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Diffstat (limited to 'gsl-1.9/specfunc/gsl_sf_legendre.h')
| -rw-r--r-- | gsl-1.9/specfunc/gsl_sf_legendre.h | 326 |
1 files changed, 326 insertions, 0 deletions
diff --git a/gsl-1.9/specfunc/gsl_sf_legendre.h b/gsl-1.9/specfunc/gsl_sf_legendre.h new file mode 100644 index 0000000..ed63262 --- /dev/null +++ b/gsl-1.9/specfunc/gsl_sf_legendre.h @@ -0,0 +1,326 @@ +/* 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 2 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); + + +#ifdef HAVE_INLINE +extern inline +int +gsl_sf_legendre_array_size(const int lmax, const int m) +{ + return lmax-m+1; +} +#endif /* HAVE_INLINE */ + + +__END_DECLS + +#endif /* __GSL_SF_LEGENDRE_H__ */ |