inifcns.cpp

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/*
 *  GiNaC Copyright (C) 1999-2023 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 "ex.h"
#include "constant.h"
#include "lst.h"
#include "fderivative.h"
#include "matrix.h"
#include "mul.h"
#include "power.h"
#include "operators.h"
#include "relational.h"
#include "pseries.h"
#include "symbol.h"
#include "symmetry.h"
#include "utils.h"
 
#include <stdexcept>
#include <vector>
 
namespace GiNaC {
 
// complex conjugate
 
static ex conjugate_evalf(const ex & arg)
{
    if (is_exactly_a<numeric>(arg)) {
        return ex_to<numeric>(arg).conjugate();
    }
    return conjugate_function(arg).hold();
}
 
static ex conjugate_eval(const ex & arg)
{
    return arg.conjugate();
}
 
static void conjugate_print_latex(const ex & arg, const print_context & c)
{
    c.s << "\\bar{"; arg.print(c); c.s << "}";
}
 
static ex conjugate_conjugate(const ex & arg)
{
    return arg;
}
 
// If x is real then U.diff(x)-I*V.diff(x) represents both conjugate(U+I*V).diff(x) 
// and conjugate((U+I*V).diff(x))
static ex conjugate_expl_derivative(const ex & arg, const symbol & s)
{
    if (s.info(info_flags::real))
        return conjugate(arg.diff(s));
    else {
        exvector vec_arg;
        vec_arg.push_back(arg);
        return fderivative(ex_to<function>(conjugate(arg)).get_serial(),0,vec_arg).hold()*arg.diff(s);
    }
}
 
static ex conjugate_real_part(const ex & arg)
{
    return arg.real_part();
}
 
static ex conjugate_imag_part(const ex & arg)
{
    return -arg.imag_part();
}
 
static bool func_arg_info(const ex & arg, unsigned inf)
{
    // for some functions we can return the info() of its argument
    // (think of conjugate())
    switch (inf) {
        case info_flags::polynomial:
        case info_flags::integer_polynomial:
        case info_flags::cinteger_polynomial:
        case info_flags::rational_polynomial:
        case info_flags::real:
        case info_flags::rational:
        case info_flags::integer:
        case info_flags::crational:
        case info_flags::cinteger:
        case info_flags::even:
        case info_flags::odd:
        case info_flags::prime:
        case info_flags::crational_polynomial:
        case info_flags::rational_function:
        case info_flags::positive:
        case info_flags::negative:
        case info_flags::nonnegative:
        case info_flags::posint:
        case info_flags::negint:
        case info_flags::nonnegint:
        case info_flags::has_indices:
            return arg.info(inf);
    }
    return false;
}
 
static bool conjugate_info(const ex & arg, unsigned inf)
{
    return func_arg_info(arg, inf);
}
 
REGISTER_FUNCTION(conjugate_function, eval_func(conjugate_eval).
                                      evalf_func(conjugate_evalf).
                                      expl_derivative_func(conjugate_expl_derivative).
                                      info_func(conjugate_info).
                                      print_func<print_latex>(conjugate_print_latex).
                                      conjugate_func(conjugate_conjugate).
                                      real_part_func(conjugate_real_part).
                                      imag_part_func(conjugate_imag_part).
                                      set_name("conjugate","conjugate"));
 
// real part
 
static ex real_part_evalf(const ex & arg)
{
    if (is_exactly_a<numeric>(arg)) {
        return ex_to<numeric>(arg).real();
    }
    return real_part_function(arg).hold();
}
 
static ex real_part_eval(const ex & arg)
{
    return arg.real_part();
}
 
static void real_part_print_latex(const ex & arg, const print_context & c)
{
    c.s << "\\Re"; arg.print(c); c.s << "";
}
 
static ex real_part_conjugate(const ex & arg)
{
    return real_part_function(arg).hold();
}
 
static ex real_part_real_part(const ex & arg)
{
    return real_part_function(arg).hold();
}
 
static ex real_part_imag_part(const ex & arg)
{
    return 0;
}
 
// If x is real then Re(e).diff(x) is equal to Re(e.diff(x)) 
static ex real_part_expl_derivative(const ex & arg, const symbol & s)
{
    if (s.info(info_flags::real))
        return real_part_function(arg.diff(s));
    else {
        exvector vec_arg;
        vec_arg.push_back(arg);
        return fderivative(ex_to<function>(real_part(arg)).get_serial(),0,vec_arg).hold()*arg.diff(s);
    }
}
 
REGISTER_FUNCTION(real_part_function, eval_func(real_part_eval).
                                      evalf_func(real_part_evalf).
                                      expl_derivative_func(real_part_expl_derivative).
                                      print_func<print_latex>(real_part_print_latex).
                                      conjugate_func(real_part_conjugate).
                                      real_part_func(real_part_real_part).
                                      imag_part_func(real_part_imag_part).
                                      set_name("real_part","real_part"));
 
