forgotten legendre.h

1 files changed, 319 insertions, 0 deletions

diff --git a/lib/Numeric/GSL/Special/gsl_sf_legendre.h b/lib/Numeric/GSL/Special/gsl_sf_legendre.h
new file mode 100644
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+++ b/lib/Numeric/GSL/Special/gsl_sf_legendre.h
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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 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);
+
+
+
+
+
+__END_DECLS
+
+#endif /* __GSL_SF_LEGENDRE_H__ */