xref: /dports/math/GiNaC/ginac-1.8.2/ginac/inifcns_trans.cpp

/** @file inifcns_trans.cpp
 *
 *  Implementation of transcendental (and trigonometric and hyperbolic)
 *  functions. */

/*
 *  GiNaC Copyright (C) 1999-2022 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 "add.h"
#include "mul.h"
#include "numeric.h"
#include "power.h"
#include "operators.h"
#include "relational.h"
#include "symbol.h"
#include "pseries.h"
#include "utils.h"

#include <stdexcept>
#include <vector>

namespace GiNaC {

//////////
// exponential function
//////////

static ex exp_evalf(const ex & x)
{
	if (is_exactly_a<numeric>(x))
		return exp(ex_to<numeric>(x));

	return exp(x).hold();
}

static ex exp_eval(const ex & x)
{
	// exp(0) -> 1
	if (x.is_zero()) {
		return _ex1;
	}

	// exp(n*Pi*I/2) -> {+1|+I|-1|-I}
	const ex TwoExOverPiI=(_ex2*x)/(Pi*I);
	if (TwoExOverPiI.info(info_flags::integer)) {
		const numeric z = mod(ex_to<numeric>(TwoExOverPiI),*_num4_p);
		if (z.is_equal(*_num0_p))
			return _ex1;
		if (z.is_equal(*_num1_p))
			return ex(I);
		if (z.is_equal(*_num2_p))
			return _ex_1;
		if (z.is_equal(*_num3_p))
			return ex(-I);
	}

	// exp(log(x)) -> x
	if (is_ex_the_function(x, log))
		return x.op(0);

	// exp(float) -> float
	if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
		return exp(ex_to<numeric>(x));

	return exp(x).hold();
}

static ex exp_expand(const ex & arg, unsigned options)
{
	ex exp_arg;
	if (options & expand_options::expand_function_args)
		exp_arg = arg.expand(options);
	else
		exp_arg=arg;

	if ((options & expand_options::expand_transcendental)
		&& is_exactly_a<add>(exp_arg)) {
		exvector prodseq;
		prodseq.reserve(exp_arg.nops());
		for (const_iterator i = exp_arg.begin(); i != exp_arg.end(); ++i)
			prodseq.push_back(exp(*i));

		return dynallocate<mul>(prodseq).setflag(status_flags::expanded);
	}

	return exp(exp_arg).hold();
}

static ex exp_deriv(const ex & x, unsigned deriv_param)
{
	GINAC_ASSERT(deriv_param==0);

	// d/dx exp(x) -> exp(x)
	return exp(x);
}

static ex exp_real_part(const ex & x)
{
	return exp(GiNaC::real_part(x))*cos(GiNaC::imag_part(x));
}

static ex exp_imag_part(const ex & x)
{
	return exp(GiNaC::real_part(x))*sin(GiNaC::imag_part(x));
}

static ex exp_conjugate(const ex & x)
{
	// conjugate(exp(x))==exp(conjugate(x))
	return exp(x.conjugate());
}

static ex exp_power(const ex & x, const ex & a)
{
	/*
	 * The power law (e^x)^a=e^(x*a) is used in two cases:
	 * a) a is an integer and x may be complex;
	 * b) both x and a are reals.
	 * Negative a is excluded to keep automatic simplifications like exp(x)/exp(x)=1.
	 */
	if (a.info(info_flags::nonnegative)
	    && (a.info(info_flags::integer) || (x.info(info_flags::real) && a.info(info_flags::real))))
		return exp(x*a);
	else if (a.info(info_flags::negative)
	         && (a.info(info_flags::integer) || (x.info(info_flags::real) && a.info(info_flags::real))))
		return power(exp(-x*a), _ex_1).hold();

	return power(exp(x), a).hold();
}

REGISTER_FUNCTION(exp, eval_func(exp_eval).
                       evalf_func(exp_evalf).
                       expand_func(exp_expand).
                       derivative_func(exp_deriv).
                       real_part_func(exp_real_part).
                       imag_part_func(exp_imag_part).
                       conjugate_func(exp_conjugate).
                       power_func(exp_power).
                       latex_name("\\exp"));

//////////
// natural logarithm
//////////

static ex log_evalf(const ex & x)
{
	if (is_exactly_a<numeric>(x))
		return log(ex_to<numeric>(x));

	return log(x).hold();
}

static ex log_eval(const ex & x)
{
	if (x.info(info_flags::numeric)) {
		if (x.is_zero())         // log(0) -> infinity
			throw(pole_error("log_eval(): log(0)",0));
		if (x.info(info_flags::rational) && x.info(info_flags::negative))
			return (log(-x)+I*Pi);
		if (x.is_equal(_ex1))  // log(1) -> 0
			return _ex0;
		if (x.is_equal(I))       // log(I) -> Pi*I/2
			return (Pi*I*_ex1_2);
		if (x.is_equal(-I))      // log(-I) -> -Pi*I/2
			return (Pi*I*_ex_1_2);

		// log(float) -> float
		if (!x.info(info_flags::crational))
			return log(ex_to<numeric>(x));
	}

	// log(exp(t)) -> t (if -Pi < t.imag() <= Pi):
	if (is_ex_the_function(x, exp)) {
		const ex &t = x.op(0);
		if (t.info(info_flags::real))
			return t;
	}

	return log(x).hold();
}

static ex log_deriv(const ex & x, unsigned deriv_param)
{
	GINAC_ASSERT(deriv_param==0);

