/** @file inifcns_trans.cpp
*
* Implementation of transcendental (and trigonometric and hyperbolic)
* functions. */
/*
* GiNaC Copyright (C) 1999-2007 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 <vector>
#include <stdexcept>
#include "inifcns.h"
#include "ex.h"
#include "constant.h"
#include "numeric.h"
#include "power.h"
#include "operators.h"
#include "relational.h"
#include "symbol.h"
#include "pseries.h"
#include "utils.h"
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_deriv(const ex & x, unsigned deriv_param)
{
GINAC_ASSERT(deriv_param==0);
// d/dx exp(x) -> exp(x)
return exp(x);
}
REGISTER_FUNCTION(exp, eval_func(exp_eval).
evalf_func(exp_evalf).
derivative_func(exp_deriv).
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 (is_a<symbol>(t) && t.info(info_flags::real)) {
return t;
}
if (t.info(info_flags::numeric)) {
const numeric &nt = ex_to<numeric>(t);
if (nt.is_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 (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) {
epvector epv;
ex acc = (new pseries(rel, epv))->setflag(status_flags::dynallocated);
epv.reserve(2);
epv.push_back(expair(-1, _ex0));
epv.push_back(expair(Order(_ex1), order));
ex rest = pseries(rel, epv).add_series(argser);
for (int i = order-1; i>0; --i) {
epvector cterm;
cterm.reserve(1);
cterm.push_back(expair(i%2 ? _ex1/i : _ex_1/i, _ex0));
acc = pseries(rel, 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, seq).add_series(ex_to<pseries>(log(newarg).series(rel, order, options)));
} else // it was a monomial
return pseries(rel, 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;
seq.push_back(expair(-I*csgn(arg*I)*Pi, _ex0));
seq.push_back(expair(Order(_ex1), order));
return series(replarg - I*Pi + pseries(rel, seq), rel, order);
}
throw do_taylor(); // caught by function::series()
}
REGISTER_FUNCTION(log, eval_func(log_eval).
evalf_func(log_evalf).
derivative_func(log_deriv).
series_func(log_series).
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);
}
REGISTER_FUNCTION(sin, eval_func(sin_eval).
evalf_func(sin_evalf).
derivative_func(sin_deriv).
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);
}
REGISTER_FUNCTION(cos, eval_func(cos_eval).
evalf_func(cos_evalf).
derivative_func(cos_deriv).
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_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);
}
REGISTER_FUNCTION(tan, eval_func(tan_eval).
evalf_func(tan_evalf).
derivative_func(tan_deriv).
series_func(tan_series).
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);
}
REGISTER_FUNCTION(asin, eval_func(asin_eval).
evalf_func(asin_evalf).
derivative_func(asin_deriv).
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);
}
REGISTER_FUNCTION(acos, eval_func(acos_eval).
evalf_func(acos_evalf).
derivative_func(acos_deriv).
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;
seq.push_back(expair(Order0correction, _ex0));
seq.push_back(expair(Order(_ex1), order));
return series(replarg - pseries(rel, seq), rel, order);
}
throw do_taylor();
}
REGISTER_FUNCTION(atan, eval_func(atan_eval).
evalf_func(atan_evalf).
derivative_func(atan_deriv).
series_func(atan_series).
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.info(info_flags::numeric) && x.info(info_flags::numeric)) {
if (y.is_zero()) {
// atan(0, 0) -> 0
if (x.is_zero())
return _ex0;
// atan(0, x), x real and positive -> 0
if (x.info(info_flags::positive))
return _ex0;
// atan(0, x), x real and negative -> Pi
if (x.info(info_flags::negative))
return Pi;
}
if (x.is_zero()) {
// atan(y, 0), y real and positive -> Pi/2
if (y.info(info_flags::positive))
return _ex1_2*Pi;
// atan(y, 0), y real and negative -> -Pi/2
if (y.info(info_flags::negative))
return _ex_1_2*Pi;
}
if (y.is_equal(x)) {
// atan(y, y), y real and positive -> Pi/4
if (y.info(info_flags::positive))
return _ex1_4*Pi;
// atan(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)) {
// atan(y, -y), y real and positive -> 3*Pi/4
if (y.info(info_flags::positive))
return numeric(3, 4)*Pi;
// atan(y, -y), y real and negative -> -Pi/4
if (y.info(info_flags::negative))
return _ex_1_4*Pi;
}
// atan(float, float) -> float
if (!y.info(info_flags::crational) && !x.info(info_flags::crational))
return atan(ex_to<numeric>(y), ex_to<numeric>(x));
// atan(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);
else if (y.info(info_flags::positive))
return atan(y/x)+Pi;
else
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 atan(y,x)
return x*power(power(x,_ex2)+power(y,_ex2),_ex_1);
}
// d/dx atan(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);
}
REGISTER_FUNCTION(sinh, eval_func(sinh_eval).
evalf_func(sinh_evalf).
derivative_func(sinh_deriv).
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);
}
REGISTER_FUNCTION(cosh, eval_func(cosh_eval).
evalf_func(cosh_evalf).
derivative_func(cosh_deriv).
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);
}
REGISTER_FUNCTION(tanh, eval_func(tanh_eval).
evalf_func(tanh_evalf).
derivative_func(tanh_deriv).
series_func(tanh_series).
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);
}
REGISTER_FUNCTION(asinh, eval_func(asinh_eval).
evalf_func(asinh_evalf).
derivative_func(asinh_deriv));
//////////
// 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);
}
REGISTER_FUNCTION(acosh, eval_func(acosh_eval).
evalf_func(acosh_evalf).
derivative_func(acosh_deriv));
//////////
// 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;
seq.push_back(expair(Order0correction, _ex0));
seq.push_back(expair(Order(_ex1), order));
return series(replarg - pseries(rel, seq), rel, order);
}
throw do_taylor();
}
REGISTER_FUNCTION(atanh, eval_func(atanh_eval).
evalf_func(atanh_evalf).
derivative_func(atanh_deriv).
series_func(atanh_series));
} // namespace GiNaC
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