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

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

The first seven Legendre polynomials are:

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

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left.P_{1}(x)\right.=x}

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

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle P_{3}(x)={\frac {1}{2}}(5x^{3}-3x)}

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

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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):

${\overline {P}}_{0}(x)=1$

${\overline {P}}_{1}(x)=2x-1$

${\overline {P}}_{2}(x)=6x^{2}-6x+1$

${\overline {P}}_{3}(x)=20x^{3}-30x^{2}+12x-1$

Powers in terms of Legendre polynomials:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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)]}

$x^{6}={\frac {1}{231}}[33P_{0}(x)+110P_{2}(x)+72P_{4}(x)+16P_{6}(x)]$

See also

• Associated Legendre function
• Legendre Polynomial -- from Wolfram MathWorld