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| :<math>x^6= \frac{1}{231}[33P_0 (x) + 110P_2(x)+ 72P_4(x)+ 16P_6(x)]</math> | | :<math>x^6= \frac{1}{231}[33P_0 (x) + 110P_2(x)+ 72P_4(x)+ 16P_6(x)]</math> |
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| Associated Legendre polynomials.
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| :<math>P_0^0 (x) =1</math>
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| :<math>P_1^0 (x) =x</math>
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| :<math>P_1^1 (x) =-(1-x^2)^{1/2}</math>
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| :<math>P_2^0 (x) =\frac{1}{2}(3x^2-1)</math>
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| :<math>P_2^1 (x) =-3x(1-x^2)^{1/2}</math>
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| :<math>P_2^2 (x) =3(1-x^2)</math>
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| ''etc''.
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| ==See also== | | ==See also== |
| *[[Associated Legendre function]] | | *[[Associated Legendre function]] |
| *[http://mathworld.wolfram.com/LegendrePolynomial.html Legendre Polynomial -- from Wolfram MathWorld] | | *[http://mathworld.wolfram.com/LegendrePolynomial.html Legendre Polynomial -- from Wolfram MathWorld] |
| [[category: mathematics]] | | [[category: mathematics]] |
Revision as of 13:00, 20 June 2008
Legendre polynomials (aka. Legendre functions of the first kind, Legendre coefficients, or zonal harmonics)
are solutions of the Legendre differential equation.
The Legendre polynomial, can be defined by the contour integral
The first seven Legendre polynomials are:
"shifted" Legendre polynomials (which obey the orthogonality relationship):
Powers in terms of Legendre polynomials:
See also