/* specfunc/gsl_sf_ellint.h

 * 

 * Copyright (C) 1996, 1997, 1998, 1999, 2000 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_ELLINT_H__

#define __GSL_SF_ELLINT_H__

#include <gsl/gsl_mode.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

/* Legendre form of complete elliptic integrals

 *

 * K(k) = Integral[1/Sqrt[1 - k^2 Sin[t]^2], {t, 0, Pi/2}]

 * E(k) = Integral[  Sqrt[1 - k^2 Sin[t]^2], {t, 0, Pi/2}]

 *

 * exceptions: GSL_EDOM

 */

int gsl_sf_ellint_Kcomp_e(double k, gsl_mode_t mode, gsl_sf_result * result);

double gsl_sf_ellint_Kcomp(double k, gsl_mode_t mode);

int gsl_sf_ellint_Ecomp_e(double k, gsl_mode_t mode, gsl_sf_result * result);

double gsl_sf_ellint_Ecomp(double k, gsl_mode_t mode);

int gsl_sf_ellint_Pcomp_e(double k, double n, gsl_mode_t mode, gsl_sf_result * result);

double gsl_sf_ellint_Pcomp(double k, double n, gsl_mode_t mode);

int gsl_sf_ellint_Dcomp_e(double k, gsl_mode_t mode, gsl_sf_result * result);

double gsl_sf_ellint_Dcomp(double k, gsl_mode_t mode);

/* Legendre form of incomplete elliptic integrals

 *

 * F(phi,k)   = Integral[1/Sqrt[1 - k^2 Sin[t]^2], {t, 0, phi}]

 * E(phi,k)   = Integral[  Sqrt[1 - k^2 Sin[t]^2], {t, 0, phi}]

 * P(phi,k,n) = Integral[(1 + n Sin[t]^2)^(-1)/Sqrt[1 - k^2 Sin[t]^2], {t, 0, phi}]

 * D(phi,k,n) = R_D(1-Sin[phi]^2, 1-k^2 Sin[phi]^2, 1.0)

 *

 * F: [Carlson, Numerische Mathematik 33 (1979) 1, (4.1)]

 * E: [Carlson, ", (4.2)]

 * P: [Carlson, ", (4.3)]

 * D: [Carlson, ", (4.4)]

 *

 * exceptions: GSL_EDOM

 */

int gsl_sf_ellint_F_e(double phi, double k, gsl_mode_t mode, gsl_sf_result * result);

double gsl_sf_ellint_F(double phi, double k, gsl_mode_t mode);

int gsl_sf_ellint_E_e(double phi, double k, gsl_mode_t mode, gsl_sf_result * result);

double gsl_sf_ellint_E(double phi, double k, gsl_mode_t mode);

int gsl_sf_ellint_P_e(double phi, double k, double n, gsl_mode_t mode, gsl_sf_result * result);

double gsl_sf_ellint_P(double phi, double k, double n, gsl_mode_t mode);

int gsl_sf_ellint_D_e(double phi, double k, gsl_mode_t mode, gsl_sf_result * result);

double gsl_sf_ellint_D(double phi, double k, gsl_mode_t mode);

/* Carlson's symmetric basis of functions

 *

 * RC(x,y)   = 1/2 Integral[(t+x)^(-1/2) (t+y)^(-1)], {t,0,Inf}]

 * RD(x,y,z) = 3/2 Integral[(t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-3/2), {t,0,Inf}]

 * RF(x,y,z) = 1/2 Integral[(t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-1/2), {t,0,Inf}]

 * RJ(x,y,z,p) = 3/2 Integral[(t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-1/2) (t+p)^(-1), {t,0,Inf}]

 *

 * exceptions: GSL_EDOM

 */

int gsl_sf_ellint_RC_e(double x, double y, gsl_mode_t mode, gsl_sf_result * result);

double gsl_sf_ellint_RC(double x, double y, gsl_mode_t mode);

int gsl_sf_ellint_RD_e(double x, double y, double z, gsl_mode_t mode, gsl_sf_result * result);

double gsl_sf_ellint_RD(double x, double y, double z, gsl_mode_t mode);

int gsl_sf_ellint_RF_e(double x, double y, double z, gsl_mode_t mode, gsl_sf_result * result);

double gsl_sf_ellint_RF(double x, double y, double z, gsl_mode_t mode);

int gsl_sf_ellint_RJ_e(double x, double y, double z, double p, gsl_mode_t mode, gsl_sf_result * result);

double gsl_sf_ellint_RJ(double x, double y, double z, double p, gsl_mode_t mode);

__END_DECLS

#endif /* __GSL_SF_ELLINT_H__ */