Editing Legendre polynomials
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'''Legendre polynomials''' ( | '''Legendre polynomials''' (aka. Legendre functions of the first kind, Legendre coefficients, or zonal harmonics) | ||
are solutions of the [[Legendre differential equation]]. | are solutions of the [[Legendre differential equation]]. | ||
The Legendre polynomial, <math>P_n (z)</math> can be defined by the contour integral | The Legendre polynomial, <math>P_n (z)</math> can be defined by the contour integral | ||
:<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> | ||
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> | ||
== | Associated Legendre polynomials. | ||
:<math>P_0^0 (x) =1</math> | |||
:<math>P_1^0 (x) =x</math> | |||
:<math>P_1^1 (x) =-(1-x^2)^{1/2}</math> | |||
:<math>P_2^0 (x) =\frac{1}{2}(3x^2-1)</math> | |||
:<math>P_2^1 (x) =-3x(1-x^2)^{1/2}</math> | |||
:<math>P_2^2 (x) =3(1-x^2)</math> | |||
''etc''. | |||
[[category: mathematics]] |