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-/* specfunc/gsl_sf_legendre.h
- *
- * Copyright (C) 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2004 Gerard Jungman
- *
- * This program is free software; you can redistribute it and/or modify
- * it under the terms of the GNU General Public License as published by
- * the Free Software Foundation; either version 3 of the License, or (at
- * your option) any later version.
- *
- * This program is distributed in the hope that it will be useful, but
- * WITHOUT ANY WARRANTY; without even the implied warranty of
- * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
- * General Public License for more details.
- *
- * You should have received a copy of the GNU General Public License
- * along with this program; if not, write to the Free Software
- * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
- */
-
-/* Author: G. Jungman */
-
-#ifndef __GSL_SF_LEGENDRE_H__
-#define __GSL_SF_LEGENDRE_H__
-
-#include <gsl/gsl_sf_result.h>
-
-#undef __BEGIN_DECLS
-#undef __END_DECLS
-#ifdef __cplusplus
-# define __BEGIN_DECLS extern "C" {
-# define __END_DECLS }
-#else
-# define __BEGIN_DECLS /* empty */
-# define __END_DECLS /* empty */
-#endif
-
-__BEGIN_DECLS
-
-
-/* P_l(x) l >= 0; |x| <= 1
- *
- * exceptions: GSL_EDOM
- */
-int gsl_sf_legendre_Pl_e(const int l, const double x, gsl_sf_result * result);
-double gsl_sf_legendre_Pl(const int l, const double x);
-
-
-/* P_l(x) for l=0,...,lmax; |x| <= 1
- *
- * exceptions: GSL_EDOM
- */
-int gsl_sf_legendre_Pl_array(
- const int lmax, const double x,
- double * result_array
- );
-
-
-/* P_l(x) and P_l'(x) for l=0,...,lmax; |x| <= 1
- *
- * exceptions: GSL_EDOM
- */
-int gsl_sf_legendre_Pl_deriv_array(
- const int lmax, const double x,
- double * result_array,
- double * result_deriv_array
- );
-
-
-/* P_l(x), l=1,2,3
- *
- * exceptions: none
- */
-int gsl_sf_legendre_P1_e(double x, gsl_sf_result * result);
-int gsl_sf_legendre_P2_e(double x, gsl_sf_result * result);
-int gsl_sf_legendre_P3_e(double x, gsl_sf_result * result);
-double gsl_sf_legendre_P1(const double x);
-double gsl_sf_legendre_P2(const double x);
-double gsl_sf_legendre_P3(const double x);
-
-
-/* Q_0(x), x > -1, x != 1
- *
- * exceptions: GSL_EDOM
- */
-int gsl_sf_legendre_Q0_e(const double x, gsl_sf_result * result);
-double gsl_sf_legendre_Q0(const double x);
-
-
-/* Q_1(x), x > -1, x != 1
- *
- * exceptions: GSL_EDOM
- */
-int gsl_sf_legendre_Q1_e(const double x, gsl_sf_result * result);
-double gsl_sf_legendre_Q1(const double x);
-
-
-/* Q_l(x), x > -1, x != 1, l >= 0
- *
- * exceptions: GSL_EDOM
- */
-int gsl_sf_legendre_Ql_e(const int l, const double x, gsl_sf_result * result);
-double gsl_sf_legendre_Ql(const int l, const double x);
-
-
-/* P_l^m(x) m >= 0; l >= m; |x| <= 1.0
- *
- * Note that this function grows combinatorially with l.
- * Therefore we can easily generate an overflow for l larger
- * than about 150.
- *
- * There is no trouble for small m, but when m and l are both large,
- * then there will be trouble. Rather than allow overflows, these
- * functions refuse to calculate when they can sense that l and m are
- * too big.
- *
- * If you really want to calculate a spherical harmonic, then DO NOT
- * use this. Instead use legendre_sphPlm() below, which uses a similar
- * recursion, but with the normalized functions.
- *
- * exceptions: GSL_EDOM, GSL_EOVRFLW
- */
-int gsl_sf_legendre_Plm_e(const int l, const int m, const double x, gsl_sf_result * result);
-double gsl_sf_legendre_Plm(const int l, const int m, const double x);
-
-
-/* P_l^m(x) m >= 0; l >= m; |x| <= 1.0
- * l=|m|,...,lmax
- *
- * exceptions: GSL_EDOM, GSL_EOVRFLW
- */
-int gsl_sf_legendre_Plm_array(
- const int lmax, const int m, const double x,
- double * result_array
- );
-
-
-/* P_l^m(x) and d(P_l^m(x))/dx; m >= 0; lmax >= m; |x| <= 1.0
- * l=|m|,...,lmax
- *
- * exceptions: GSL_EDOM, GSL_EOVRFLW
- */
-int gsl_sf_legendre_Plm_deriv_array(
- const int lmax, const int m, const double x,
- double * result_array,
- double * result_deriv_array
- );
-
-
-/* P_l^m(x), normalized properly for use in spherical harmonics
- * m >= 0; l >= m; |x| <= 1.0
- *
- * There is no overflow problem, as there is for the
- * standard normalization of P_l^m(x).
