Legendre function

The general Legendre equation reads \left(1 - x^2\right) y'' - 2xy' + \left[\lambda(\lambda+1) - \frac{\mu^2}{1-x^2}\right] y = 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_{\lambda}^{\mu}(z) = \frac{1}{\Gamma(1-\mu)} \left[\frac{z+1}{z-1}\right]^{\mu/2} \,_2F_1 \left(-\lambda, \lambda+1; 1-\mu; \frac{1-z}{2}\right),\qquad \text{for } \ |1-z| 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}} \,_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. 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. Positive integer order For positive integer \mu = m \in \N^+ the evaluation of P^\mu_\lambda above involves cancellation of singular terms. We can find the limit valid for m \in \N_0 as P^m_\lambda(z) = \lim_{\mu \to m} P^\mu_\lambda (z) = \frac{(-\lambda )_m (\lambda + 1)_m}{m!} \left[\frac{1-z}{1+z}\right]^{m/2} \,_2F_1 \left(-\lambda, \lambda+1; 1+m; \frac{1-z}{2}\right), with (\lambda)_{n} the (rising) Pochhammer symbol. ==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{n!}{1\cdot3\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\cdot4(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} x Q_{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 \N_0 , and \mu = m \in \N_0 is given by Q_n^{m}(x) = (-1)^m (1-x^2)^\frac{m}{2} \frac{d^m}{dx^m}Q_n(x)\,. ==Integral representations==

The Legendre functions can be written as contour integrals. For example, P_\lambda(z) =P^0_\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 winds around the points and in the positive direction and does not wind around . For real , 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\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{f} where \hat{f}(s)=\int_1^\infty f(x)P_s(x)dx,\qquad -1\leq\Re(s)\leq 0 ==Singularities of Legendre functions of the first kind () as a consequence of symmetry ==

Legendre functions of non-integer degree are unbounded at the interval [-1, 1] . In applications in physics, this often provides a selection criterion. Indeed, because Legendre functions of the second kind are always unbounded, in order to have a bounded solution of Legendre's equation at all, the degree must be integer valued: only for integer degree, Legendre functions of the first kind reduce to Legendre polynomials, which are bounded on [-1, 1]. It can be shown that the singularity of the Legendre functions for non-integer degree is a consequence of the mirror symmetry of Legendre's equation. Thus there is a symmetry under the selection rule just mentioned. == See also ==