inifcns_gamma.cpp

Go to the documentation of this file.

 
/*
 *  GiNaC Copyright (C) 1999-2020 Johannes Gutenberg University Mainz, Germany
 *
 *  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
 */
 
#include "inifcns.h"
#include "constant.h"
#include "pseries.h"
#include "numeric.h"
#include "power.h"
#include "relational.h"
#include "operators.h"
#include "symbol.h"
#include "symmetry.h"
#include "utils.h"
 
#include <stdexcept>
#include <vector>
 
namespace GiNaC {
 
// Logarithm of Gamma function
 
static ex lgamma_evalf(const ex & x)
{
    if (is_exactly_a<numeric>(x)) {
        try {
            return lgamma(ex_to<numeric>(x));
        } catch (const dunno &e) { }
    }
    
    return lgamma(x).hold();
}
 
 
static ex lgamma_eval(const ex & x)
{
    if (x.info(info_flags::numeric)) {
        // trap integer arguments:
        if (x.info(info_flags::integer)) {
            // lgamma(n) -> log((n-1)!) for postitive n
            if (x.info(info_flags::posint))
                return log(factorial(x + _ex_1));
            else
                throw (pole_error("lgamma_eval(): logarithmic pole",0));
        }
        if (!ex_to<numeric>(x).is_rational())
            return lgamma(ex_to<numeric>(x));
    }
    
    return lgamma(x).hold();
}
 
 
static ex lgamma_deriv(const ex & x, unsigned deriv_param)
{
    GINAC_ASSERT(deriv_param==0);
    
    // d/dx  lgamma(x) -> psi(x)
    return psi(x);
}
 
 
static ex lgamma_series(const ex & arg,
                        const relational & rel,
                        int order,
                        unsigned options)
{
    // method:
    // Taylor series where there is no pole falls back to psi function
    // evaluation.
    // On a pole at -m we could use the recurrence relation
    //   lgamma(x) == lgamma(x+1)-log(x)
    // from which follows
    //   series(lgamma(x),x==-m,order) ==
    //   series(lgamma(x+m+1)-log(x)...-log(x+m)),x==-m,order);
    const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
    if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive))
        throw do_taylor();  // caught by function::series()
    // if we got here we have to care for a simple pole of tgamma(-m):
    numeric m = -ex_to<numeric>(arg_pt);
    ex recur;
    for (numeric p = 0; p<=m; ++p)
        recur += log(arg+p);
    return (lgamma(arg+m+_ex1)-recur).series(rel, order, options);
}
 
 
static ex lgamma_conjugate(const ex & x)
{
    // conjugate(lgamma(x))==lgamma(conjugate(x)) unless on the branch cut
    // which runs along the negative real axis.
    if (x.info(info_flags::positive)) {
        return lgamma(x);
    }
    if (is_exactly_a<numeric>(x) &&
        !x.imag_part().is_zero()) {
        return lgamma(x.conjugate());
    }
    return conjugate_function(lgamma(x)).hold();
}
 
 
REGISTER_FUNCTION(lgamma, eval_func(lgamma_eval).
                          evalf_func(lgamma_evalf).
                          derivative_func(lgamma_deriv).
                          series_func(lgamma_series).
                          conjugate_func(lgamma_conjugate).
                          latex_name("\\log \\Gamma"));
 
 
// true Gamma function
 
static ex tgamma_evalf(const ex & x)
{
    if (is_exactly_a<numeric>(x)) {
        try {
            return tgamma(ex_to<numeric>(x));
        } catch (const dunno &e) { }
    }
    