// imag part
 
static ex imag_part_evalf(const ex & arg)
{
    if (is_exactly_a<numeric>(arg)) {
        return ex_to<numeric>(arg).imag();
    }
    return imag_part_function(arg).hold();
}
 
static ex imag_part_eval(const ex & arg)
{
    return arg.imag_part();
}
 
static void imag_part_print_latex(const ex & arg, const print_context & c)
{
    c.s << "\\Im"; arg.print(c); c.s << "";
}
 
static ex imag_part_conjugate(const ex & arg)
{
    return imag_part_function(arg).hold();
}
 
static ex imag_part_real_part(const ex & arg)
{
    return imag_part_function(arg).hold();
}
 
static ex imag_part_imag_part(const ex & arg)
{
    return 0;
}
 
// If x is real then Im(e).diff(x) is equal to Im(e.diff(x)) 
static ex imag_part_expl_derivative(const ex & arg, const symbol & s)
{
    if (s.info(info_flags::real))
        return imag_part_function(arg.diff(s));
    else {
        exvector vec_arg;
        vec_arg.push_back(arg);
        return fderivative(ex_to<function>(imag_part(arg)).get_serial(),0,vec_arg).hold()*arg.diff(s);
    }
}
 
REGISTER_FUNCTION(imag_part_function, eval_func(imag_part_eval).
                                      evalf_func(imag_part_evalf).
                                      expl_derivative_func(imag_part_expl_derivative).
                                      print_func<print_latex>(imag_part_print_latex).
                                      conjugate_func(imag_part_conjugate).
                                      real_part_func(imag_part_real_part).
                                      imag_part_func(imag_part_imag_part).
                                      set_name("imag_part","imag_part"));
 
// absolute value
 
static ex abs_evalf(const ex & arg)
{
    if (is_exactly_a<numeric>(arg))
        return abs(ex_to<numeric>(arg));
    
    return abs(arg).hold();
}
 
static ex abs_eval(const ex & arg)
{
    if (is_exactly_a<numeric>(arg))
        return abs(ex_to<numeric>(arg));
 
    if (arg.info(info_flags::nonnegative))
        return arg;
 
    if (arg.info(info_flags::negative) || (-arg).info(info_flags::nonnegative))
        return -arg;
 
    if (is_ex_the_function(arg, abs))
        return arg;
 
    if (is_ex_the_function(arg, exp))
        return exp(arg.op(0).real_part());
 
    if (is_exactly_a<power>(arg)) {
        const ex& base = arg.op(0);
        const ex& exponent = arg.op(1);
        if (base.info(info_flags::positive) || exponent.info(info_flags::real))
            return pow(abs(base), exponent.real_part());
    }
 
    if (is_ex_the_function(arg, conjugate_function))
        return abs(arg.op(0));
 
    if (is_ex_the_function(arg, step))
        return arg;
 
    return abs(arg).hold();
}
 
static ex abs_expand(const ex & arg, unsigned options)
{
    if ((options & expand_options::expand_transcendental)
        && is_exactly_a<mul>(arg)) {
        exvector prodseq;
        prodseq.reserve(arg.nops());
        for (const_iterator i = arg.begin(); i != arg.end(); ++i) {
            if (options & expand_options::expand_function_args)
                prodseq.push_back(abs(i->expand(options)));
            else
                prodseq.push_back(abs(*i));
        }
        return dynallocate<mul>(prodseq).setflag(status_flags::expanded);
    }
 
    if (options & expand_options::expand_function_args)
        return abs(arg.expand(options)).hold();
    else
        return abs(arg).hold();
}
 
static ex abs_expl_derivative(const ex & arg, const symbol & s)
{
    ex diff_arg = arg.diff(s);
    return (diff_arg*arg.conjugate()+arg*diff_arg.conjugate())/2/abs(arg);
}
 
static void abs_print_latex(const ex & arg, const print_context & c)
{
    c.s << "{|"; arg.print(c); c.s << "|}";
}
 
static void abs_print_csrc_float(const ex & arg, const print_context & c)
{
    c.s << "fabs("; arg.print(c); c.s << ")";
}
 
static ex abs_conjugate(const ex & arg)
{
    return abs(arg).hold();
}
 
static ex abs_real_part(const ex & arg)
{
    return abs(arg).hold();
}
 
static ex abs_imag_part(const ex& arg)
{
    return 0;
}
 
static ex abs_power(const ex & arg, const ex & exp)
{
    if ((is_a<numeric>(exp) && ex_to<numeric>(exp).is_even()) || exp.info(info_flags::even)) {
        if (arg.info(info_flags::real) || arg.is_equal(arg.conjugate()))
            return pow(arg, exp);
        else
            return pow(arg, exp/2) * pow(arg.conjugate(), exp/2);
    } else
        return power(abs(arg), exp).hold();
}
 
bool abs_info(const ex & arg, unsigned inf)
{
    switch (inf) {
        case info_flags::integer:
        case info_flags::even:
        case info_flags::odd:
        case info_flags::prime:
            return arg.info(inf);
        case info_flags::nonnegint:
            return arg.info(info_flags::integer);
        case info_flags::nonnegative:
        case info_flags::real:
            return true;
        case info_flags::negative:
            return false;
        case info_flags::positive:
            return arg.info(info_flags::positive) || arg.info(info_flags::negative);
        case info_flags::has_indices: {
            if (arg.info(info_flags::has_indices))
                return true;
            else
                return false;
        }
    }
    return false;
}
 
