Legendre polynomials

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, $P_{n}(z)$ can be defined by the contour integral

$P_{n}(z)={\frac {1}{2\pi i}}\oint (1-2tz+t^{2})^{1/2}~t^{-n-1}{\rm {d}}t$

Legendre polynomials can also be defined (Ref 1) using Rodrigues formula, used for producing a series of orthogonal polynomials, as:

$P_{n}(x)={\frac {1}{2^{n}n!}}{\frac {d^{n}}{dx^{n}}}(x^{2}-1)^{n}$

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

$\int _{-1}^{1}P_{n}(x)P_{m}(x)dx=0,$ for $m\neq n$

whereas

$\int _{-1}^{1}P_{n}(x)P_{n}(x)dx={\frac {2}{2n+1}}$

The first seven Legendre polynomials are:

$\left.P_{0}(x)\right.=1$

$\left.P_{1}(x)\right.=x$

$P_{2}(x)={\frac {1}{2}}(3x^{2}-1)$

$P_{3}(x)={\frac {1}{2}}(5x^{3}-3x)$

$P_{4}(x)={\frac {1}{8}}(35x^{4}-30x^{2}+3)$

$P_{5}(x)={\frac {1}{8}}(63x^{5}-70x^{3}+15x)$

$P_{6}(x)={\frac {1}{16}}(231x^{6}-315x^{4}+105x^{2}-5)$

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

${\overline {P}}_{0}(x)=1$

${\overline {P}}_{1}(x)=2x-1$

${\overline {P}}_{2}(x)=6x^{2}-6x+1$

${\overline {P}}_{3}(x)=20x^{3}-30x^{2}+12x-1$

Powers in terms of Legendre polynomials:

$\left.x\right.=P_{1}(x)$

$x^{2}={\frac {1}{3}}[P_{0}(x)+2P_{2}(x)]$

$x^{3}={\frac {1}{5}}[3P_{1}(x)+2P_{3}(x)]$

$x^{4}={\frac {1}{35}}[7P_{0}(x)+20P_{2}(x)+8P_{4}(x)]$

$x^{5}={\frac {1}{63}}[27P_{1}(x)+28P_{3}(x)+8P_{5}(x)]$

$x^{6}={\frac {1}{231}}[33P_{0}(x)+110P_{2}(x)+72P_{4}(x)+16P_{6}(x)]$

Applications in statistical mechanics

• Computational implementation of integral equations
• Order parameters
• Lebwohl-Lasher model
• Rotational relaxation

See also

• Associated Legendre function
• Legendre Polynomial -- from Wolfram MathWorld

References

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