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{{For|the most common case of integer degree|Legendre polynomials|associated Legendre polynomials}}
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{{no footnotes|date=January 2013}}
 
In mathematics, the '''Legendre functions''' ''P''<sub>λ</sub>, ''Q''<sub>λ</sub> and '''associated Legendre functions''' ''P''{{su|p=&mu;|b=&lambda;}}, ''Q''{{su|p=&mu;|b=&lambda;}} are generalizations of [[Legendre polynomials]] to non-integer degree.
[[File:Associated Legendre Poly.svg|thumb|500px|Associated Legendre polynomial curves for l=5.]]
 
==Differential equation==
Associated Legendre functions are  solutions of the '''general Legendre equation'''
 
:<math>(1-x^2)\,y'' -2xy' + \left[\lambda(\lambda+1) - \frac{\mu^2}{1-x^2}\right]\,y = 0,\,</math>
 
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, &minus;1, and ∞). Like all such equations, it can be converted into the [[hypergeometric differential equation]] by a change of variable, and its solutions can be expressed using [[hypergeometric function]]s.
 
==Definition==
These functions may actually be defined for general complex parameters and argument:
 
:<math>P_{\lambda}^{\mu}(z) = \frac{1}{\Gamma(1-\mu)} \left[\frac{1+z}{1-z}\right]^{\mu/2} \,_2F_1 (-\lambda, \lambda+1; 1-\mu; \frac{1-z}{2}),\qquad \text{for } \  |1-z|<2</math>
 
where <math>\Gamma</math> is the [[gamma function]] and <math> _2F_1</math> is the [[hypergeometric function]].
 
The second order differential equation has a second solution,
<math>Q_\lambda^{\mu}(z)</math>, defined as:
 
:<math>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}} \,_2F_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.</math>
 
==Integral representations==
The Legendre functions can be written as contour integrals. For example
:<math>P_\lambda(z) =\frac{1}{2\pi i}
\int_{1,z} \frac{(t^2-1)^\lambda}{2^\lambda(t-z)^{\lambda+1}}dt</math>
where the contour circles around the points 1 and ''z'' in the positive direction and does not circle around &minus;1.
For real x, we have
:<math>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}</math>
 
==Legendre function as characters==
The real integral representation of <math>P_s</math> are very useful in the study of harmonic analysis on <math>L^1(G//K)</math> where <math>G//K</math> is the [[Homogeneous space|double coset space]] of <math>SL(2,\mathbb{R})</math> (see [[Zonal spherical function]]). Actually the Fourier transform on <math>L^1(G//K)</math> is given by
:<math>L^1(G//K)\ni f\mapsto \hat{f}</math>
where
:<math>\hat{f}(s)=\int_1^\infty f(x)P_s(x)dx,\qquad -1\leq\Re(s)\leq 0 </math>
 
==References==
 
* {{AS ref|8|332}}
* {{citation|first1=Richard|last1=Courant|authorlink1=Richard Courant|first2=David|last2=Hilbert|authorlink2=David Hilbert|year=1953|title=Methods of Mathematical Physics, Volume 1|publisher=Interscience Publisher, Inc|publication-place=New York}}.
*{{dlmf|first=T. M. |last=Dunster|id=14|title=Legendre and Related Functions}}
*{{eom|id=L/l058030|first=A.B.|last= Ivanov}}
*{{Citation | last1=Snow | first1=Chester | title=Hypergeometric and Legendre functions with applications to integral equations of potential theory | url=http://babel.hathitrust.org/cgi/pt?id=mdp.39015023896346 | publisher=U. S. Government Printing Office | location=Washington, D.C. | series=National Bureau of Standards Applied Mathematics Series, No. 19 | mr=0048145 | year=1952|origyear=1942}}
*{{Citation | last1=Whittaker | first1=E. T. |authorlink1=E. T. Whittaker| last2=Watson | first2=G. N. | authorlink2=G. N. Watson|title=A Course in Modern Analysis | publisher=[[Cambridge University Press]] | isbn=978-0-521-58807-2 | year=1963 }}
 
==External links==
*[http://functions.wolfram.com/HypergeometricFunctions/LegendrePGeneral/ Legendre function P] on the Wolfram functions site.
*[http://functions.wolfram.com/HypergeometricFunctions/LegendreQGeneral/ Legendre function Q] on the Wolfram functions site.
*[http://functions.wolfram.com/HypergeometricFunctions/LegendreP2General/ Associated Legendre function P] on the Wolfram functions site.
*[http://functions.wolfram.com/HypergeometricFunctions/LegendreQ2General/ Associated Legendre function Q] on the Wolfram functions site.
 
[[Category:Hypergeometric functions]]

Latest revision as of 15:22, 22 November 2014

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