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| :<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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| | Legendre polynomials form an orthogonal system in the range [-1:1], i.e.: |
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| | :<math> \int_{-1}^{1} P_n(x) P_m(x) d x = 0, </math> for <math> m \ne n </math> |
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| The first seven Legendre polynomials are: | | The first seven Legendre polynomials are: |
Revision as of 18:52, 20 June 2008
Legendre polynomials (also known as 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
Legendre polynomials can also be defined using Rodrigues formula as:
Legendre polynomials form an orthogonal system in the range [-1:1], i.e.:
- for
The first seven Legendre polynomials are:
"shifted" Legendre polynomials (which obey the orthogonality relationship):
Powers in terms of Legendre polynomials:
See also