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In mathematics, the Legendre functions Pλ, Qλ and associated Legendre functions Pμ
λ are generalizations of Legendre polynomials to non-integer degree.
Associated Legendre functions are solutions of the general Legendre equation
where the complex numbers λ and μ are called the degree and order of the associated Legendre functions, respectively. The Legendre polynomials are the associated Legendre functions of order μ=0.
This is a second order linear equation with three regular singular points (at 1, −1, and ∞). Like all such equations, it can be converted into a hypergeometric differential equation by a change of variable, and its solutions can be expressed using hypergeometric functions.
These functions may actually be defined for general complex parameters and argument:
The second order differential equation has a second solution, , defined as:
- A useful relation between Legendre P and Q functions is Whipple's formula.
The Legendre functions can be written as contour integrals. For example,
where the contour winds around the points 1 and z in the positive direction and does not wind around −1. For real x, we have
Legendre function as characters
The real integral representation of are very useful in the study of harmonic analysis on where is the double coset space of (see Zonal spherical function). Actually the Fourier transform on is given by
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- Dunster, T. M. (2010), "Legendre and Related Functions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR2723248
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- Whittaker, E. T.; Watson, G. N. (1963), A Course in Modern Analysis, Cambridge University Press, ISBN 978-0-521-58807-2