REGISTER_FUNCTION(abs, eval_func(abs_eval).
                       evalf_func(abs_evalf).
                       expand_func(abs_expand).
                       expl_derivative_func(abs_expl_derivative).
                       info_func(abs_info).
                       print_func<print_latex>(abs_print_latex).
                       print_func<print_csrc_float>(abs_print_csrc_float).
                       print_func<print_csrc_double>(abs_print_csrc_float).
                       conjugate_func(abs_conjugate).
                       real_part_func(abs_real_part).
                       imag_part_func(abs_imag_part).
                       power_func(abs_power));
 
// Step function
 
static ex step_evalf(const ex & arg)
{
    if (is_exactly_a<numeric>(arg))
        return step(ex_to<numeric>(arg));
    
    return step(arg).hold();
}
 
static ex step_eval(const ex & arg)
{
    if (is_exactly_a<numeric>(arg))
        return step(ex_to<numeric>(arg));
    
    else if (is_exactly_a<mul>(arg) &&
             is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
        numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
        if (oc.is_real()) {
            if (oc > 0)
                // step(42*x) -> step(x)
                return step(arg/oc).hold();
            else
                // step(-42*x) -> step(-x)
                return step(-arg/oc).hold();
        }
        if (oc.real().is_zero()) {
            if (oc.imag() > 0)
                // step(42*I*x) -> step(I*x)
                return step(I*arg/oc).hold();
            else
                // step(-42*I*x) -> step(-I*x)
                return step(-I*arg/oc).hold();
        }
    }
    
    return step(arg).hold();
}
 
static ex step_series(const ex & arg,
                      const relational & rel,
                      int order,
                      unsigned options)
{
    const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
    if (arg_pt.info(info_flags::numeric)
        && ex_to<numeric>(arg_pt).real().is_zero()
        && !(options & series_options::suppress_branchcut))
        throw (std::domain_error("step_series(): on imaginary axis"));
    
    epvector seq { expair(step(arg_pt), _ex0) };
    return pseries(rel, std::move(seq));
}
 
static ex step_conjugate(const ex& arg)
{
    return step(arg).hold();
}
 
static ex step_real_part(const ex& arg)
{
    return step(arg).hold();
}
 
static ex step_imag_part(const ex& arg)
{
    return 0;
}
 
REGISTER_FUNCTION(step, eval_func(step_eval).
                        evalf_func(step_evalf).
                        series_func(step_series).
                        conjugate_func(step_conjugate).
                        real_part_func(step_real_part).
                        imag_part_func(step_imag_part));
 
// Complex sign
 
static ex csgn_evalf(const ex & arg)
{
    if (is_exactly_a<numeric>(arg))
        return csgn(ex_to<numeric>(arg));
    
    return csgn(arg).hold();
}
 
static ex csgn_eval(const ex & arg)
{
    if (is_exactly_a<numeric>(arg))
        return csgn(ex_to<numeric>(arg));
    
    else if (is_exactly_a<mul>(arg) &&
             is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
        numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
        if (oc.is_real()) {
            if (oc > 0)
                // csgn(42*x) -> csgn(x)
                return csgn(arg/oc).hold();
            else
                // csgn(-42*x) -> -csgn(x)
                return -csgn(arg/oc).hold();
        }
        if (oc.real().is_zero()) {
            if (oc.imag() > 0)
                // csgn(42*I*x) -> csgn(I*x)
                return csgn(I*arg/oc).hold();
            else
                // csgn(-42*I*x) -> -csgn(I*x)
                return -csgn(I*arg/oc).hold();
        }
    }
    
    return csgn(arg).hold();
}
 
static ex csgn_series(const ex & arg,
                      const relational & rel,
                      int order,
                      unsigned options)
{
    const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
    if (arg_pt.info(info_flags::numeric)
        && ex_to<numeric>(arg_pt).real().is_zero()
        && !(options & series_options::suppress_branchcut))
        throw (std::domain_error("csgn_series(): on imaginary axis"));
    
    epvector seq { expair(csgn(arg_pt), _ex0) };
    return pseries(rel, std::move(seq));
}
 
static ex csgn_conjugate(const ex& arg)
{
    return csgn(arg).hold();
}
 
static ex csgn_real_part(const ex& arg)
{
    return csgn(arg).hold();
}
 
static ex csgn_imag_part(const ex& arg)
{
    return 0;
}
 
static ex csgn_power(const ex & arg, const ex & exp)
{
    if (is_a<numeric>(exp) && exp.info(info_flags::positive) && ex_to<numeric>(exp).is_integer()) {
        if (ex_to<numeric>(exp).is_odd())
            return csgn(arg).hold();
        else
            return power(csgn(arg), _ex2).hold();
    } else
        return power(csgn(arg), exp).hold();
}
 
 
REGISTER_FUNCTION(csgn, eval_func(csgn_eval).
                        evalf_func(csgn_evalf).
                        series_func(csgn_series).
                        conjugate_func(csgn_conjugate).
                        real_part_func(csgn_real_part).
                        imag_part_func(csgn_imag_part).
                        power_func(csgn_power));
 
