/* specfunc/gsl_sf_gamma.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 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., 675 Mass Ave, Cambridge, MA 02139, USA.

 */

/* Author:  G. Jungman */

#ifndef __GSL_SF_GAMMA_H__

#define __GSL_SF_GAMMA_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

/* Log[Gamma(x)], x not a negative integer

 * Uses real Lanczos method.

 * Returns the real part of Log[Gamma[x]] when x < 0,

 * i.e. Log[|Gamma[x]|].

 *

 * exceptions: GSL_EDOM, GSL_EROUND

 */

int gsl_sf_lngamma_e(double x, gsl_sf_result * result);

double gsl_sf_lngamma(const double x);

/* Log[Gamma(x)], x not a negative integer

 * Uses real Lanczos method. Determines

 * the sign of Gamma[x] as well as Log[|Gamma[x]|] for x < 0.

 * So Gamma[x] = sgn * Exp[result_lg].

 *

 * exceptions: GSL_EDOM, GSL_EROUND

 */

int gsl_sf_lngamma_sgn_e(double x, gsl_sf_result * result_lg, double *sgn);

/* Gamma(x), x not a negative integer

 * Uses real Lanczos method.

 *

 * exceptions: GSL_EDOM, GSL_EOVRFLW, GSL_EROUND

 */

int gsl_sf_gamma_e(const double x, gsl_sf_result * result);

double gsl_sf_gamma(const double x);

/* Regulated Gamma Function, x > 0

 * Gamma^*(x) = Gamma(x)/(Sqrt[2Pi] x^(x-1/2) exp(-x))

 *            = (1 + 1/(12x) + ...),  x->Inf

 * A useful suggestion of Temme.

 *

 * exceptions: GSL_EDOM

 */

int gsl_sf_gammastar_e(const double x, gsl_sf_result * result);

double gsl_sf_gammastar(const double x);

/* 1/Gamma(x)

 * Uses real Lanczos method.

 *

 * exceptions: GSL_EUNDRFLW, GSL_EROUND

 */

int gsl_sf_gammainv_e(const double x, gsl_sf_result * result);

double gsl_sf_gammainv(const double x);

/* Log[Gamma(z)] for z complex, z not a negative integer

 * Uses complex Lanczos method. Note that the phase part (arg)

 * is not well-determined when |z| is very large, due

 * to inevitable roundoff in restricting to (-Pi,Pi].

 * This will raise the GSL_ELOSS exception when it occurs.

 * The absolute value part (lnr), however, never suffers.

 *

 * Calculates:

 *   lnr = log|Gamma(z)|

 *   arg = arg(Gamma(z))  in (-Pi, Pi]

 *

 * exceptions: GSL_EDOM, GSL_ELOSS

 */

int gsl_sf_lngamma_complex_e(double zr, double zi, gsl_sf_result * lnr, gsl_sf_result * arg);

/* x^n / n!

 *

 * x >= 0.0, n >= 0

 * exceptions: GSL_EDOM, GSL_EOVRFLW, GSL_EUNDRFLW

 */

int gsl_sf_taylorcoeff_e(const int n, const double x, gsl_sf_result * result);

double gsl_sf_taylorcoeff(const int n, const double x);

/* n!

 *

 * exceptions: GSL_EDOM, GSL_OVRFLW

 */

int gsl_sf_fact_e(const unsigned int n, gsl_sf_result * result);

double gsl_sf_fact(const unsigned int n);

/* n!! = n(n-2)(n-4) ... 

 *

 * exceptions: GSL_EDOM, GSL_OVRFLW

 */

int gsl_sf_doublefact_e(const unsigned int n, gsl_sf_result * result);

double gsl_sf_doublefact(const unsigned int n);

/* log(n!) 

 * Faster than ln(Gamma(n+1)) for n < 170; defers for larger n.

 *

 * exceptions: none

 */

int gsl_sf_lnfact_e(const unsigned int n, gsl_sf_result * result);

double gsl_sf_lnfact(const unsigned int n);

/* log(n!!) 

