Difference between revisions of "Legendre polynomials"

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==See also==
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*[http://mathworld.wolfram.com/LegendrePolynomial.html Legendre Polynomial -- from Wolfram MathWorld]
 
[[category: mathematics]]
 
[[category: mathematics]]

Revision as of 11:17, 31 May 2007

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)


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:

\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)]


x^4= \frac{1}{35}[7P_0 (x) + 20P_2(x)+ 8P_4(x)]


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.

See also