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

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

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

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

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

$\underset{-1}{\overset{1}{\int }}{P}_n(x)P_m(x)dx=0,$ for $m\ne n$

whereas

$\underset{-1}{\overset{1}{\int }}{P}_n(x)P_n(x)dx=\frac{2}{2n+1}$

The first seven Legendre polynomials are:

$P_0(x)=1$

$P_1(x)=x$

$P_2(x)=\frac{1}{2}(3x^2-1)$

$P_3(x)=\frac{1}{2}(5x^3-3x)$

$P_4(x)=\frac{1}{8}(35x^4-30x^2+3)$

$P_5(x)=\frac{1}{8}(63x^5-70x^3+15x)$

$P_6(x)=\frac{1}{16}(231x^6-315x^4+105x^2-5)$

"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:

$x=P_1(x)$

$x^2=\frac{1}{3}[P_0(x)+2P_2(x)]$

$x^3=\frac{1}{5}[3P_1(x)+2P_3(x)]$

$x^4=\frac{1}{35}[7P_0(x)+20P_2(x)+8P_4(x)]$

$x^5=\frac{1}{63}[27P_1(x)+28P_3(x)+8P_5(x)]$

$x^6=\frac{1}{231}[33P_0(x)+110P_2(x)+72P_4(x)+16P_6(x)]$

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)