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, Pn(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) d x = 0, for m \ne n

whereas

\int_{-1}^{1} P_n(x) P_n(x) d x = \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)]

[edit] Applications in statistical mechanics

[edit] See also

[edit] References

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