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, 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)} 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 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:

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_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)]}

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) (original in Russian)