 
// Eta function: eta(x,y) == log(x*y) - log(x) - log(y).
// This function is closely related to the unwinding number K, sometimes found
// in modern literature: K(z) == (z-log(exp(z)))/(2*Pi*I).
 
static ex eta_evalf(const ex &x, const ex &y)
{
    // It seems like we basically have to replicate the eval function here,
    // since the expression might not be fully evaluated yet.
    if (x.info(info_flags::positive) || y.info(info_flags::positive))
        return _ex0;
 
    if (x.info(info_flags::numeric) &&  y.info(info_flags::numeric)) {
        const numeric nx = ex_to<numeric>(x);
        const numeric ny = ex_to<numeric>(y);
        const numeric nxy = ex_to<numeric>(x*y);
        int cut = 0;
        if (nx.is_real() && nx.is_negative())
            cut -= 4;
        if (ny.is_real() && ny.is_negative())
            cut -= 4;
        if (nxy.is_real() && nxy.is_negative())
            cut += 4;
        return evalf(I/4*Pi)*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
                              (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
    }
 
    return eta(x,y).hold();
}
 
static ex eta_eval(const ex &x, const ex &y)
{
    // trivial:  eta(x,c) -> 0  if c is real and positive
    if (x.info(info_flags::positive) || y.info(info_flags::positive))
        return _ex0;
 
    if (x.info(info_flags::numeric) &&  y.info(info_flags::numeric)) {
        // don't call eta_evalf here because it would call Pi.evalf()!
        const numeric nx = ex_to<numeric>(x);
        const numeric ny = ex_to<numeric>(y);
        const numeric nxy = ex_to<numeric>(x*y);
        int cut = 0;
        if (nx.is_real() && nx.is_negative())
            cut -= 4;
        if (ny.is_real() && ny.is_negative())
            cut -= 4;
        if (nxy.is_real() && nxy.is_negative())
            cut += 4;
        return (I/4)*Pi*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
                         (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
    }
    
    return eta(x,y).hold();
}
 
static ex eta_series(const ex & x, const ex & y,
                     const relational & rel,
                     int order,
                     unsigned options)
{
    const ex x_pt = x.subs(rel, subs_options::no_pattern);
    const ex y_pt = y.subs(rel, subs_options::no_pattern);
    if ((x_pt.info(info_flags::numeric) && x_pt.info(info_flags::negative)) ||
        (y_pt.info(info_flags::numeric) && y_pt.info(info_flags::negative)) ||
        ((x_pt*y_pt).info(info_flags::numeric) && (x_pt*y_pt).info(info_flags::negative)))
            throw (std::domain_error("eta_series(): on discontinuity"));
    epvector seq { expair(eta(x_pt,y_pt), _ex0) };
    return pseries(rel, std::move(seq));
}
 
static ex eta_conjugate(const ex & x, const ex & y)
{
    return -eta(x, y).hold();
}
 
static ex eta_real_part(const ex & x, const ex & y)
{
    return 0;
}
 
static ex eta_imag_part(const ex & x, const ex & y)
{
    return -I*eta(x, y).hold();
}
 
REGISTER_FUNCTION(eta, eval_func(eta_eval).
                       evalf_func(eta_evalf).
                       series_func(eta_series).
                       latex_name("\\eta").
                       set_symmetry(sy_symm(0, 1)).
                       conjugate_func(eta_conjugate).
                       real_part_func(eta_real_part).
                       imag_part_func(eta_imag_part));
 
 
// dilogarithm
 
static ex Li2_evalf(const ex & x)
{
    if (is_exactly_a<numeric>(x))
        return Li2(ex_to<numeric>(x));
    
    return Li2(x).hold();
}
 
static ex Li2_eval(const ex & x)
{
    if (x.info(info_flags::numeric)) {
        // Li2(0) -> 0
        if (x.is_zero())
            return _ex0;
        // Li2(1) -> Pi^2/6
        if (x.is_equal(_ex1))
            return power(Pi,_ex2)/_ex6;
        // Li2(1/2) -> Pi^2/12 - log(2)^2/2
        if (x.is_equal(_ex1_2))
            return power(Pi,_ex2)/_ex12 + power(log(_ex2),_ex2)*_ex_1_2;
        // Li2(-1) -> -Pi^2/12
        if (x.is_equal(_ex_1))
            return -power(Pi,_ex2)/_ex12;
        // Li2(I) -> -Pi^2/48+Catalan*I
        if (x.is_equal(I))
            return power(Pi,_ex2)/_ex_48 + Catalan*I;
        // Li2(-I) -> -Pi^2/48-Catalan*I
        if (x.is_equal(-I))
            return power(Pi,_ex2)/_ex_48 - Catalan*I;
        // Li2(float)
        if (!x.info(info_flags::crational))
            return Li2(ex_to<numeric>(x));
    }
    
    return Li2(x).hold();
}
 
static ex Li2_deriv(const ex & x, unsigned deriv_param)
{
    GINAC_ASSERT(deriv_param==0);
    