	// d/dx log(x) -> 1/x
	return power(x, _ex_1);
}

static ex log_series(const ex &arg,
                     const relational &rel,
                     int order,
                     unsigned options)
{
	GINAC_ASSERT(is_a<symbol>(rel.lhs()));
	ex arg_pt;
	bool must_expand_arg = false;
	// maybe substitution of rel into arg fails because of a pole
	try {
		arg_pt = arg.subs(rel, subs_options::no_pattern);
	} catch (pole_error &) {
		must_expand_arg = true;
	}
	// or we are at the branch point anyways
	if (arg_pt.is_zero())
		must_expand_arg = true;

	if (arg.diff(ex_to<symbol>(rel.lhs())).is_zero()) {
		throw do_taylor();
	}

	if (must_expand_arg) {
		// method:
		// This is the branch point: Series expand the argument first, then
		// trivially factorize it to isolate that part which has constant
		// leading coefficient in this fashion:
		//   x^n + x^(n+1) +...+ Order(x^(n+m))  ->  x^n * (1 + x +...+ Order(x^m)).
		// Return a plain n*log(x) for the x^n part and series expand the
		// other part.  Add them together and reexpand again in order to have
		// one unnested pseries object.  All this also works for negative n.
		pseries argser;          // series expansion of log's argument
		unsigned extra_ord = 0;  // extra expansion order
		do {
			// oops, the argument expanded to a pure Order(x^something)...
			argser = ex_to<pseries>(arg.series(rel, order+extra_ord, options));
			++extra_ord;
		} while (!argser.is_terminating() && argser.nops()==1);

		const symbol &s = ex_to<symbol>(rel.lhs());
		const ex &point = rel.rhs();
		const int n = argser.ldegree(s);
		epvector seq;
		// construct what we carelessly called the n*log(x) term above
		const ex coeff = argser.coeff(s, n);
		// expand the log, but only if coeff is real and > 0, since otherwise
		// it would make the branch cut run into the wrong direction
		if (coeff.info(info_flags::positive))
			seq.push_back(expair(n*log(s-point)+log(coeff), _ex0));
		else
			seq.push_back(expair(log(coeff*pow(s-point, n)), _ex0));

		if (!argser.is_terminating() || argser.nops()!=1) {
			// in this case n more (or less) terms are needed
			// (sadly, to generate them, we have to start from the beginning)
			if (n == 0 && coeff == 1) {
				ex rest = pseries(rel, epvector{expair(-1, _ex0), expair(Order(_ex1), order)}).add_series(argser);
				ex acc = dynallocate<pseries>(rel, epvector());
				for (int i = order-1; i>0; --i) {
					epvector cterm { expair(i%2 ? _ex1/i : _ex_1/i, _ex0) };
					acc = pseries(rel, std::move(cterm)).add_series(ex_to<pseries>(acc));
					acc = (ex_to<pseries>(rest)).mul_series(ex_to<pseries>(acc));
				}
				return acc;
			}
			const ex newarg = ex_to<pseries>((arg/coeff).series(rel, order+n, options)).shift_exponents(-n).convert_to_poly(true);
			return pseries(rel, std::move(seq)).add_series(ex_to<pseries>(log(newarg).series(rel, order, options)));
		} else  // it was a monomial
			return pseries(rel, std::move(seq));
	}
	if (!(options & series_options::suppress_branchcut) &&
	     arg_pt.info(info_flags::negative)) {
		// 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;
		const ex replarg = series(log(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
		epvector seq;
		if (order > 0) {
			seq.reserve(2);
			seq.push_back(expair(-I*csgn(arg*I)*Pi, _ex0));
		}
		seq.push_back(expair(Order(_ex1), order));
		return series(replarg - I*Pi + pseries(rel, std::move(seq)), rel, order);
	}
	throw do_taylor();  // caught by function::series()
}

static ex log_real_part(const ex & x)
{
	if (x.info(info_flags::nonnegative))
		return log(x).hold();
	return log(abs(x));
}

static ex log_imag_part(const ex & x)
{
	if (x.info(info_flags::nonnegative))
		return 0;
	return atan2(GiNaC::imag_part(x), GiNaC::real_part(x));
}

static ex log_expand(const ex & arg, unsigned options)
{
	if ((options & expand_options::expand_transcendental)
		&& is_exactly_a<mul>(arg) && !arg.info(info_flags::indefinite)) {
		exvector sumseq;
		exvector prodseq;
		sumseq.reserve(arg.nops());
		prodseq.reserve(arg.nops());
		bool possign=true;

		// searching for positive/negative factors
		for (const_iterator i = arg.begin(); i != arg.end(); ++i) {
			ex e;
			if (options & expand_options::expand_function_args)
				e=i->expand(options);
			else
				e=*i;
			if (e.info(info_flags::positive))
				sumseq.push_back(log(e));
			else if (e.info(info_flags::negative)) {
				sumseq.push_back(log(-e));
				possign = !possign;
			} else
				prodseq.push_back(e);
		}

		if (sumseq.size() > 0) {
			ex newarg;
			if (options & expand_options::expand_function_args)
				newarg=((possign?_ex1:_ex_1)*mul(prodseq)).expand(options);
			else {
				newarg=(possign?_ex1:_ex_1)*mul(prodseq);
				ex_to<basic>(newarg).setflag(status_flags::purely_indefinite);
			}
			return add(sumseq)+log(newarg);
		} else {
			if (!(options & expand_options::expand_function_args))
				ex_to<basic>(arg).setflag(status_flags::purely_indefinite);
		}
	}

	if (options & expand_options::expand_function_args)
		return log(arg.expand(options)).hold();
	else
		return log(arg).hold();
}

static ex log_conjugate(const ex & x)
{
	// conjugate(log(x))==log(conjugate(x)) unless on the branch cut which
	// runs along the negative real axis.
	if (x.info(info_flags::positive)) {
		return log(x);
	}
	if (is_exactly_a<numeric>(x) &&
	    !x.imag_part().is_zero()) {
		return log(x.conjugate());
	}
	return conjugate_function(log(x)).hold();
}

REGISTER_FUNCTION(log, eval_func(log_eval).
                       evalf_func(log_evalf).
                       expand_func(log_expand).
                       derivative_func(log_deriv).
                       series_func(log_series).
                       real_part_func(log_real_part).
                       imag_part_func(log_imag_part).
                       conjugate_func(log_conjugate).
                       latex_name("\\ln"));