    return tgamma(x).hold();
}
 
 
static ex tgamma_eval(const ex & x)
{
    if (x.info(info_flags::numeric)) {
        // trap integer arguments:
        const numeric two_x = (*_num2_p)*ex_to<numeric>(x);
        if (two_x.is_even()) {
            // tgamma(n) -> (n-1)! for postitive n
            if (two_x.is_positive()) {
                return factorial(ex_to<numeric>(x).sub(*_num1_p));
            } else {
                throw (pole_error("tgamma_eval(): simple pole",1));
            }
        }
        // trap half integer arguments:
        if (two_x.is_integer()) {
            // trap positive x==(n+1/2)
            // tgamma(n+1/2) -> Pi^(1/2)*(1*3*..*(2*n-1))/(2^n)
            if (two_x.is_positive()) {
                const numeric n = ex_to<numeric>(x).sub(*_num1_2_p);
                return (doublefactorial(n.mul(*_num2_p).sub(*_num1_p)).div(pow(*_num2_p,n))) * sqrt(Pi);
            } else {
                // trap negative x==(-n+1/2)
                // tgamma(-n+1/2) -> Pi^(1/2)*(-2)^n/(1*3*..*(2*n-1))
                const numeric n = abs(ex_to<numeric>(x).sub(*_num1_2_p));
                return (pow(*_num_2_p, n).div(doublefactorial(n.mul(*_num2_p).sub(*_num1_p))))*sqrt(Pi);
            }
        }
        if (!ex_to<numeric>(x).is_rational())
            return tgamma(ex_to<numeric>(x));
    }
    
    return tgamma(x).hold();
}
 
 
static ex tgamma_deriv(const ex & x, unsigned deriv_param)
{
    GINAC_ASSERT(deriv_param==0);
    
    // d/dx  tgamma(x) -> psi(x)*tgamma(x)
    return psi(x)*tgamma(x);
}
 
 
static ex tgamma_series(const ex & arg,
                        const relational & rel,
                        int order,
                        unsigned options)
{
    // method:
    // Taylor series where there is no pole falls back to psi function
    // evaluation.
    // On a pole at -m use the recurrence relation
    //   tgamma(x) == tgamma(x+1) / x
    // from which follows
    //   series(tgamma(x),x==-m,order) ==
    //   series(tgamma(x+m+1)/(x*(x+1)*...*(x+m)),x==-m,order);
    const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
    if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive))
        throw do_taylor();  // caught by function::series()
    // if we got here we have to care for a simple pole at -m:
    const numeric m = -ex_to<numeric>(arg_pt);
    ex ser_denom = _ex1;
    for (numeric p; p<=m; ++p)
        ser_denom *= arg+p;
    return (tgamma(arg+m+_ex1)/ser_denom).series(rel, order, options);
}
 
 
static ex tgamma_conjugate(const ex & x)
{
    // conjugate(tgamma(x))==tgamma(conjugate(x))
    return tgamma(x.conjugate());
}
 
 
REGISTER_FUNCTION(tgamma, eval_func(tgamma_eval).
                          evalf_func(tgamma_evalf).
                          derivative_func(tgamma_deriv).
                          series_func(tgamma_series).
                          conjugate_func(tgamma_conjugate).
                          latex_name("\\Gamma"));
 
 
// beta-function
 
static ex beta_evalf(const ex & x, const ex & y)
{
    if (is_exactly_a<numeric>(x) && is_exactly_a<numeric>(y)) {
        try {
            return exp(lgamma(ex_to<numeric>(x))+lgamma(ex_to<numeric>(y))-lgamma(ex_to<numeric>(x+y)));
        } catch (const dunno &e) { }
    }
    
    return beta(x,y).hold();
}
 
 
static ex beta_eval(const ex & x, const ex & y)
{
    if (x.is_equal(_ex1))
        return 1/y;
    if (y.is_equal(_ex1))
        return 1/x;
    if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) {
        // treat all problematic x and y that may not be passed into tgamma,
        // because they would throw there although beta(x,y) is well-defined
        // using the formula beta(x,y) == (-1)^y * beta(1-x-y, y)
        const numeric &nx = ex_to<numeric>(x);
        const numeric &ny = ex_to<numeric>(y);
        if (nx.is_real() && nx.is_integer() &&
            ny.is_real() && ny.is_integer()) {
            if (nx.is_negative()) {
                if (nx<=-ny)
                    return pow(*_num_1_p, ny)*beta(1-x-y, y);
                else
                    throw (pole_error("beta_eval(): simple pole",1));
            }
            if (ny.is_negative()) {
                if (ny<=-nx)
                    return pow(*_num_1_p, nx)*beta(1-y-x, x);
                else
                    throw (pole_error("beta_eval(): simple pole",1));
            }
            return tgamma(x)*tgamma(y)/tgamma(x+y);
        }
        // no problem in numerator, but denominator has pole:
        if ((nx+ny).is_real() &&
            (nx+ny).is_integer() &&
           !(nx+ny).is_positive())
             return _ex0;
        if (!ex_to<numeric>(x).is_rational() || !ex_to<numeric>(x).is_rational())
            return evalf(beta(x, y).hold());
    }
    