    // d/dx Li2(x) -> -log(1-x)/x
    return -log(_ex1-x)/x;
}
 
static ex Li2_series(const ex &x, const relational &rel, int order, unsigned options)
{
    const ex x_pt = x.subs(rel, subs_options::no_pattern);
    if (x_pt.info(info_flags::numeric)) {
        // First special case: x==0 (derivatives have poles)
        if (x_pt.is_zero()) {
            // method:
            // The problem is that in d/dx Li2(x==0) == -log(1-x)/x we cannot 
            // simply substitute x==0.  The limit, however, exists: it is 1.
            // We also know all higher derivatives' limits:
            // (d/dx)^n Li2(x) == n!/n^2.
            // So the primitive series expansion is
            // Li2(x==0) == x + x^2/4 + x^3/9 + ...
            // and so on.
            // We first construct such a primitive series expansion manually in
            // a dummy symbol s and then insert the argument's series expansion
            // for s.  Reexpanding the resulting series returns the desired
            // result.
            const symbol s;
            ex ser;
            // manually construct the primitive expansion
            for (int i=1; i<order; ++i)
                ser += pow(s,i) / pow(numeric(i), *_num2_p);
            // substitute the argument's series expansion
            ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
            // maybe that was terminating, so add a proper order term
            epvector nseq { expair(Order(_ex1), order) };
            ser += pseries(rel, std::move(nseq));
            // reexpanding it will collapse the series again
            return ser.series(rel, order);
            // NB: Of course, this still does not allow us to compute anything
            // like sin(Li2(x)).series(x==0,2), since then this code here is
            // not reached and the derivative of sin(Li2(x)) doesn't allow the
            // substitution x==0.  Probably limits *are* needed for the general
            // cases.  In case L'Hospital's rule is implemented for limits and
            // basic::series() takes care of this, this whole block is probably
            // obsolete!
        }
        // second special case: x==1 (branch point)
        if (x_pt.is_equal(_ex1)) {
            // method:
            // construct series manually in a dummy symbol s
            const symbol s;
            ex ser = zeta(_ex2);
            // manually construct the primitive expansion
            for (int i=1; i<order; ++i)
                ser += pow(1-s,i) * (numeric(1,i)*(I*Pi+log(s-1)) - numeric(1,i*i));
            // substitute the argument's series expansion
            ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
            // maybe that was terminating, so add a proper order term
            epvector nseq { expair(Order(_ex1), order) };
            ser += pseries(rel, std::move(nseq));
            // reexpanding it will collapse the series again
            return ser.series(rel, order);
        }
        // third special case: x real, >=1 (branch cut)
        if (!(options & series_options::suppress_branchcut) &&
            ex_to<numeric>(x_pt).is_real() && ex_to<numeric>(x_pt)>1) {
            // method:
            // This is the branch cut: assemble the primitive series manually
            // and then add the corresponding complex step function.
            const symbol &s = ex_to<symbol>(rel.lhs());
            const ex point = rel.rhs();
            const symbol foo;
            epvector seq;
            // zeroth order term:
            seq.push_back(expair(Li2(x_pt), _ex0));
            // compute the intermediate terms:
            ex replarg = series(Li2(x), s==foo, order);
            for (size_t i=1; i<replarg.nops()-1; ++i)
                seq.push_back(expair((replarg.op(i)/power(s-foo,i)).series(foo==point,1,options).op(0).subs(foo==s, subs_options::no_pattern),i));
            // append an order term:
            seq.push_back(expair(Order(_ex1), replarg.nops()-1));
            return pseries(rel, std::move(seq));
        }
    }
    // all other cases should be safe, by now:
    throw do_taylor();  // caught by function::series()
}
 
static ex Li2_conjugate(const ex & x)
{
    // conjugate(Li2(x))==Li2(conjugate(x)) unless on the branch cuts which
    // run along the positive real axis beginning at 1.
    if (x.info(info_flags::negative)) {
        return Li2(x).hold();
    }
    if (is_exactly_a<numeric>(x) &&
        (!x.imag_part().is_zero() || x < *_num1_p)) {
        return Li2(x.conjugate());
    }
    return conjugate_function(Li2(x)).hold();
}
 
REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
                       evalf_func(Li2_evalf).
                       derivative_func(Li2_deriv).
                       series_func(Li2_series).
                       conjugate_func(Li2_conjugate).
                       latex_name("\\mathrm{Li}_2"));
 
// trilogarithm
 
static ex Li3_eval(const ex & x)
{
    if (x.is_zero())
        return x;
    return Li3(x).hold();
}
 
REGISTER_FUNCTION(Li3, eval_func(Li3_eval).
                       latex_name("\\mathrm{Li}_3"));
 
// Derivatives of Riemann's Zeta-function  zetaderiv(0,x)==zeta(x)
 
static ex zetaderiv_eval(const ex & n, const ex & x)
{
    if (n.info(info_flags::numeric)) {
        // zetaderiv(0,x) -> zeta(x)
        if (n.is_zero())
            return zeta(x).hold();
    }
    
    return zetaderiv(n, x).hold();
}
 
static ex zetaderiv_deriv(const ex & n, const ex & x, unsigned deriv_param)
{
    GINAC_ASSERT(deriv_param<2);
    
    if (deriv_param==0) {
        // d/dn zeta(n,x)
        throw(std::logic_error("cannot diff zetaderiv(n,x) with respect to n"));
    }
    // d/dx psi(n,x)
    return zetaderiv(n+1,x);
}
 
