-n-1} {\rm d}t</math>
Legendre polynomials can also be defined (Ref 1) using Rodrigues formula, used for  producing a series of orthogonal polynomials, as:
:<math> P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} (x^2 - 1)^n </math>
Legendre polynomials form an orthogonal system in the range [-1:1], i.e.:
:<math> \int_{-1}^{1} P_n(x) P_m(x) d x = 0, </math>  for <math> m \ne n </math>
whereas
:<math>\int_{-1}^{1} P_n(x) P_n(x) d x = \frac{2}{2n+1} </math>


The first seven  Legendre polynomials are:
The first seven  Legendre polynomials are:
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:<math>P_6 (x) =\frac{1}{16}(231x^6 -315x^4 + 105x^2 -5)</math>
:<math>P_6 (x) =\frac{1}{16}(231x^6 -315x^4 + 105x^2 -5)</math>


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


:<math>\overline{P}_0 (x) =1</math>
:<math>\overline{P}_0 (x) =1</math>
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:<math>x^6= \frac{1}{231}[33P_0 (x) + 110P_2(x)+ 72P_4(x)+ 16P_6(x)]</math>  
:<math>x^6= \frac{1}{231}[33P_0 (x) + 110P_2(x)+ 72P_4(x)+ 16P_6(x)]</math>  
 
==Applications in statistical mechanics==
*[[Computational implementation of integral equations]]
*[[Order parameters]]
*[[Lebwohl-Lasher model]]
*[[Rotational relaxation]]
==See also==
==See also==
*[[Associated Legendre function]]
*[[Associated Legendre function]]
*[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) (translated from Russian text)

Latest revision as of 11:06, 7 July 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, used for producing a series of orthogonal polynomials, 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:






Applications in statistical mechanics[edit]

See also[edit]

References[edit]

  1. B. P. Demidotwitsch, I. A. Maron, and E. S. Schuwalowa, "Métodos numéricos de Análisis", Ed. Paraninfo, Madrid (1980) (translated from Russian text)