    return beta(x,y).hold();
}
 
 
static ex beta_deriv(const ex & x, const ex & y, unsigned deriv_param)
{
    GINAC_ASSERT(deriv_param<2);
    ex retval;
    
    // d/dx beta(x,y) -> (psi(x)-psi(x+y)) * beta(x,y)
    if (deriv_param==0)
        retval = (psi(x)-psi(x+y))*beta(x,y);
    // d/dy beta(x,y) -> (psi(y)-psi(x+y)) * beta(x,y)
    if (deriv_param==1)
        retval = (psi(y)-psi(x+y))*beta(x,y);
    return retval;
}
 
 
static ex beta_series(const ex & arg1,
                      const ex & arg2,
                      const relational & rel,
                      int order,
                      unsigned options)
{
    // method:
    // Taylor series where there is no pole of one of the tgamma functions
    // falls back to beta function evaluation.  Otherwise, fall back to
    // tgamma series directly.
    const ex arg1_pt = arg1.subs(rel, subs_options::no_pattern);
    const ex arg2_pt = arg2.subs(rel, subs_options::no_pattern);
    GINAC_ASSERT(is_a<symbol>(rel.lhs()));
    const symbol &s = ex_to<symbol>(rel.lhs());
    ex arg1_ser, arg2_ser, arg1arg2_ser;
    if ((!arg1_pt.info(info_flags::integer) || arg1_pt.info(info_flags::positive)) &&
        (!arg2_pt.info(info_flags::integer) || arg2_pt.info(info_flags::positive)))
        throw do_taylor();  // caught by function::series()
    // trap the case where arg1 is on a pole:
    if (arg1.info(info_flags::integer) && !arg1.info(info_flags::positive))
        arg1_ser = tgamma(arg1+s);
    else
        arg1_ser = tgamma(arg1);
    // trap the case where arg2 is on a pole:
    if (arg2.info(info_flags::integer) && !arg2.info(info_flags::positive))
        arg2_ser = tgamma(arg2+s);
    else
        arg2_ser = tgamma(arg2);
    // trap the case where arg1+arg2 is on a pole:
    if ((arg1+arg2).info(info_flags::integer) && !(arg1+arg2).info(info_flags::positive))
        arg1arg2_ser = tgamma(arg2+arg1+s);
    else
        arg1arg2_ser = tgamma(arg2+arg1);
    // compose the result (expanding all the terms):
    return (arg1_ser*arg2_ser/arg1arg2_ser).series(rel, order, options).expand();
}
 
 
REGISTER_FUNCTION(beta, eval_func(beta_eval).
                        evalf_func(beta_evalf).
                        derivative_func(beta_deriv).
                        series_func(beta_series).
                        latex_name("\\mathrm{B}").
                        set_symmetry(sy_symm(0, 1)));
 
 
// Psi-function (aka digamma-function)
 
static ex psi1_evalf(const ex & x)
{
    if (is_exactly_a<numeric>(x)) {
        try {
            return psi(ex_to<numeric>(x));
        } catch (const dunno &e) { }
    }
    
