Legendre polynomials

Legendre polynomials (aka. 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:

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

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

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

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

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

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)}

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle P_{6}(x)={\frac {1}{16}}(231x^{6}-315x^{4}+105x^{2}-5)}

"shifted" Legendre polynomials (which obey the orthogonality relationship):

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

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\overline {P}}_{1}(x)=2x-1}

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

$x^{2}={\frac {1}{3}}[P_{0}(x)+2P_{2}(x)]$

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

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x^{4}={\frac {1}{35}}[7P_{0}(x)+20P_{2}(x)+8P_{4}(x)]}

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x^{5}={\frac {1}{63}}[27P_{1}(x)+28P_{3}(x)+8P_{5}(x)]}

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

Associated Legendre polynomials.

$P_{0}^{0}(x)=1$

$P_{1}^{0}(x)=x$

$P_{1}^{1}(x)=-(1-x^{2})^{1/2}$

$P_{2}^{0}(x)={\frac {1}{2}}(3x^{2}-1)$

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

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

etc.