Legendre functions

In physical science and mathematics, the Legendre functions , and associated Legendre functions , , and Legendre functions of the second kind, , 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
\left(1 - x^2\right) y'' - 2xy' + \left lambda(\lambda+1) - \frac\righty = 0,
where the numbers and may be complex, and are called the degree and order of the relevant function, respectively. The polynomial solutions when is an integer (denoted ), and are the Legendre polynomials ; and when
is an integer (denoted ), and is also an integer with are the associated Legendre polynomials. All other cases of and can be discussed as one, and the solutions are written , . If , the superscript is omitted, and one writes just , . However, the solution when is an integer is often discussed separately as Legendre's function of the second kind, and denoted .

This is a second order linear equation with three regular singular points (at , , 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.

Solutions of the differential equation

Since the differential equation is linear, homogeneous (the right hand side =zero) and of second order, it has two linearly independent solutions, which can both be expressed in terms of the hypergeometric function, $_2F_1$. With $\Gamma$ being the gamma function, the first solution is
P_^(z) = \frac \left frac\right \,_2F_1 \left(-\lambda, \lambda+1; 1-\mu; \frac\right),\qquad \text \ , 1-z, <2
and the second is,
$Q_^(z) = \frac\frac \,_2F_1 \left(\frac, \frac; \lambda+\frac; \frac\right),\qquad \text\ \ , z, >1.$

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

Legendre functions of the second kind ()

The nonpolynomial solution for the special case of integer degree $\lambda = n \in \N_0$, and $\mu = 0$, is often discussed separately.
It is given by
$Q_n(x)=\frac\left(x^+\fracx^+\fracx^+\cdots\right)$

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
$Q_n(x) = \begin \frac \log \frac & n = 0 \\ P_1(x) Q_0(x) - 1 & n = 1 \\ \frac x Q_(x) - \frac Q_(x) & n \geq 2 \,. \end$

Associated Legendre functions of the second kind

The nonpolynomial solution for the special case of integer degree $\lambda = n \in \N_0$, and $\mu = m \in \N_0$ is given by
$Q_n^(x) = (-1)^m (1-x^2)^\frac \fracQ_n(x)\,.$

Integral representations

The Legendre functions can be written as contour integrals. For example,
$P_\lambda(z) =P^0_\lambda(z) = \frac \int_ \fracdt$
where the contour winds around the points and in the positive direction and does not wind around .
For real , we have
$P_s(x) = \frac\int_^\left(x+\sqrt\cos\theta\right)^s d\theta = \frac\int_0^1\left(x+\sqrt(2t-1)\right)^s\frac,\qquad s\in\Complex$

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 $SL(2,\R)$ (see Zonal spherical function). Actually the Fourier transform on $L^1(G//K)$ is given by
$L^1(G//K)\ni f\mapsto \hat$
where
$\hat(s)=\int_1^\infty f(x)P_s(x)dx,\qquad -1\leq\Re(s)\leq 0$

See also

* Ferrers function

References

*
* .
*
*
*
*

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.

{{Authority control

Hypergeometric functions