    return psi(x).hold();
}
 
static ex psi1_eval(const ex & x)
{
    if (x.info(info_flags::numeric)) {
        const numeric &nx = ex_to<numeric>(x);
        if (nx.is_integer()) {
            // integer case 
            if (nx.is_positive()) {
                // psi(n) -> 1 + 1/2 +...+ 1/(n-1) - Euler
                numeric rat = 0;
                for (numeric i(nx+(*_num_1_p)); i>0; --i)
                    rat += i.inverse();
                return rat-Euler;
            } else {
                // for non-positive integers there is a pole:
                throw (pole_error("psi_eval(): simple pole",1));
            }
        }
        if (((*_num2_p)*nx).is_integer()) {
            // half integer case
            if (nx.is_positive()) {
                // psi((2m+1)/2) -> 2/(2m+1) + 2/2m +...+ 2/1 - Euler - 2log(2)
                numeric rat = 0;
                for (numeric i = (nx+(*_num_1_p))*(*_num2_p); i>0; i-=(*_num2_p))
                    rat += (*_num2_p)*i.inverse();
                return rat-Euler-_ex2*log(_ex2);
            } else {
                // use the recurrence relation
                //   psi(-m-1/2) == psi(-m-1/2+1) - 1 / (-m-1/2)
                // to relate psi(-m-1/2) to psi(1/2):
                //   psi(-m-1/2) == psi(1/2) + r
                // where r == ((-1/2)^(-1) + ... + (-m-1/2)^(-1))
                numeric recur = 0;
                for (numeric p = nx; p<0; ++p)
                    recur -= pow(p, *_num_1_p);
                return recur+psi(_ex1_2);
            }
        }
        //  psi1_evalf should be called here once it becomes available
    }
    
    return psi(x).hold();
}
 
static ex psi1_deriv(const ex & x, unsigned deriv_param)
{
    GINAC_ASSERT(deriv_param==0);
    
    // d/dx psi(x) -> psi(1,x)
    return psi(_ex1, x);
}
 
static ex psi1_series(const ex & arg,
                      const relational & rel,
                      int order,
                      unsigned options)
{
    // method:
    // Taylor series where there is no pole falls back to polygamma function
    // evaluation.
    // On a pole at -m use the recurrence relation
    //   psi(x) == psi(x+1) - 1/z
    // from which follows
    //   series(psi(x),x==-m,order) ==
    //   series(psi(x+m+1) - 1/x - 1/(x+1) - 1/(x+m)),x==-m,order);
    const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
    if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive))
        throw do_taylor();  // caught by function::series()
    // if we got here we have to care for a simple pole at -m:
    const numeric m = -ex_to<numeric>(arg_pt);
    ex recur;
    for (numeric p; p<=m; ++p)
        recur += power(arg+p,_ex_1);
    return (psi(arg+m+_ex1)-recur).series(rel, order, options);
}
 
unsigned psi1_SERIAL::serial =
    function::register_new(function_options("psi", 1).
                           eval_func(psi1_eval).
                           evalf_func(psi1_evalf).
                           derivative_func(psi1_deriv).
                           series_func(psi1_series).
                           latex_name("\\psi").
                           overloaded(2));
 
// Psi-functions (aka polygamma-functions)  psi(0,x)==psi(x)
 
static ex psi2_evalf(const ex & n, const ex & x)
{
    if (is_exactly_a<numeric>(n) && is_exactly_a<numeric>(x)) {
        try {
            return psi(ex_to<numeric>(n),ex_to<numeric>(x));
        } catch (const dunno &e) { }
    }
    
