Legendre function

In physical science and mathematics, the Legendre functions $P λ$ , $Q λ$ and associated Legendre functions $P μ λ$ , $Q μ λ$ , and Legendre functions of the second kind, $Q n$ , are all solutions of Legendre's differential equation. The Legendre polynomials and the associated Legendre polynomials are also solutions of the differential equation in special cases, which, by virtue of being polynomials, have a large number of additional properties, mathematical structure, and applications. For these polynomial solutions, see the separate Wikipedia articles.

Legendre's differential equation

The general Legendre equation reads

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

where the numbers $λ$ and $μ$ may be complex, and are called the degree and order of the relevant function, respectively. The polynomial solutions when $λ$ is an integer (denoted $n$ ), and $μ = 0$ are the Legendre polynomials $P n$ ; and when $λ$ is an integer (denoted $n$ ), and $μ = m$ is also an integer with $| m | < n$ are the associated Legendre polynomials. All other cases of $λ$ and $μ$ can be discussed as one, and the solutions are written $P μ λ$ , $Q μ λ$ . If $μ = 0$ , the superscript is omitted, and one writes just $P λ$ , $Q λ$ . However, the solution $Q λ$ when $λ$ is an integer is often discussed separately as Legendre's function of the second kind, and denoted $Q n$ .

This is a second order linear equation with three regular singular points (at $1, -1$ , and $\infty$ ). 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.

Solutions of the differential equation

Since the differential equation is linear and of second order, it has two linearly independent solutions, which can both be expressed in terms of the hypergeometric function, $_{2}F_{1}$ . With $\Gamma$ being the gamma function, the first solution is

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

and the second is,

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.

These are generally known as Legendre functions of the first and second kind of noninteger degree, with the additional qualifier 'associated' if $μ$ is non-zero. A useful relation between the $P$ and $Q$ solutions is Whipple's formula.

Legendre functions of the second kind ( $Q n$ )

The nonpolynomial solution for the special case of integer degree $\lambda =n\in \mathbb {N} _{0}$ , and $\mu =0$ , is often discussed separately. It is given by

Q_{n}(x)={\frac {n!}{1\cdot 3\cdots (2n+1)}}\left(x^{-(n+1)}+{\frac {(n+1)(n+2)}{2(2n+3)}}x^{-(n+3)}+{\frac {(n+1)(n+2)(n+3)(n+4)}{2\cdot 4(2n+3)(2n+5)}}x^{-(n+5)}+\cdots \right)

This solution is necessarily singular when $x=\pm 1$ .

The Legendre functions of the second kind can also be defined recursively via Bonnet's recursion formula

Q_{n}(x)={\begin{cases}{\frac {1}{2}}\log {\frac {1+x}{1-x}}&n=0\\P_{1}(x)Q_{0}(x)-1&n=1\\{\frac {2n-1}{n}}xQ_{n-1}(x)-{\frac {n-1}{n}}Q_{n-2}(x)&n\geq 2\,.\end{cases}}

Associated Legendre functions of the second kind

The nonpolynomial solution for the special case of integer degree $\lambda =n\in \mathbb {N} _{0}$ , and $\mu =m\in \mathbb {N} _{0}$ is given by

Q_{n}^{m}(x)=(-1)^{m}(1-x^{2})^{\frac {m}{2}}{\frac {\mathrm {d} ^{m}}{\mathrm {d} x^{m}}}Q_{n}(x)\,.

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. Vol. 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. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.
Ivanov, A.B. (2001) [1994], "Legendre function", Encyclopedia of Mathematics, EMS Press
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, hdl:2027/mdp.39015011416826, 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.