//////////
// sine (trigonometric function)
//////////

static ex sin_evalf(const ex & x)
{
	if (is_exactly_a<numeric>(x))
		return sin(ex_to<numeric>(x));

	return sin(x).hold();
}

static ex sin_eval(const ex & x)
{
	// sin(n/d*Pi) -> { all known non-nested radicals }
	const ex SixtyExOverPi = _ex60*x/Pi;
	ex sign = _ex1;
	if (SixtyExOverPi.info(info_flags::integer)) {
		numeric z = mod(ex_to<numeric>(SixtyExOverPi),*_num120_p);
		if (z>=*_num60_p) {
			// wrap to interval [0, Pi)
			z -= *_num60_p;
			sign = _ex_1;
		}
		if (z>*_num30_p) {
			// wrap to interval [0, Pi/2)
			z = *_num60_p-z;
		}
		if (z.is_equal(*_num0_p))  // sin(0)       -> 0
			return _ex0;
		if (z.is_equal(*_num5_p))  // sin(Pi/12)   -> sqrt(6)/4*(1-sqrt(3)/3)
			return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex_1_3*sqrt(_ex3));
		if (z.is_equal(*_num6_p))  // sin(Pi/10)   -> sqrt(5)/4-1/4
			return sign*(_ex1_4*sqrt(_ex5)+_ex_1_4);
		if (z.is_equal(*_num10_p)) // sin(Pi/6)    -> 1/2
			return sign*_ex1_2;
		if (z.is_equal(*_num15_p)) // sin(Pi/4)    -> sqrt(2)/2
			return sign*_ex1_2*sqrt(_ex2);
		if (z.is_equal(*_num18_p)) // sin(3/10*Pi) -> sqrt(5)/4+1/4
			return sign*(_ex1_4*sqrt(_ex5)+_ex1_4);
		if (z.is_equal(*_num20_p)) // sin(Pi/3)    -> sqrt(3)/2
			return sign*_ex1_2*sqrt(_ex3);
		if (z.is_equal(*_num25_p)) // sin(5/12*Pi) -> sqrt(6)/4*(1+sqrt(3)/3)
			return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex1_3*sqrt(_ex3));
		if (z.is_equal(*_num30_p)) // sin(Pi/2)    -> 1
			return sign;
	}

	if (is_exactly_a<function>(x)) {
		const ex &t = x.op(0);

		// sin(asin(x)) -> x
		if (is_ex_the_function(x, asin))
			return t;

		// sin(acos(x)) -> sqrt(1-x^2)
		if (is_ex_the_function(x, acos))
			return sqrt(_ex1-power(t,_ex2));

		// sin(atan(x)) -> x/sqrt(1+x^2)
		if (is_ex_the_function(x, atan))
			return t*power(_ex1+power(t,_ex2),_ex_1_2);
	}

	// sin(float) -> float
	if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
		return sin(ex_to<numeric>(x));

	// sin() is odd
	if (x.info(info_flags::negative))
		return -sin(-x);

	return sin(x).hold();
}

static ex sin_deriv(const ex & x, unsigned deriv_param)
{
	GINAC_ASSERT(deriv_param==0);

	// d/dx sin(x) -> cos(x)
	return cos(x);
}

static ex sin_real_part(const ex & x)
{
	return cosh(GiNaC::imag_part(x))*sin(GiNaC::real_part(x));
}

static ex sin_imag_part(const ex & x)
{
	return sinh(GiNaC::imag_part(x))*cos(GiNaC::real_part(x));
}

static ex sin_conjugate(const ex & x)
{
	// conjugate(sin(x))==sin(conjugate(x))
	return sin(x.conjugate());
}

REGISTER_FUNCTION(sin, eval_func(sin_eval).
                       evalf_func(sin_evalf).
                       derivative_func(sin_deriv).
                       real_part_func(sin_real_part).
                       imag_part_func(sin_imag_part).
                       conjugate_func(sin_conjugate).
                       latex_name("\\sin"));

//////////
// cosine (trigonometric function)
//////////

static ex cos_evalf(const ex & x)
{
	if (is_exactly_a<numeric>(x))
		return cos(ex_to<numeric>(x));

	return cos(x).hold();
}

static ex cos_eval(const ex & x)
{
	// cos(n/d*Pi) -> { all known non-nested radicals }
	const ex SixtyExOverPi = _ex60*x/Pi;
	ex sign = _ex1;
	if (SixtyExOverPi.info(info_flags::integer)) {
		numeric z = mod(ex_to<numeric>(SixtyExOverPi),*_num120_p);
		if (z>=*_num60_p) {
			// wrap to interval [0, Pi)
			z = *_num120_p-z;
		}
		if (z>=*_num30_p) {
			// wrap to interval [0, Pi/2)
			z = *_num60_p-z;
			sign = _ex_1;
		}
		if (z.is_equal(*_num0_p))  // cos(0)       -> 1
			return sign;
		if (z.is_equal(*_num5_p))  // cos(Pi/12)   -> sqrt(6)/4*(1+sqrt(3)/3)
			return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex1_3*sqrt(_ex3));
		if (z.is_equal(*_num10_p)) // cos(Pi/6)    -> sqrt(3)/2
			return sign*_ex1_2*sqrt(_ex3);
		if (z.is_equal(*_num12_p)) // cos(Pi/5)    -> sqrt(5)/4+1/4
			return sign*(_ex1_4*sqrt(_ex5)+_ex1_4);
		if (z.is_equal(*_num15_p)) // cos(Pi/4)    -> sqrt(2)/2
			return sign*_ex1_2*sqrt(_ex2);
		if (z.is_equal(*_num20_p)) // cos(Pi/3)    -> 1/2
			return sign*_ex1_2;
		if (z.is_equal(*_num24_p)) // cos(2/5*Pi)  -> sqrt(5)/4-1/4x
			return sign*(_ex1_4*sqrt(_ex5)+_ex_1_4);
		if (z.is_equal(*_num25_p)) // cos(5/12*Pi) -> sqrt(6)/4*(1-sqrt(3)/3)
			return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex_1_3*sqrt(_ex3));
		if (z.is_equal(*_num30_p)) // cos(Pi/2)    -> 0
			return _ex0;
	}

	if (is_exactly_a<function>(x)) {
		const ex &t = x.op(0);

		// cos(acos(x)) -> x
		if (is_ex_the_function(x, acos))
			return t;

		// cos(asin(x)) -> sqrt(1-x^2)
		if (is_ex_the_function(x, asin))
			return sqrt(_ex1-power(t,_ex2));

		// cos(atan(x)) -> 1/sqrt(1+x^2)
		if (is_ex_the_function(x, atan))
			return power(_ex1+power(t,_ex2),_ex_1_2);
	}

	// cos(float) -> float
	if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
		return cos(ex_to<numeric>(x));

	// cos() is even
	if (x.info(info_flags::negative))
		return cos(-x);

	return cos(x).hold();
}

static ex cos_deriv(const ex & x, unsigned deriv_param)
{
	GINAC_ASSERT(deriv_param==0);