 *

 * exceptions: none

 */

int gsl_sf_lndoublefact_e(const unsigned int n, gsl_sf_result * result);

double gsl_sf_lndoublefact(const unsigned int n);

/* log(n choose m)

 *

 * exceptions: GSL_EDOM 

 */

int gsl_sf_lnchoose_e(unsigned int n, unsigned int m, gsl_sf_result * result);

double gsl_sf_lnchoose(unsigned int n, unsigned int m);

/* n choose m

 *

 * exceptions: GSL_EDOM, GSL_EOVRFLW

 */

int gsl_sf_choose_e(unsigned int n, unsigned int m, gsl_sf_result * result);

double gsl_sf_choose(unsigned int n, unsigned int m);

/* Logarithm of Pochhammer (Apell) symbol

 *   log( (a)_x )

 *   where (a)_x := Gamma[a + x]/Gamma[a]

 *

 * a > 0, a+x > 0

 *

 * exceptions:  GSL_EDOM

 */

int gsl_sf_lnpoch_e(const double a, const double x, gsl_sf_result * result);

double gsl_sf_lnpoch(const double a, const double x);

/* Logarithm of Pochhammer (Apell) symbol, with sign information.

 *   result = log( |(a)_x| )

 *   sgn    = sgn( (a)_x )

 *   where (a)_x := Gamma[a + x]/Gamma[a]

 *

 * a != neg integer, a+x != neg integer

 *

 * exceptions:  GSL_EDOM

 */

int gsl_sf_lnpoch_sgn_e(const double a, const double x, gsl_sf_result * result, double * sgn);

/* Pochhammer (Apell) symbol

 *   (a)_x := Gamma[a + x]/Gamma[x]

 *

 * a != neg integer, a+x != neg integer

 *

 * exceptions:  GSL_EDOM, GSL_EOVRFLW

 */

int gsl_sf_poch_e(const double a, const double x, gsl_sf_result * result);

double gsl_sf_poch(const double a, const double x);

/* Relative Pochhammer (Apell) symbol

 *   ((a,x) - 1)/x

 *   where (a,x) = (a)_x := Gamma[a + x]/Gamma[a]

 *

 * exceptions:  GSL_EDOM

 */

int gsl_sf_pochrel_e(const double a, const double x, gsl_sf_result * result);

double gsl_sf_pochrel(const double a, const double x);

/* Normalized Incomplete Gamma Function

 *

 * Q(a,x) = 1/Gamma(a) Integral[ t^(a-1) e^(-t), {t,x,Infinity} ]

 *

 * a >= 0, x >= 0

 *   Q(a,0) := 1

 *   Q(0,x) := 0, x != 0

 *

 * exceptions: GSL_EDOM

 */

int gsl_sf_gamma_inc_Q_e(const double a, const double x, gsl_sf_result * result);

double gsl_sf_gamma_inc_Q(const double a, const double x);

/* Complementary Normalized Incomplete Gamma Function

 *

 * P(a,x) = 1/Gamma(a) Integral[ t^(a-1) e^(-t), {t,0,x} ]

 *

 * a > 0, x >= 0

 *

 * exceptions: GSL_EDOM

 */

int gsl_sf_gamma_inc_P_e(const double a, const double x, gsl_sf_result * result);

double gsl_sf_gamma_inc_P(const double a, const double x);

/* Non-normalized Incomplete Gamma Function

 *

 * Gamma(a,x) := Integral[ t^(a-1) e^(-t), {t,x,Infinity} ]

 *

 * x >= 0.0

 *   Gamma(a, 0) := Gamma(a)

 *

 * exceptions: GSL_EDOM

 */

int gsl_sf_gamma_inc_e(const double a, const double x, gsl_sf_result * result);

double gsl_sf_gamma_inc(const double a, const double x);

/* Logarithm of Beta Function

 * Log[B(a,b)]

 *

 * a > 0, b > 0

 * exceptions: GSL_EDOM

 */

int gsl_sf_lnbeta_e(const double a, const double b, gsl_sf_result * result);

double gsl_sf_lnbeta(const double a, const double b);

/* Beta Function

 * B(a,b)

 *

 * a > 0, b > 0

 * exceptions: GSL_EDOM, GSL_EOVRFLW, GSL_EUNDRFLW

 */

int gsl_sf_beta_e(const double a, const double b, gsl_sf_result * result);

double gsl_sf_beta(const double a, const double b);

/* Normalized Incomplete Beta Function

 * B_x(a,b)/B(a,b)

 *

 * a > 0, b > 0, 0 <= x <= 1

 * exceptions: GSL_EDOM, GSL_EUNDRFLW

 */

int gsl_sf_beta_inc_e(const double a, const double b, const double x, gsl_sf_result * result);

double gsl_sf_beta_inc(const double a, const double b, const double x);

/* The maximum x such that gamma(x) is not

 * considered an overflow.

 */

#define GSL_SF_GAMMA_XMAX  171.0

__END_DECLS

#endif /* __GSL_SF_GAMMA_H__ */