REGISTER_FUNCTION(zetaderiv, eval_func(zetaderiv_eval).
                             derivative_func(zetaderiv_deriv).
                             latex_name("\\zeta^\\prime"));
 
// factorial
 
static ex factorial_evalf(const ex & x)
{
    return factorial(x).hold();
}
 
static ex factorial_eval(const ex & x)
{
    if (is_exactly_a<numeric>(x))
        return factorial(ex_to<numeric>(x));
    else
        return factorial(x).hold();
}
 
static void factorial_print_dflt_latex(const ex & x, const print_context & c)
{
    if (is_exactly_a<symbol>(x) ||
        is_exactly_a<constant>(x) ||
        is_exactly_a<function>(x)) {
        x.print(c); c.s << "!";
    } else {
        c.s << "("; x.print(c); c.s << ")!";
    }
}
 
static ex factorial_conjugate(const ex & x)
{
    return factorial(x).hold();
}
 
static ex factorial_real_part(const ex & x)
{
    return factorial(x).hold();
}
 
static ex factorial_imag_part(const ex & x)
{
    return 0;
}
 
REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
                             evalf_func(factorial_evalf).
                             print_func<print_dflt>(factorial_print_dflt_latex).
                             print_func<print_latex>(factorial_print_dflt_latex).
                             conjugate_func(factorial_conjugate).
                             real_part_func(factorial_real_part).
                             imag_part_func(factorial_imag_part));
 
// binomial
 
static ex binomial_evalf(const ex & x, const ex & y)
{
    return binomial(x, y).hold();
}
 
static ex binomial_sym(const ex & x, const numeric & y)
{
    if (y.is_integer()) {
        if (y.is_nonneg_integer()) {
            const unsigned N = y.to_int();
            if (N == 0) return _ex1;
            if (N == 1) return x;
            ex t = x.expand();
            for (unsigned i = 2; i <= N; ++i)
                t = (t * (x + i - y - 1)).expand() / i;
            return t;
        } else
            return _ex0;
    }
 
    return binomial(x, y).hold();
}
 
static ex binomial_eval(const ex & x, const ex &y)
{
    if (is_exactly_a<numeric>(y)) {
        if (is_exactly_a<numeric>(x) && ex_to<numeric>(x).is_integer())
            return binomial(ex_to<numeric>(x), ex_to<numeric>(y));
        else
            return binomial_sym(x, ex_to<numeric>(y));
    } else
        return binomial(x, y).hold();
}
 
// At the moment the numeric evaluation of a binomial function always
// gives a real number, but if this would be implemented using the gamma
// function, also complex conjugation should be changed (or rather, deleted).
static ex binomial_conjugate(const ex & x, const ex & y)
{
    return binomial(x,y).hold();
}
 
static ex binomial_real_part(const ex & x, const ex & y)
{
    return binomial(x,y).hold();
}
 
static ex binomial_imag_part(const ex & x, const ex & y)
{
    return 0;
}
 
REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
                            evalf_func(binomial_evalf).
                            conjugate_func(binomial_conjugate).
                            real_part_func(binomial_real_part).
                            imag_part_func(binomial_imag_part));
 
// Order term function (for truncated power series)
 
static ex Order_eval(const ex & x)
{
    if (is_exactly_a<numeric>(x)) {
        // O(c) -> O(1) or 0
        if (!x.is_zero())
            return Order(_ex1).hold();
        else
            return _ex0;
    } else if (is_exactly_a<mul>(x)) {
        const mul &m = ex_to<mul>(x);
        // O(c*expr) -> O(expr)
        if (is_exactly_a<numeric>(m.op(m.nops() - 1)))
            return Order(x / m.op(m.nops() - 1)).hold();
    }
    return Order(x).hold();
}
 
static ex Order_series(const ex & x, const relational & r, int order, unsigned options)
{
    // Just wrap the function into a pseries object
    GINAC_ASSERT(is_a<symbol>(r.lhs()));
    const symbol &s = ex_to<symbol>(r.lhs());
    epvector new_seq { expair(Order(_ex1), numeric(std::min(x.ldegree(s), order))) };
    return pseries(r, std::move(new_seq));
}
 
static ex Order_conjugate(const ex & x)
{
    return Order(x).hold();
}
 
static ex Order_real_part(const ex & x)
{
    return Order(x).hold();
}
 
static ex Order_imag_part(const ex & x)
{
    if(x.info(info_flags::real))
        return 0;
    return Order(x).hold();
}
 
static ex Order_power(const ex & x, const ex & e)
{
    // Order(x)^e -> Order(x^e) for positive integer e
    if (is_exactly_a<numeric>(e) && e.info(info_flags::posint))
        return Order(pow(x, e));
    // NB: For negative exponents, the above could be wrong.
    // This is because series() produces Order(x^n) to denote the order where
    // it gave up. So, Order(x^n) can also be an x^(n+1) term if the x^n term
    // vanishes. In this situation, 1/Order(x^n) can also be a x^(-n-1) term.
    // Transforming it to Order(x^-n) would miss that.
 
