Legendre function

In mathematics, the Legendre functions P_λ, Q_λ and associated Legendre functions P^μ
_λ, Q^μ
_λ are generalizations of Legendre polynomials to non-integer degree.

Associated Legendre polynomial curves for l=5.

Differential equation

Associated Legendre functions are solutions of the general Legendre equation

(1-x^{2})\,y''-2xy'+\left[\lambda (\lambda +1)-{\frac {\mu ^{2}}{1-x^{2}}}\right]\,y=0,\,

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.

Definition

These functions may actually be defined for general complex parameters and argument:

P_{\lambda }^{\mu }(z)={\frac {1}{\Gamma (1-\mu )}}\left[{\frac {1+z}{1-z}}\right]^{\mu /2}\,_{2}F_{1}(-\lambda ,\lambda +1;1-\mu ;{\frac {1-z}{2}}),\qquad {\text{for }}\ |1-z|<2

where $\Gamma$ is the gamma function and $_{2}F_{1}$ is the hypergeometric function.

The second order differential equation has a second solution, $Q_{\lambda }^{\mu }(z)$ , defined as:

Q_{\lambda }^{\mu }(z)={\frac {{\sqrt {\pi }}\ \Gamma (\lambda +\mu +1)}{2^{\lambda +1}\Gamma (\lambda +3/2)}}{\frac {e^{i\mu \pi }(z^{2}-1)^{\mu /2}}{z^{\lambda +\mu +1}}}\,_{2}F_{1}\left({\frac {\lambda +\mu +1}{2}},{\frac {\lambda +\mu +2}{2}};\lambda +{\frac {3}{2}};{\frac {1}{z^{2}}}\right),\qquad {\text{for}}\ \ |z|>1.

A useful relation between Legendre P and Q functions is Whipple's formula.

Integral representations

The Legendre functions can be written as contour integrals. For example,

P_{\lambda }(z)=P_{\lambda }^{0}(z)={\frac {1}{2\pi i}}\int _{1,z}{\frac {(t^{2}-1)^{\lambda }}{2^{\lambda }(t-z)^{\lambda +1}}}dt

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

P_{s}(x)={\frac {1}{2\pi }}\int _{-\pi }^{\pi }\left(x+{\sqrt {x^{2}-1}}\cos \theta \right)^{s}d\theta ={\frac {1}{\pi }}\int _{0}^{1}\left(x+{\sqrt {x^{2}-1}}(2t-1)\right)^{s}{\frac {dt}{\sqrt {t(1-t)}}},\qquad s\in \mathbb {C}

Legendre function as characters

The real integral representation of $P_{s}$ are very useful in the study of harmonic analysis on $L^{1}(G//K)$ where $G//K$ is the double coset space of $SL(2,\mathbb {R} )$ (see Zonal spherical function). Actually the Fourier transform on $L^{1}(G//K)$ is given by

L^{1}(G//K)\ni f\mapsto {\hat {f}}

where

{\hat {f}}(s)=\int _{1}^{\infty }f(x)P_{s}(x)dx,\qquad -1\leq \Re (s)\leq 0

References

Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 8". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 332. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.
Courant, Richard; Hilbert, David (1953), Methods of Mathematical Physics, Volume 1, New York: Interscience Publisher, Inc .
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, MR 2723248
Ivanov, A.B. (2001) [1994], "L/l058030", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Snow, Chester (1952) [1942], Hypergeometric and Legendre functions with applications to integral equations of potential theory, National Bureau of Standards Applied Mathematics Series, No. 19, Washington, D.C.: U. S. Government Printing Office, MR 0048145
Whittaker, E. T.; Watson, G. N. (1963), A Course in Modern Analysis, Cambridge University Press, ISBN 978-0-521-58807-2

External links

Legendre function P on the Wolfram functions site.
Legendre function Q on the Wolfram functions site.
Associated Legendre function P on the Wolfram functions site.
Associated Legendre function Q on the Wolfram functions site.

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