Editing Legendre polynomials
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:<math>P_n (z) = \frac{1}{2 \pi i} \oint ( 1-2tz + t^2)^{1/2}~t^{-n-1} {\rm d}t</math> | :<math>P_n (z) = \frac{1}{2 \pi i} \oint ( 1-2tz + t^2)^{1/2}~t^{-n-1} {\rm d}t</math> | ||
Legendre polynomials can also be defined | '''Legendre polynomials''' can also be defined using '''Rodrigues formula''' as: | ||
:<math> P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} (x^2 - 1)^n </math> | :<math> P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} (x^2 - 1)^n </math> | ||
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:<math> \int_{-1}^{1} P_n(x) P_m(x) d x = 0, </math> for <math> m \ne n </math> | :<math> \int_{-1}^{1} P_n(x) P_m(x) d x = 0, </math> for <math> m \ne n </math> | ||
The first seven Legendre polynomials are: | The first seven Legendre polynomials are: | ||
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:<math>P_6 (x) =\frac{1}{16}(231x^6 -315x^4 + 105x^2 -5)</math> | :<math>P_6 (x) =\frac{1}{16}(231x^6 -315x^4 + 105x^2 -5)</math> | ||
"shifted" Legendre polynomials (which obey the orthogonality relationship | "shifted" Legendre polynomials (which obey the orthogonality relationship): | ||
:<math>\overline{P}_0 (x) =1</math> | :<math>\overline{P}_0 (x) =1</math> | ||
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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> | ||
==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]] | ||