    return power(Order(x), e).hold();
}
 
static ex Order_expl_derivative(const ex & arg, const symbol & s)
{
    return Order(arg.diff(s));
}
 
REGISTER_FUNCTION(Order, eval_func(Order_eval).
                         series_func(Order_series).
                         latex_name("\\mathcal{O}").
                         expl_derivative_func(Order_expl_derivative).
                         conjugate_func(Order_conjugate).
                         real_part_func(Order_real_part).
                         imag_part_func(Order_imag_part).
                         power_func(Order_power));
 
// Solve linear system
 
class symbolset {
    exset s;
    void insert_symbols(const ex &e)
    {
        if (is_a<symbol>(e)) {
            s.insert(e);
        } else {
            for (const ex &sube : e) {
                insert_symbols(sube);
            }
        }
    }
public:
    explicit symbolset(const ex &e)
    {
        insert_symbols(e);
    }
    bool has(const ex &e) const
    {
        return s.find(e) != s.end();
    }
};
 
ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
{
    // solve a system of linear equations
    if (eqns.info(info_flags::relation_equal)) {
        if (!symbols.info(info_flags::symbol))
            throw(std::invalid_argument("lsolve(): 2nd argument must be a symbol"));
        const ex sol = lsolve(lst{eqns}, lst{symbols});
        
        GINAC_ASSERT(sol.nops()==1);
        GINAC_ASSERT(is_exactly_a<relational>(sol.op(0)));
        
        return sol.op(0).op(1); // return rhs of first solution
    }
    
    // syntax checks
    if (!(eqns.info(info_flags::list) || eqns.info(info_flags::exprseq))) {
        throw(std::invalid_argument("lsolve(): 1st argument must be a list, a sequence, or an equation"));
    }
    for (size_t i=0; i<eqns.nops(); i++) {
        if (!eqns.op(i).info(info_flags::relation_equal)) {
            throw(std::invalid_argument("lsolve(): 1st argument must be a list of equations"));
        }
    }
    if (!(symbols.info(info_flags::list) || symbols.info(info_flags::exprseq))) {
        throw(std::invalid_argument("lsolve(): 2nd argument must be a list, a sequence, or a symbol"));
    }
    for (size_t i=0; i<symbols.nops(); i++) {
        if (!symbols.op(i).info(info_flags::symbol)) {
            throw(std::invalid_argument("lsolve(): 2nd argument must be a list or a sequence of symbols"));
        }
    }
    
    // build matrix from equation system
    matrix sys(eqns.nops(),symbols.nops());
    matrix rhs(eqns.nops(),1);
    matrix vars(symbols.nops(),1);
    
    for (size_t r=0; r<eqns.nops(); r++) {
        const ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
        const symbolset syms(eq);
        ex linpart = eq;
        for (size_t c=0; c<symbols.nops(); c++) {
            if (!syms.has(symbols.op(c)))
                continue;
            const ex co = eq.coeff(ex_to<symbol>(symbols.op(c)),1);
            linpart -= co*symbols.op(c);
            sys(r,c) = co;
        }
        linpart = linpart.expand();
        rhs(r,0) = -linpart;
    }
    
    // test if system is linear and fill vars matrix
    const symbolset sys_syms(sys);
    const symbolset rhs_syms(rhs);
    for (size_t i=0; i<symbols.nops(); i++) {
        vars(i,0) = symbols.op(i);
        if (sys_syms.has(symbols.op(i)))
            throw(std::logic_error("lsolve: system is not linear"));
        if (rhs_syms.has(symbols.op(i)))
            throw(std::logic_error("lsolve: system is not linear"));
    }
    
    matrix solution;
    try {
        solution = sys.solve(vars,rhs,options);
    } catch (const std::runtime_error & e) {
        // Probably singular matrix or otherwise overdetermined system:
        // It is consistent to return an empty list
        return lst{};
    }
    GINAC_ASSERT(solution.cols()==1);
    GINAC_ASSERT(solution.rows()==symbols.nops());
    
    // return list of equations of the form lst{var1==sol1,var2==sol2,...}
    lst sollist;
    for (size_t i=0; i<symbols.nops(); i++)
        sollist.append(symbols.op(i)==solution(i,0));
    
    return sollist;
}
 
// Find real root of f(x) numerically
 
const numeric
fsolve(const ex& f_in, const symbol& x, const numeric& x1, const numeric& x2)
{
    if (!x1.is_real() || !x2.is_real()) {
        throw std::runtime_error("fsolve(): interval not bounded by real numbers");
    }
    if (x1==x2) {
        throw std::runtime_error("fsolve(): vanishing interval");
    }
    // xx[0] == left interval limit, xx[1] == right interval limit.
    // fx[0] == f(xx[0]), fx[1] == f(xx[1]).
    // We keep the root bracketed: xx[0]<xx[1] and fx[0]*fx[1]<0.
    numeric xx[2] = { x1<x2 ? x1 : x2,
                      x1<x2 ? x2 : x1 };
    ex f;
    if (is_a<relational>(f_in)) {
        f = f_in.lhs()-f_in.rhs();
    } else {
        f = f_in;
    }
    const ex fx_[2] = { f.subs(x==xx[0]).evalf(),
                        f.subs(x==xx[1]).evalf() };
    if (!is_a<numeric>(fx_[0]) || !is_a<numeric>(fx_[1])) {
        throw std::runtime_error("fsolve(): function does not evaluate numerically");
    }
    numeric fx[2] = { ex_to<numeric>(fx_[0]),
                      ex_to<numeric>(fx_[1]) };
    if (!fx[0].is_real() || !fx[1].is_real()) {
        throw std::runtime_error("fsolve(): function evaluates to complex values at interval boundaries");
    }
    if (fx[0]*fx[1]>=0) {
        throw std::runtime_error("fsolve(): function does not change sign at interval boundaries");
    }
 