- *
- * Specifically, it returns:
- *
- * sqrt((2l+1)/(4pi)) sqrt((l-m)!/(l+m)!) P_l^m(x)
- *
- * exceptions: GSL_EDOM
- */
-int gsl_sf_legendre_sphPlm_e(const int l, int m, const double x, gsl_sf_result * result);
-double gsl_sf_legendre_sphPlm(const int l, const int m, const double x);
-
-
-/* sphPlm(l,m,x) values
- * m >= 0; l >= m; |x| <= 1.0
- * l=|m|,...,lmax
- *
- * exceptions: GSL_EDOM
- */
-int gsl_sf_legendre_sphPlm_array(
- const int lmax, int m, const double x,
- double * result_array
- );
-
-
-/* sphPlm(l,m,x) and d(sphPlm(l,m,x))/dx values
- * m >= 0; l >= m; |x| <= 1.0
- * l=|m|,...,lmax
- *
- * exceptions: GSL_EDOM
- */
-int gsl_sf_legendre_sphPlm_deriv_array(
- const int lmax, const int m, const double x,
- double * result_array,
- double * result_deriv_array
- );
-
-
-
-/* size of result_array[] needed for the array versions of Plm
- * (lmax - m + 1)
- */
-int gsl_sf_legendre_array_size(const int lmax, const int m);
-
-/* Irregular Spherical Conical Function
- * P^{1/2}_{-1/2 + I lambda}(x)
- *
- * x > -1.0
- * exceptions: GSL_EDOM
- */
-int gsl_sf_conicalP_half_e(const double lambda, const double x, gsl_sf_result * result);
-double gsl_sf_conicalP_half(const double lambda, const double x);
-
-
-/* Regular Spherical Conical Function
- * P^{-1/2}_{-1/2 + I lambda}(x)
- *
- * x > -1.0
- * exceptions: GSL_EDOM
- */
-int gsl_sf_conicalP_mhalf_e(const double lambda, const double x, gsl_sf_result * result);
-double gsl_sf_conicalP_mhalf(const double lambda, const double x);
-
-
-/* Conical Function
- * P^{0}_{-1/2 + I lambda}(x)
- *
- * x > -1.0
- * exceptions: GSL_EDOM
- */
-int gsl_sf_conicalP_0_e(const double lambda, const double x, gsl_sf_result * result);
-double gsl_sf_conicalP_0(const double lambda, const double x);
-
-
-/* Conical Function
- * P^{1}_{-1/2 + I lambda}(x)
- *
- * x > -1.0
- * exceptions: GSL_EDOM
- */
-int gsl_sf_conicalP_1_e(const double lambda, const double x, gsl_sf_result * result);
-double gsl_sf_conicalP_1(const double lambda, const double x);
-
-
-/* Regular Spherical Conical Function
- * P^{-1/2-l}_{-1/2 + I lambda}(x)
- *
- * x > -1.0, l >= -1
- * exceptions: GSL_EDOM
- */
-int gsl_sf_conicalP_sph_reg_e(const int l, const double lambda, const double x, gsl_sf_result * result);
-double gsl_sf_conicalP_sph_reg(const int l, const double lambda, const double x);
-
-
-/* Regular Cylindrical Conical Function
- * P^{-m}_{-1/2 + I lambda}(x)
- *
- * x > -1.0, m >= -1
- * exceptions: GSL_EDOM
- */
-int gsl_sf_conicalP_cyl_reg_e(const int m, const double lambda, const double x, gsl_sf_result * result);
-double gsl_sf_conicalP_cyl_reg(const int m, const double lambda, const double x);
-
-
-/* The following spherical functions are specializations
- * of Legendre functions which give the regular eigenfunctions
- * of the Laplacian on a 3-dimensional hyperbolic space.
- * Of particular interest is the flat limit, which is
- * Flat-Lim := {lambda->Inf, eta->0, lambda*eta fixed}.
- */
-
-/* Zeroth radial eigenfunction of the Laplacian on the
- * 3-dimensional hyperbolic space.