	// d/dx cos(x) -> -sin(x)
	return -sin(x);
}

static ex cos_real_part(const ex & x)
{
	return cosh(GiNaC::imag_part(x))*cos(GiNaC::real_part(x));
}

static ex cos_imag_part(const ex & x)
{
	return -sinh(GiNaC::imag_part(x))*sin(GiNaC::real_part(x));
}

static ex cos_conjugate(const ex & x)
{
	// conjugate(cos(x))==cos(conjugate(x))
	return cos(x.conjugate());
}

REGISTER_FUNCTION(cos, eval_func(cos_eval).
                       evalf_func(cos_evalf).
                       derivative_func(cos_deriv).
                       real_part_func(cos_real_part).
                       imag_part_func(cos_imag_part).
                       conjugate_func(cos_conjugate).
                       latex_name("\\cos"));

//////////
// tangent (trigonometric function)
//////////

static ex tan_evalf(const ex & x)
{
	if (is_exactly_a<numeric>(x))
		return tan(ex_to<numeric>(x));

	return tan(x).hold();
}

static ex tan_eval(const ex & x)
{
	// tan(n/d*Pi) -> { all known non-nested radicals }
	const ex SixtyExOverPi = _ex60*x/Pi;
	ex sign = _ex1;
	if (SixtyExOverPi.info(info_flags::integer)) {
		numeric z = mod(ex_to<numeric>(SixtyExOverPi),*_num60_p);
		if (z>=*_num60_p) {
			// wrap to interval [0, Pi)
			z -= *_num60_p;
		}
		if (z>=*_num30_p) {
			// wrap to interval [0, Pi/2)
			z = *_num60_p-z;
			sign = _ex_1;
		}
		if (z.is_equal(*_num0_p))  // tan(0)       -> 0
			return _ex0;
		if (z.is_equal(*_num5_p))  // tan(Pi/12)   -> 2-sqrt(3)
			return sign*(_ex2-sqrt(_ex3));
		if (z.is_equal(*_num10_p)) // tan(Pi/6)    -> sqrt(3)/3
			return sign*_ex1_3*sqrt(_ex3);
		if (z.is_equal(*_num15_p)) // tan(Pi/4)    -> 1
			return sign;
		if (z.is_equal(*_num20_p)) // tan(Pi/3)    -> sqrt(3)
			return sign*sqrt(_ex3);
		if (z.is_equal(*_num25_p)) // tan(5/12*Pi) -> 2+sqrt(3)
			return sign*(sqrt(_ex3)+_ex2);
		if (z.is_equal(*_num30_p)) // tan(Pi/2)    -> infinity
			throw (pole_error("tan_eval(): simple pole",1));
	}

	if (is_exactly_a<function>(x)) {
		const ex &t = x.op(0);

		// tan(atan(x)) -> x
		if (is_ex_the_function(x, atan))
			return t;

		// tan(asin(x)) -> x/sqrt(1+x^2)
		if (is_ex_the_function(x, asin))
			return t*power(_ex1-power(t,_ex2),_ex_1_2);

		// tan(acos(x)) -> sqrt(1-x^2)/x
		if (is_ex_the_function(x, acos))
			return power(t,_ex_1)*sqrt(_ex1-power(t,_ex2));
	}

	// tan(float) -> float
	if (x.info(info_flags::numeric) && !x.info(info_flags::crational)) {
		return tan(ex_to<numeric>(x));
	}

	// tan() is odd
	if (x.info(info_flags::negative))
		return -tan(-x);

	return tan(x).hold();
}

static ex tan_deriv(const ex & x, unsigned deriv_param)
{
	GINAC_ASSERT(deriv_param==0);

	// d/dx tan(x) -> 1+tan(x)^2;
	return (_ex1+power(tan(x),_ex2));
}

static ex tan_real_part(const ex & x)
{
	ex a = GiNaC::real_part(x);
	ex b = GiNaC::imag_part(x);
	return tan(a)/(1+power(tan(a),2)*power(tan(b),2));
}

static ex tan_imag_part(const ex & x)
{
	ex a = GiNaC::real_part(x);
	ex b = GiNaC::imag_part(x);
	return tanh(b)/(1+power(tan(a),2)*power(tan(b),2));
}

static ex tan_series(const ex &x,
                     const relational &rel,
                     int order,
                     unsigned options)
{
	GINAC_ASSERT(is_a<symbol>(rel.lhs()));
	// method:
	// Taylor series where there is no pole falls back to tan_deriv.
	// On a pole simply expand sin(x)/cos(x).
	const ex x_pt = x.subs(rel, subs_options::no_pattern);
	if (!(2*x_pt/Pi).info(info_flags::odd))
		throw do_taylor();  // caught by function::series()
	// if we got here we have to care for a simple pole
	return (sin(x)/cos(x)).series(rel, order, options);
}

static ex tan_conjugate(const ex & x)
{
	// conjugate(tan(x))==tan(conjugate(x))
	return tan(x.conjugate());
}

REGISTER_FUNCTION(tan, eval_func(tan_eval).
                       evalf_func(tan_evalf).
                       derivative_func(tan_deriv).
                       series_func(tan_series).
                       real_part_func(tan_real_part).
                       imag_part_func(tan_imag_part).
                       conjugate_func(tan_conjugate).
                       latex_name("\\tan"));

//////////
// inverse sine (arc sine)
//////////

static ex asin_evalf(const ex & x)
{
	if (is_exactly_a<numeric>(x))
		return asin(ex_to<numeric>(x));

	return asin(x).hold();
}

static ex asin_eval(const ex & x)
{
	if (x.info(info_flags::numeric)) {

		// asin(0) -> 0
		if (x.is_zero())
			return x;

		// asin(1/2) -> Pi/6
		if (x.is_equal(_ex1_2))
			return numeric(1,6)*Pi;

		// asin(1) -> Pi/2
		if (x.is_equal(_ex1))
			return _ex1_2*Pi;

		// asin(-1/2) -> -Pi/6
		if (x.is_equal(_ex_1_2))
			return numeric(-1,6)*Pi;

		// asin(-1) -> -Pi/2
		if (x.is_equal(_ex_1))
			return _ex_1_2*Pi;

		// asin(float) -> float
		if (!x.info(info_flags::crational))
			return asin(ex_to<numeric>(x));

		// asin() is odd
		if (x.info(info_flags::negative))
			return -asin(-x);
	}

	return asin(x).hold();
}

static ex asin_deriv(const ex & x, unsigned deriv_param)
{
	GINAC_ASSERT(deriv_param==0);

