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Legendre polynomials

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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


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

The first seven Legendre polynomials are:



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:


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[edit]

See also[edit]


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