Legendre function

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In mathematics, the Legendre functions P_λ, Q_λ and associated Legendre functions PTemplate:Su, QTemplate:Su 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:

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

Integral representations

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

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

where the contour circles around the points 1 and z in the positive direction and does not circle 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

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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.