	// d/dx asin(x) -> 1/sqrt(1-x^2)
	return power(1-power(x,_ex2),_ex_1_2);
}

static ex asin_conjugate(const ex & x)
{
	// conjugate(asin(x))==asin(conjugate(x)) unless on the branch cuts which
	// run along the real axis outside the interval [-1, +1].
	if (is_exactly_a<numeric>(x) &&
	    (!x.imag_part().is_zero() || (x > *_num_1_p && x < *_num1_p))) {
		return asin(x.conjugate());
	}
	return conjugate_function(asin(x)).hold();
}

REGISTER_FUNCTION(asin, eval_func(asin_eval).
                        evalf_func(asin_evalf).
                        derivative_func(asin_deriv).
                        conjugate_func(asin_conjugate).
                        latex_name("\\arcsin"));

//////////
// inverse cosine (arc cosine)
//////////

static ex acos_evalf(const ex & x)
{
	if (is_exactly_a<numeric>(x))
		return acos(ex_to<numeric>(x));

	return acos(x).hold();
}

static ex acos_eval(const ex & x)
{
	if (x.info(info_flags::numeric)) {

		// acos(1) -> 0
		if (x.is_equal(_ex1))
			return _ex0;

		// acos(1/2) -> Pi/3
		if (x.is_equal(_ex1_2))
			return _ex1_3*Pi;

		// acos(0) -> Pi/2
		if (x.is_zero())
			return _ex1_2*Pi;

		// acos(-1/2) -> 2/3*Pi
		if (x.is_equal(_ex_1_2))
			return numeric(2,3)*Pi;

		// acos(-1) -> Pi
		if (x.is_equal(_ex_1))
			return Pi;

		// acos(float) -> float
		if (!x.info(info_flags::crational))
			return acos(ex_to<numeric>(x));

		// acos(-x) -> Pi-acos(x)
		if (x.info(info_flags::negative))
			return Pi-acos(-x);
	}

	return acos(x).hold();
}

static ex acos_deriv(const ex & x, unsigned deriv_param)
{
	GINAC_ASSERT(deriv_param==0);

	// d/dx acos(x) -> -1/sqrt(1-x^2)
	return -power(1-power(x,_ex2),_ex_1_2);
}

static ex acos_conjugate(const ex & x)
{
	// conjugate(acos(x))==acos(conjugate(x)) unless on the branch cuts which
	// run along the real axis outside the interval [-1, +1].
	if (is_exactly_a<numeric>(x) &&
	    (!x.imag_part().is_zero() || (x > *_num_1_p && x < *_num1_p))) {
		return acos(x.conjugate());
	}
	return conjugate_function(acos(x)).hold();
}

REGISTER_FUNCTION(acos, eval_func(acos_eval).
                        evalf_func(acos_evalf).
                        derivative_func(acos_deriv).
                        conjugate_func(acos_conjugate).
                        latex_name("\\arccos"));

//////////
// inverse tangent (arc tangent)
//////////

static ex atan_evalf(const ex & x)
{
	if (is_exactly_a<numeric>(x))
		return atan(ex_to<numeric>(x));

	return atan(x).hold();
}

static ex atan_eval(const ex & x)
{
	if (x.info(info_flags::numeric)) {

		// atan(0) -> 0
		if (x.is_zero())
			return _ex0;

		// atan(1) -> Pi/4
		if (x.is_equal(_ex1))
			return _ex1_4*Pi;

		// atan(-1) -> -Pi/4
		if (x.is_equal(_ex_1))
			return _ex_1_4*Pi;

		if (x.is_equal(I) || x.is_equal(-I))
			throw (pole_error("atan_eval(): logarithmic pole",0));

		// atan(float) -> float
		if (!x.info(info_flags::crational))
			return atan(ex_to<numeric>(x));

		// atan() is odd
		if (x.info(info_flags::negative))
			return -atan(-x);
	}

	return atan(x).hold();
}

static ex atan_deriv(const ex & x, unsigned deriv_param)
{
	GINAC_ASSERT(deriv_param==0);

	// d/dx atan(x) -> 1/(1+x^2)
	return power(_ex1+power(x,_ex2), _ex_1);
}

static ex atan_series(const ex &arg,
                      const relational &rel,
                      int order,
                      unsigned options)
{
	GINAC_ASSERT(is_a<symbol>(rel.lhs()));
	// method:
	// Taylor series where there is no pole or cut falls back to atan_deriv.
	// There are two branch cuts, one runnig from I up the imaginary axis and
	// one running from -I down the imaginary axis.  The points I and -I are
	// poles.
	// On the branch cuts and the poles series expand
	//     (log(1+I*x)-log(1-I*x))/(2*I)
	// instead.
	const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
	if (!(I*arg_pt).info(info_flags::real))
		throw do_taylor();     // Re(x) != 0
	if ((I*arg_pt).info(info_flags::real) && abs(I*arg_pt)<_ex1)
		throw do_taylor();     // Re(x) == 0, but abs(x)<1
	// care for the poles, using the defining formula for atan()...
	if (arg_pt.is_equal(I) || arg_pt.is_equal(-I))
		return ((log(1+I*arg)-log(1-I*arg))/(2*I)).series(rel, order, options);
	if (!(options & series_options::suppress_branchcut)) {
		// 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;
		const ex replarg = series(atan(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
		ex Order0correction = replarg.op(0)+csgn(arg)*Pi*_ex_1_2;
		if ((I*arg_pt)<_ex0)
			Order0correction += log((I*arg_pt+_ex_1)/(I*arg_pt+_ex1))*I*_ex_1_2;
		else
			Order0correction += log((I*arg_pt+_ex1)/(I*arg_pt+_ex_1))*I*_ex1_2;
		epvector seq;
		if (order > 0) {
			seq.reserve(2);
			seq.push_back(expair(Order0correction, _ex0));
		}
		seq.push_back(expair(Order(_ex1), order));
		return series(replarg - pseries(rel, std::move(seq)), rel, order);
	}
	throw do_taylor();
}

static ex atan_conjugate(const ex & x)
{
	// conjugate(atan(x))==atan(conjugate(x)) unless on the branch cuts which
	// run along the imaginary axis outside the interval [-I, +I].
	if (x.info(info_flags::real))
		return atan(x);
	if (is_exactly_a<numeric>(x)) {
		const numeric x_re = ex_to<numeric>(x.real_part());
		const numeric x_im = ex_to<numeric>(x.imag_part());
		if (!x_re.is_zero() ||
		    (x_im > *_num_1_p && x_im < *_num1_p))
			return atan(x.conjugate());
	}
	return conjugate_function(atan(x)).hold();
}