    return psi(n,x).hold();
}
 
static ex psi2_eval(const ex & n, const ex & x)
{
    // psi(0,x) -> psi(x)
    if (n.is_zero())
        return psi(x);
    // psi(-1,x) -> log(tgamma(x))
    if (n.is_equal(_ex_1))
        return log(tgamma(x));
    if (n.info(info_flags::numeric) && n.info(info_flags::posint) &&
        x.info(info_flags::numeric)) {
        const numeric &nn = ex_to<numeric>(n);
        const numeric &nx = ex_to<numeric>(x);
        if (nx.is_integer()) {
            // integer case 
            if (nx.is_equal(*_num1_p))
                // use psi(n,1) == (-)^(n+1) * n! * zeta(n+1)
                return pow(*_num_1_p,nn+(*_num1_p))*factorial(nn)*zeta(ex(nn+(*_num1_p)));
            if (nx.is_positive()) {
                // use the recurrence relation
                //   psi(n,m) == psi(n,m+1) - (-)^n * n! / m^(n+1)
                // to relate psi(n,m) to psi(n,1):
                //   psi(n,m) == psi(n,1) + r
                // where r == (-)^n * n! * (1^(-n-1) + ... + (m-1)^(-n-1))
                numeric recur = 0;
                for (numeric p = 1; p<nx; ++p)
                    recur += pow(p, -nn+(*_num_1_p));
                recur *= factorial(nn)*pow((*_num_1_p), nn);
                return recur+psi(n,_ex1);
            } else {
                // for non-positive integers there is a pole:
                throw (pole_error("psi2_eval(): pole",1));
            }
        }
        if (((*_num2_p)*nx).is_integer()) {
            // half integer case
            if (nx.is_equal(*_num1_2_p))
                // use psi(n,1/2) == (-)^(n+1) * n! * (2^(n+1)-1) * zeta(n+1)
                return pow(*_num_1_p,nn+(*_num1_p))*factorial(nn)*(pow(*_num2_p,nn+(*_num1_p)) + (*_num_1_p))*zeta(ex(nn+(*_num1_p)));
            if (nx.is_positive()) {
                const numeric m = nx - (*_num1_2_p);
                // use the multiplication formula
                //   psi(n,2*m) == (psi(n,m) + psi(n,m+1/2)) / 2^(n+1)
                // to revert to positive integer case
                return psi(n,(*_num2_p)*m)*pow((*_num2_p),nn+(*_num1_p))-psi(n,m);
            } else {
                // use the recurrence relation
                //   psi(n,-m-1/2) == psi(n,-m-1/2+1) - (-)^n * n! / (-m-1/2)^(n+1)
                // to relate psi(n,-m-1/2) to psi(n,1/2):
                //   psi(n,-m-1/2) == psi(n,1/2) + r
                // where r == (-)^(n+1) * n! * ((-1/2)^(-n-1) + ... + (-m-1/2)^(-n-1))
                numeric recur = 0;
                for (numeric p = nx; p<0; ++p)
                    recur += pow(p, -nn+(*_num_1_p));
                recur *= factorial(nn)*pow(*_num_1_p, nn+(*_num_1_p));
                return recur+psi(n,_ex1_2);
            }
        }
        //  psi2_evalf should be called here once it becomes available
    }
    
    return psi(n, x).hold();
}    
 
static ex psi2_deriv(const ex & n, const ex & x, unsigned deriv_param)
{
    GINAC_ASSERT(deriv_param<2);
    
    if (deriv_param==0) {
        // d/dn psi(n,x)
        throw(std::logic_error("cannot diff psi(n,x) with respect to n"));
    }
    // d/dx psi(n,x) -> psi(n+1,x)
    return psi(n+_ex1, x);
}
 
static ex psi2_series(const ex & n,
                      const ex & arg,
                      const relational & rel,
                      int order,
                      unsigned options)
{
    // method:
    // Taylor series where there is no pole falls back to polygamma function
    // evaluation.
    // On a pole at -m use the recurrence relation
    //   psi(n,x) == psi(n,x+1) - (-)^n * n! / x^(n+1)
    // from which follows
    //   series(psi(x),x==-m,order) == 
    //   series(psi(x+m+1) - (-1)^n * n! * ((x)^(-n-1) + (x+1)^(-n-1) + ...
    //                                      ... + (x+m)^(-n-1))),x==-m,order);
    const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
    if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive))
        throw do_taylor();  // caught by function::series()
    // if we got here we have to care for a pole of order n+1 at -m:
    const numeric m = -ex_to<numeric>(arg_pt);
    ex recur;
    for (numeric p; p<=m; ++p)
        recur += power(arg+p,-n+_ex_1);
    recur *= factorial(n)*power(_ex_1,n);
    return (psi(n, arg+m+_ex1)-recur).series(rel, order, options);
}
 
unsigned psi2_SERIAL::serial =
    function::register_new(function_options("psi", 2).
                           eval_func(psi2_eval).
                           evalf_func(psi2_evalf).
                           derivative_func(psi2_deriv).
                           series_func(psi2_series).
                           latex_name("\\psi").
                           overloaded(2));
 
 
} // namespace GiNaC