- *
- * legendre_H3d_0(lambda,eta) := sin(lambda*eta)/(lambda*sinh(eta))
- *
- * Normalization:
- * Flat-Lim legendre_H3d_0(lambda,eta) = j_0(lambda*eta)
- *
- * eta >= 0.0
- * exceptions: GSL_EDOM
- */
-int gsl_sf_legendre_H3d_0_e(const double lambda, const double eta, gsl_sf_result * result);
-double gsl_sf_legendre_H3d_0(const double lambda, const double eta);
-
-
-/* First radial eigenfunction of the Laplacian on the
- * 3-dimensional hyperbolic space.
- *
- * legendre_H3d_1(lambda,eta) :=
- * 1/sqrt(lambda^2 + 1) sin(lam eta)/(lam sinh(eta))
- * (coth(eta) - lambda cot(lambda*eta))
- *
- * Normalization:
- * Flat-Lim legendre_H3d_1(lambda,eta) = j_1(lambda*eta)
- *
- * eta >= 0.0
- * exceptions: GSL_EDOM
- */
-int gsl_sf_legendre_H3d_1_e(const double lambda, const double eta, gsl_sf_result * result);
-double gsl_sf_legendre_H3d_1(const double lambda, const double eta);
-
-
-/* l'th radial eigenfunction of the Laplacian on the
- * 3-dimensional hyperbolic space.
- *
- * Normalization:
- * Flat-Lim legendre_H3d_l(l,lambda,eta) = j_l(lambda*eta)
- *
- * eta >= 0.0, l >= 0
- * exceptions: GSL_EDOM
- */
-int gsl_sf_legendre_H3d_e(const int l, const double lambda, const double eta, gsl_sf_result * result);
-double gsl_sf_legendre_H3d(const int l, const double lambda, const double eta);
-
-
-/* Array of H3d(ell), 0 <= ell <= lmax
- */
-int gsl_sf_legendre_H3d_array(const int lmax, const double lambda, const double eta, double * result_array);
-
-/* associated legendre P_{lm} routines */
-
-typedef enum
-{
- GSL_SF_LEGENDRE_SCHMIDT,
- GSL_SF_LEGENDRE_SPHARM,
- GSL_SF_LEGENDRE_FULL,
- GSL_SF_LEGENDRE_NONE
-} gsl_sf_legendre_t;
-
-int gsl_sf_legendre_array(const gsl_sf_legendre_t norm,
- const size_t lmax, const double x,
- double result_array[]);
-int gsl_sf_legendre_array_e(const gsl_sf_legendre_t norm,
- const size_t lmax, const double x,
- const double csphase,
- double result_array[]);
-int gsl_sf_legendre_deriv_array(const gsl_sf_legendre_t norm,
- const size_t lmax, const double x,
- double result_array[],
- double result_deriv_array[]);
-int gsl_sf_legendre_deriv_array_e(const gsl_sf_legendre_t norm,
- const size_t lmax, const double x,
- const double csphase,
- double result_array[],
- double result_deriv_array[]);
-int gsl_sf_legendre_deriv_alt_array(const gsl_sf_legendre_t norm,
- const size_t lmax, const double x,
- double result_array[],
- double result_deriv_array[]);
-int gsl_sf_legendre_deriv_alt_array_e(const gsl_sf_legendre_t norm,
- const size_t lmax, const double x,
- const double csphase,
- double result_array[],
- double result_deriv_array[]);
-int gsl_sf_legendre_deriv2_array(const gsl_sf_legendre_t norm,
- const size_t lmax, const double x,
- double result_array[],
- double result_deriv_array[],
- double result_deriv2_array[]);
-int gsl_sf_legendre_deriv2_array_e(const gsl_sf_legendre_t norm,
- const size_t lmax, const double x,
- const double csphase,
- double result_array[],
- double result_deriv_array[],
- double result_deriv2_array[]);
-int gsl_sf_legendre_deriv2_alt_array(const gsl_sf_legendre_t norm,
- const size_t lmax, const double x,
- double result_array[],
- double result_deriv_array[],
- double result_deriv2_array[]);
-int gsl_sf_legendre_deriv2_alt_array_e(const gsl_sf_legendre_t norm,
- const size_t lmax, const double x,
- const double csphase,
- double result_array[],
- double result_deriv_array[],
- double result_deriv2_array[]);
-size_t gsl_sf_legendre_array_n(const size_t lmax);
-size_t gsl_sf_legendre_array_index(const size_t l, const size_t m);
-size_t gsl_sf_legendre_nlm(const size_t lmax);
-
-__END_DECLS
-
-#endif /* __GSL_SF_LEGENDRE_H__ */
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