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The following functions compute the associated Legendre Polynomials
P_l^m(x). Note that this function grows combinatorially with
l and can overflow for l larger than about 150. There is
no trouble for small m, but overflow occurs when m and
l are both large. Rather than allow overflows, these functions
refuse to calculate P_l^m(x) and return GSL_EOVRFLW when
they can sense that l and m are too big.
If you want to calculate a spherical harmonic, then do not use
these functions. Instead use gsl_sf_legendre_sphPlm() below,
which uses a similar recursion, but with the normalized functions.
These routines compute the associated Legendre polynomial P_l^m(x) for m >= 0, l >= m, |x| <= 1.
These functions compute an array of Legendre polynomials P_l^m(x), and optionally their derivatives dP_l^m(x)/dx, for m >= 0, l = |m|, ..., lmax, |x| <= 1.
These routines compute the normalized associated Legendre polynomial $\sqrt{(2l+1)/(4\pi)} \sqrt{(l-m)!/(l+m)!} P_l^m(x)$ suitable for use in spherical harmonics. The parameters must satisfy m >= 0, l >= m, |x| <= 1. Theses routines avoid the overflows that occur for the standard normalization of P_l^m(x).
These functions compute an array of normalized associated Legendre functions $\sqrt{(2l+1)/(4\pi)} \sqrt{(l-m)!/(l+m)!} P_l^m(x)$, and optionally their derivatives, for m >= 0, l = |m|, ..., lmax, |x| <= 1.0