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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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.:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{-1}^{1} P_n(x) P_m(x) d x = 0, } for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \ne n }

whereas

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{-1}^{1} P_n(x) P_n(x) d x = \frac{2}{2n+1} }

The first seven Legendre polynomials are:

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. P_1 (x) \right.=x}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_2 (x) =\frac{1}{2}(3x^2-1)}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_3 (x) =\frac{1}{2}(5x^3 -3x)}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_4 (x) =\frac{1}{8}(35x^4 - 30x^2 +3)}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_5 (x) =\frac{1}{8}(63x^5 - 70x^3 + 15x)}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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]):

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{P}_0 (x) =1}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{P}_1 (x) =2x -1}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{P}_2 (x) =6x^2 -6x +1}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{P}_3 (x) =20x^3 - 30x^2 +12x -1}

Powers in terms of Legendre polynomials:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. x \right.= P_1 (x)}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2= \frac{1}{3}[P_0 (x) + 2P_2(x)]}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^3= \frac{1}{5}[3P_1 (x) + 2P_3(x)]}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^4= \frac{1}{35}[7P_0 (x) + 20P_2(x)+ 8P_4(x)]}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^5= \frac{1}{63}[27P_1 (x) + 28P_3(x)+ 8P_5(x)]}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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)