    // The Newton-Raphson method has quadratic convergence!  Simply put, it
    // replaces x with x-f(x)/f'(x) at each step.  -f/f' is the delta:
    const ex ff = normal(-f/f.diff(x));
    int side = 0;  // Start at left interval limit.
    numeric xxprev;
    numeric fxprev;
    do {
        xxprev = xx[side];
        fxprev = fx[side];
        ex dx_ = ff.subs(x == xx[side]).evalf();
        if (!is_a<numeric>(dx_))
            throw std::runtime_error("fsolve(): function derivative does not evaluate numerically");
        xx[side] += ex_to<numeric>(dx_);
        // Now check if Newton-Raphson method shot out of the interval 
        bool bad_shot = (side == 0 && xx[0] < xxprev) || 
                (side == 1 && xx[1] > xxprev) || xx[0] > xx[1];
        if (!bad_shot) {
            // Compute f(x) only if new x is inside the interval.
            // The function might be difficult to compute numerically
            // or even ill defined outside the interval. Also it's
            // a small optimization. 
            ex f_x = f.subs(x == xx[side]).evalf();
            if (!is_a<numeric>(f_x))
                throw std::runtime_error("fsolve(): function does not evaluate numerically");
            fx[side] = ex_to<numeric>(f_x);
        }
        if (bad_shot) {
            // Oops, Newton-Raphson method shot out of the interval.
            // Restore, and try again with the other side instead!
            xx[side] = xxprev;
            fx[side] = fxprev;
            side = !side;
            xxprev = xx[side];
            fxprev = fx[side];
 
            ex dx_ = ff.subs(x == xx[side]).evalf();
            if (!is_a<numeric>(dx_))
                throw std::runtime_error("fsolve(): function derivative does not evaluate numerically [2]");
            xx[side] += ex_to<numeric>(dx_);
 
            ex f_x = f.subs(x==xx[side]).evalf();
            if (!is_a<numeric>(f_x))
                throw std::runtime_error("fsolve(): function does not evaluate numerically [2]");
            fx[side] = ex_to<numeric>(f_x);
        }
        if ((fx[side]<0 && fx[!side]<0) || (fx[side]>0 && fx[!side]>0)) {
            // Oops, the root isn't bracketed any more.
            // Restore, and perform a bisection!
            xx[side] = xxprev;
            fx[side] = fxprev;
 
            // Ah, the bisection! Bisections converge linearly. Unfortunately,
            // they occur pretty often when Newton-Raphson arrives at an x too
            // close to the result on one side of the interval and
            // f(x-f(x)/f'(x)) turns out to have the same sign as f(x) due to
            // precision errors! Recall that this function does not have a
            // precision goal as one of its arguments but instead relies on
            // x converging to a fixed point. We speed up the (safe but slow)
            // bisection method by mixing in a dash of the (unsafer but faster)
            // secant method: Instead of splitting the interval at the
            // arithmetic mean (bisection), we split it nearer to the root as
            // determined by the secant between the values xx[0] and xx[1].
            // Don't set the secant_weight to one because that could disturb
            // the convergence in some corner cases!
            constexpr double secant_weight = 0.984375;  // == 63/64 < 1
            numeric xxmid = (1-secant_weight)*0.5*(xx[0]+xx[1])
                + secant_weight*(xx[0]+fx[0]*(xx[0]-xx[1])/(fx[1]-fx[0]));
            ex fxmid_ = f.subs(x == xxmid).evalf();
            if (!is_a<numeric>(fxmid_))
                throw std::runtime_error("fsolve(): function does not evaluate numerically [3]");
            numeric fxmid = ex_to<numeric>(fxmid_);
            if (fxmid.is_zero()) {
                // Luck strikes...
                return xxmid;
            }
            if ((fxmid<0 && fx[side]>0) || (fxmid>0 && fx[side]<0)) {
                side = !side;
            }
            xxprev = xx[side];
            fxprev = fx[side];
            xx[side] = xxmid;
            fx[side] = fxmid;
        }
    } while (xxprev!=xx[side]);
    return xxprev;
}
 
 
/* Force inclusion of functions from inifcns_gamma and inifcns_zeta
 * for static lib (so ginsh will see them). */
unsigned force_include_tgamma = tgamma_SERIAL::serial;
unsigned force_include_zeta1 = zeta1_SERIAL::serial;
 
} // namespace GiNaC