Legendre polynomials: Difference between revisions

From SklogWiki
Jump to navigation Jump to search
m (Style: expanded acronym)
m (Rodrigues)
Line 4: Line 4:


:<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 using '''Rodrigues formula''' as:
:<math> P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} (x^2 - 1)^n </math>


The first seven  Legendre polynomials are:
The first seven  Legendre polynomials are:

Revision as of 17:47, 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:

The first seven Legendre polynomials are:







"shifted" Legendre polynomials (which obey the orthogonality relationship):




Powers in terms of Legendre polynomials:






See also