REGISTER_FUNCTION(atan, eval_func(atan_eval).
                        evalf_func(atan_evalf).
                        derivative_func(atan_deriv).
                        series_func(atan_series).
                        conjugate_func(atan_conjugate).
                        latex_name("\\arctan"));

//////////
// inverse tangent (atan2(y,x))
//////////

static ex atan2_evalf(const ex &y, const ex &x)
{
	if (is_exactly_a<numeric>(y) && is_exactly_a<numeric>(x))
		return atan(ex_to<numeric>(y), ex_to<numeric>(x));

	return atan2(y, x).hold();
}

static ex atan2_eval(const ex & y, const ex & x)
{
	if (y.is_zero()) {

		// atan2(0, 0) -> 0
		if (x.is_zero())
			return _ex0;

		// atan2(0, x), x real and positive -> 0
		if (x.info(info_flags::positive))
			return _ex0;

		// atan2(0, x), x real and negative -> Pi
		if (x.info(info_flags::negative))
			return Pi;
	}

	if (x.is_zero()) {

		// atan2(y, 0), y real and positive -> Pi/2
		if (y.info(info_flags::positive))
			return _ex1_2*Pi;

		// atan2(y, 0), y real and negative -> -Pi/2
		if (y.info(info_flags::negative))
			return _ex_1_2*Pi;
	}

	if (y.is_equal(x)) {

		// atan2(y, y), y real and positive -> Pi/4
		if (y.info(info_flags::positive))
			return _ex1_4*Pi;

		// atan2(y, y), y real and negative -> -3/4*Pi
		if (y.info(info_flags::negative))
			return numeric(-3, 4)*Pi;
	}

	if (y.is_equal(-x)) {

		// atan2(y, -y), y real and positive -> 3*Pi/4
		if (y.info(info_flags::positive))
			return numeric(3, 4)*Pi;

		// atan2(y, -y), y real and negative -> -Pi/4
		if (y.info(info_flags::negative))
			return _ex_1_4*Pi;
	}

	// atan2(float, float) -> float
	if (is_a<numeric>(y) && !y.info(info_flags::crational) &&
	    is_a<numeric>(x) && !x.info(info_flags::crational))
		return atan(ex_to<numeric>(y), ex_to<numeric>(x));

	// atan2(real, real) -> atan(y/x) +/- Pi
	if (y.info(info_flags::real) && x.info(info_flags::real)) {
		if (x.info(info_flags::positive))
			return atan(y/x);

		if (x.info(info_flags::negative)) {
			if (y.info(info_flags::positive))
				return atan(y/x)+Pi;
			if (y.info(info_flags::negative))
				return atan(y/x)-Pi;
		}
	}

	return atan2(y, x).hold();
}

static ex atan2_deriv(const ex & y, const ex & x, unsigned deriv_param)
{
	GINAC_ASSERT(deriv_param<2);

	if (deriv_param==0) {
		// d/dy atan2(y,x)
		return x*power(power(x,_ex2)+power(y,_ex2),_ex_1);
	}
	// d/dx atan2(y,x)
	return -y*power(power(x,_ex2)+power(y,_ex2),_ex_1);
}

REGISTER_FUNCTION(atan2, eval_func(atan2_eval).
                         evalf_func(atan2_evalf).
                         derivative_func(atan2_deriv));

//////////
// hyperbolic sine (trigonometric function)
//////////

static ex sinh_evalf(const ex & x)
{
	if (is_exactly_a<numeric>(x))
		return sinh(ex_to<numeric>(x));

	return sinh(x).hold();
}

static ex sinh_eval(const ex & x)
{
	if (x.info(info_flags::numeric)) {

		// sinh(0) -> 0
		if (x.is_zero())
			return _ex0;

		// sinh(float) -> float
		if (!x.info(info_flags::crational))
			return sinh(ex_to<numeric>(x));

		// sinh() is odd
		if (x.info(info_flags::negative))
			return -sinh(-x);
	}

	if ((x/Pi).info(info_flags::numeric) &&
		ex_to<numeric>(x/Pi).real().is_zero())  // sinh(I*x) -> I*sin(x)
		return I*sin(x/I);

	if (is_exactly_a<function>(x)) {
		const ex &t = x.op(0);

		// sinh(asinh(x)) -> x
		if (is_ex_the_function(x, asinh))
			return t;

		// sinh(acosh(x)) -> sqrt(x-1) * sqrt(x+1)
		if (is_ex_the_function(x, acosh))
			return sqrt(t-_ex1)*sqrt(t+_ex1);

		// sinh(atanh(x)) -> x/sqrt(1-x^2)
		if (is_ex_the_function(x, atanh))
			return t*power(_ex1-power(t,_ex2),_ex_1_2);
	}

	return sinh(x).hold();
}

static ex sinh_deriv(const ex & x, unsigned deriv_param)
{
	GINAC_ASSERT(deriv_param==0);

	// d/dx sinh(x) -> cosh(x)
	return cosh(x);
}

static ex sinh_real_part(const ex & x)
{
	return sinh(GiNaC::real_part(x))*cos(GiNaC::imag_part(x));
}

static ex sinh_imag_part(const ex & x)
{
	return cosh(GiNaC::real_part(x))*sin(GiNaC::imag_part(x));
}

static ex sinh_conjugate(const ex & x)
{
	// conjugate(sinh(x))==sinh(conjugate(x))
	return sinh(x.conjugate());
}

REGISTER_FUNCTION(sinh, eval_func(sinh_eval).
                        evalf_func(sinh_evalf).
                        derivative_func(sinh_deriv).
                        real_part_func(sinh_real_part).
                        imag_part_func(sinh_imag_part).
                        conjugate_func(sinh_conjugate).
                        latex_name("\\sinh"));

//////////
// hyperbolic cosine (trigonometric function)
//////////

static ex cosh_evalf(const ex & x)
{
	if (is_exactly_a<numeric>(x))
		return cosh(ex_to<numeric>(x));

	return cosh(x).hold();
}

static ex cosh_eval(const ex & x)
{
	if (x.info(info_flags::numeric)) {

		// cosh(0) -> 1
		if (x.is_zero())
			return _ex1;

		// cosh(float) -> float
		if (!x.info(info_flags::crational))
			return cosh(ex_to<numeric>(x));

		// cosh() is even
		if (x.info(info_flags::negative))
			return cosh(-x);
	}

	if ((x/Pi).info(info_flags::numeric) &&
		ex_to<numeric>(x/Pi).real().is_zero())  // cosh(I*x) -> cos(x)
		return cos(x/I);

	if (is_exactly_a<function>(x)) {
		const ex &t = x.op(0);

		// cosh(acosh(x)) -> x
		if (is_ex_the_function(x, acosh))
			return t;

		// cosh(asinh(x)) -> sqrt(1+x^2)
		if (is_ex_the_function(x, asinh))
			return sqrt(_ex1+power(t,_ex2));

		// cosh(atanh(x)) -> 1/sqrt(1-x^2)
		if (is_ex_the_function(x, atanh))
			return power(_ex1-power(t,_ex2),_ex_1_2);
	}

	return cosh(x).hold();
}

static ex cosh_deriv(const ex & x, unsigned deriv_param)
{
	GINAC_ASSERT(deriv_param==0);

	// d/dx cosh(x) -> sinh(x)
	return sinh(x);
}

static ex cosh_real_part(const ex & x)
{
	return cosh(GiNaC::real_part(x))*cos(GiNaC::imag_part(x));
}

static ex cosh_imag_part(const ex & x)
{
	return sinh(GiNaC::real_part(x))*sin(GiNaC::imag_part(x));
}

static ex cosh_conjugate(const ex & x)
{
	// conjugate(cosh(x))==cosh(conjugate(x))
	return cosh(x.conjugate());
}

REGISTER_FUNCTION(cosh, eval_func(cosh_eval).
                        evalf_func(cosh_evalf).
                        derivative_func(cosh_deriv).
                        real_part_func(cosh_real_part).
                        imag_part_func(cosh_imag_part).
                        conjugate_func(cosh_conjugate).
                        latex_name("\\cosh"));

//////////
// hyperbolic tangent (trigonometric function)
//////////

static ex tanh_evalf(const ex & x)
{
	if (is_exactly_a<numeric>(x))
		return tanh(ex_to<numeric>(x));

	return tanh(x).hold();
}

static ex tanh_eval(const ex & x)
{
	if (x.info(info_flags::numeric)) {

		// tanh(0) -> 0
		if (x.is_zero())
			return _ex0;

		// tanh(float) -> float
		if (!x.info(info_flags::crational))
			return tanh(ex_to<numeric>(x));

		// tanh() is odd
		if (x.info(info_flags::negative))
			return -tanh(-x);
	}

	if ((x/Pi).info(info_flags::numeric) &&
		ex_to<numeric>(x/Pi).real().is_zero())  // tanh(I*x) -> I*tan(x);
		return I*tan(x/I);

	if (is_exactly_a<function>(x)) {
		const ex &t = x.op(0);

		// tanh(atanh(x)) -> x
		if (is_ex_the_function(x, atanh))
			return t;

		// tanh(asinh(x)) -> x/sqrt(1+x^2)
		if (is_ex_the_function(x, asinh))
			return t*power(_ex1+power(t,_ex2),_ex_1_2);

		// tanh(acosh(x)) -> sqrt(x-1)*sqrt(x+1)/x
		if (is_ex_the_function(x, acosh))
			return sqrt(t-_ex1)*sqrt(t+_ex1)*power(t,_ex_1);
	}

	return tanh(x).hold();
}

static ex tanh_deriv(const ex & x, unsigned deriv_param)
{
	GINAC_ASSERT(deriv_param==0);

	// d/dx tanh(x) -> 1-tanh(x)^2
	return _ex1-power(tanh(x),_ex2);
}

static ex tanh_series(const ex &x,
                      const relational &rel,
                      int order,
                      unsigned options)
{
	GINAC_ASSERT(is_a<symbol>(rel.lhs()));
	// method:
	// Taylor series where there is no pole falls back to tanh_deriv.
	// On a pole simply expand sinh(x)/cosh(x).
	const ex x_pt = x.subs(rel, subs_options::no_pattern);
	if (!(2*I*x_pt/Pi).info(info_flags::odd))
		throw do_taylor();  // caught by function::series()
	// if we got here we have to care for a simple pole
	return (sinh(x)/cosh(x)).series(rel, order, options);
}

static ex tanh_real_part(const ex & x)
{
	ex a = GiNaC::real_part(x);
	ex b = GiNaC::imag_part(x);
	return tanh(a)/(1+power(tanh(a),2)*power(tan(b),2));
}

static ex tanh_imag_part(const ex & x)
{
	ex a = GiNaC::real_part(x);
	ex b = GiNaC::imag_part(x);
	return tan(b)/(1+power(tanh(a),2)*power(tan(b),2));
}

static ex tanh_conjugate(const ex & x)
{
	// conjugate(tanh(x))==tanh(conjugate(x))
	return tanh(x.conjugate());
}

REGISTER_FUNCTION(tanh, eval_func(tanh_eval).
                        evalf_func(tanh_evalf).
                        derivative_func(tanh_deriv).
                        series_func(tanh_series).
                        real_part_func(tanh_real_part).
                        imag_part_func(tanh_imag_part).
                        conjugate_func(tanh_conjugate).
                        latex_name("\\tanh"));

//////////
// inverse hyperbolic sine (trigonometric function)
//////////

static ex asinh_evalf(const ex & x)
{
	if (is_exactly_a<numeric>(x))
		return asinh(ex_to<numeric>(x));

	return asinh(x).hold();
}

static ex asinh_eval(const ex & x)
{
	if (x.info(info_flags::numeric)) {

		// asinh(0) -> 0
		if (x.is_zero())
			return _ex0;

		// asinh(float) -> float
		if (!x.info(info_flags::crational))
			return asinh(ex_to<numeric>(x));

		// asinh() is odd
		if (x.info(info_flags::negative))
			return -asinh(-x);
	}

	return asinh(x).hold();
}

static ex asinh_deriv(const ex & x, unsigned deriv_param)
{
	GINAC_ASSERT(deriv_param==0);

	// d/dx asinh(x) -> 1/sqrt(1+x^2)
	return power(_ex1+power(x,_ex2),_ex_1_2);
}

static ex asinh_conjugate(const ex & x)
{
	// conjugate(asinh(x))==asinh(conjugate(x)) unless on the branch cuts which
	// run along the imaginary axis outside the interval [-I, +I].
	if (x.info(info_flags::real))
		return asinh(x);
	if (is_exactly_a<numeric>(x)) {
		const numeric x_re = ex_to<numeric>(x.real_part());
		const numeric x_im = ex_to<numeric>(x.imag_part());
		if (!x_re.is_zero() ||
		    (x_im > *_num_1_p && x_im < *_num1_p))
			return asinh(x.conjugate());
	}
	return conjugate_function(asinh(x)).hold();
}

REGISTER_FUNCTION(asinh, eval_func(asinh_eval).
                         evalf_func(asinh_evalf).
                         derivative_func(asinh_deriv).
                         conjugate_func(asinh_conjugate));

//////////
// inverse hyperbolic cosine (trigonometric function)
//////////

static ex acosh_evalf(const ex & x)
{
	if (is_exactly_a<numeric>(x))
		return acosh(ex_to<numeric>(x));

	return acosh(x).hold();
}

static ex acosh_eval(const ex & x)
{
	if (x.info(info_flags::numeric)) {

		// acosh(0) -> Pi*I/2
		if (x.is_zero())
			return Pi*I*numeric(1,2);

		// acosh(1) -> 0
		if (x.is_equal(_ex1))
			return _ex0;

		// acosh(-1) -> Pi*I
		if (x.is_equal(_ex_1))
			return Pi*I;

		// acosh(float) -> float
		if (!x.info(info_flags::crational))
			return acosh(ex_to<numeric>(x));

		// acosh(-x) -> Pi*I-acosh(x)
		if (x.info(info_flags::negative))
			return Pi*I-acosh(-x);
	}

	return acosh(x).hold();
}

static ex acosh_deriv(const ex & x, unsigned deriv_param)
{
	GINAC_ASSERT(deriv_param==0);

	// d/dx acosh(x) -> 1/(sqrt(x-1)*sqrt(x+1))
	return power(x+_ex_1,_ex_1_2)*power(x+_ex1,_ex_1_2);
}

static ex acosh_conjugate(const ex & x)
{
	// conjugate(acosh(x))==acosh(conjugate(x)) unless on the branch cut
	// which runs along the real axis from +1 to -inf.
	if (is_exactly_a<numeric>(x) &&
	    (!x.imag_part().is_zero() || x > *_num1_p)) {
		return acosh(x.conjugate());
	}
	return conjugate_function(acosh(x)).hold();
}

REGISTER_FUNCTION(acosh, eval_func(acosh_eval).
                         evalf_func(acosh_evalf).
                         derivative_func(acosh_deriv).
                         conjugate_func(acosh_conjugate));

//////////
// inverse hyperbolic tangent (trigonometric function)
//////////

static ex atanh_evalf(const ex & x)
{
	if (is_exactly_a<numeric>(x))
		return atanh(ex_to<numeric>(x));

	return atanh(x).hold();
}

static ex atanh_eval(const ex & x)
{
	if (x.info(info_flags::numeric)) {

		// atanh(0) -> 0
		if (x.is_zero())
			return _ex0;

		// atanh({+|-}1) -> throw
		if (x.is_equal(_ex1) || x.is_equal(_ex_1))
			throw (pole_error("atanh_eval(): logarithmic pole",0));

		// atanh(float) -> float
		if (!x.info(info_flags::crational))
			return atanh(ex_to<numeric>(x));

		// atanh() is odd
		if (x.info(info_flags::negative))
			return -atanh(-x);
	}

	return atanh(x).hold();
}

static ex atanh_deriv(const ex & x, unsigned deriv_param)
{
	GINAC_ASSERT(deriv_param==0);

	// d/dx atanh(x) -> 1/(1-x^2)
	return power(_ex1-power(x,_ex2),_ex_1);
}

static ex atanh_series(const ex &arg,
                       const relational &rel,
                       int order,
                       unsigned options)
{
	GINAC_ASSERT(is_a<symbol>(rel.lhs()));
	// method:
	// Taylor series where there is no pole or cut falls back to atanh_deriv.
	// There are two branch cuts, one runnig from 1 up the real axis and one
	// one running from -1 down the real axis.  The points 1 and -1 are poles
	// On the branch cuts and the poles series expand
	//     (log(1+x)-log(1-x))/2
	// instead.
	const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
	if (!(arg_pt).info(info_flags::real))
		throw do_taylor();     // Im(x) != 0
	if ((arg_pt).info(info_flags::real) && abs(arg_pt)<_ex1)
		throw do_taylor();     // Im(x) == 0, but abs(x)<1
	// care for the poles, using the defining formula for atanh()...
	if (arg_pt.is_equal(_ex1) || arg_pt.is_equal(_ex_1))
		return ((log(_ex1+arg)-log(_ex1-arg))*_ex1_2).series(rel, order, options);
	// ...and the branch cuts (the discontinuity at the cut being just I*Pi)
	if (!(options & series_options::suppress_branchcut)) {
		// 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;
		const ex replarg = series(atanh(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
		ex Order0correction = replarg.op(0)+csgn(I*arg)*Pi*I*_ex1_2;
		if (arg_pt<_ex0)
			Order0correction += log((arg_pt+_ex_1)/(arg_pt+_ex1))*_ex1_2;
		else
			Order0correction += log((arg_pt+_ex1)/(arg_pt+_ex_1))*_ex_1_2;
		epvector seq;
		if (order > 0) {
			seq.reserve(2);
			seq.push_back(expair(Order0correction, _ex0));
		}
		seq.push_back(expair(Order(_ex1), order));
		return series(replarg - pseries(rel, std::move(seq)), rel, order);
	}
	throw do_taylor();
}

static ex atanh_conjugate(const ex & x)
{
	// conjugate(atanh(x))==atanh(conjugate(x)) unless on the branch cuts which
	// run along the real axis outside the interval [-1, +1].
	if (is_exactly_a<numeric>(x) &&
	    (!x.imag_part().is_zero() || (x > *_num_1_p && x < *_num1_p))) {
		return atanh(x.conjugate());
	}
	return conjugate_function(atanh(x)).hold();
}

REGISTER_FUNCTION(atanh, eval_func(atanh_eval).
                         evalf_func(atanh_evalf).
                         derivative_func(atanh_deriv).
                         series_func(atanh_series).
                         conjugate_func(atanh_conjugate));


} // namespace GiNaC