* functions. */
/*
- * GiNaC Copyright (C) 1999-2000 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2005 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
*
* 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., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
+ * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
*/
#include <vector>
#include "constant.h"
#include "numeric.h"
#include "power.h"
+#include "operators.h"
#include "relational.h"
#include "symbol.h"
+#include "pseries.h"
#include "utils.h"
-#ifndef NO_NAMESPACE_GINAC
namespace GiNaC {
-#endif // ndef NO_NAMESPACE_GINAC
//////////
// exponential function
static ex exp_evalf(const ex & x)
{
- BEGIN_TYPECHECK
- TYPECHECK(x,numeric)
- END_TYPECHECK(exp(x))
-
- return exp(ex_to_numeric(x)); // -> numeric exp(numeric)
+ 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}
- ex TwoExOverPiI=(_ex2()*x)/(Pi*I);
- if (TwoExOverPiI.info(info_flags::integer)) {
- numeric z=mod(ex_to_numeric(TwoExOverPiI),_num4());
- if (z.is_equal(_num0()))
- return _ex1();
- if (z.is_equal(_num1()))
- return ex(I);
- if (z.is_equal(_num2()))
- return _ex_1();
- if (z.is_equal(_num3()))
- return ex(-I);
- }
- // exp(log(x)) -> x
- if (is_ex_the_function(x, log))
- return x.op(0);
-
- // exp(float)
- if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
- return exp_evalf(x);
-
- return exp(x).hold();
+ // 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);
+ GINAC_ASSERT(deriv_param==0);
- // d/dx exp(x) -> exp(x)
- return exp(x);
+ // d/dx exp(x) -> exp(x)
+ return exp(x);
}
-REGISTER_FUNCTION(exp, exp_eval, exp_evalf, exp_deriv, NULL);
+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)
{
- BEGIN_TYPECHECK
- TYPECHECK(x,numeric)
- END_TYPECHECK(log(x))
-
- return log(ex_to_numeric(x)); // -> numeric log(numeric)
+ 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_equal(_ex1())) // log(1) -> 0
- return _ex0();
- if (x.is_equal(_ex_1())) // log(-1) -> I*Pi
- return (I*Pi);
- if (x.is_equal(I)) // log(I) -> Pi*I/2
- return (Pi*I*_num1_2());
- if (x.is_equal(-I)) // log(-I) -> -Pi*I/2
- return (Pi*I*_num_1_2());
- if (x.is_equal(_ex0())) // log(0) -> infinity
- throw(std::domain_error("log_eval(): log(0)"));
- // log(float)
- if (!x.info(info_flags::crational))
- return log_evalf(x);
- }
- // log(exp(t)) -> t (if -Pi < t.imag() <= Pi):
- if (is_ex_the_function(x, exp)) {
- ex t = x.op(0);
- if (t.info(info_flags::numeric)) {
- numeric nt = ex_to_numeric(t);
- if (nt.is_real())
- return t;
- }
- }
-
- return log(x).hold();
+ 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);
+ GINAC_ASSERT(deriv_param==0);
+
+ // d/dx log(x) -> 1/x
+ return power(x, _ex_1);
+}
- // 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, log_eval, log_evalf, log_deriv, NULL);
+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)
{
- BEGIN_TYPECHECK
- TYPECHECK(x,numeric)
- END_TYPECHECK(sin(x))
-
- return sin(ex_to_numeric(x)); // -> numeric sin(numeric)
+ 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 }
- ex SixtyExOverPi = _ex60()*x/Pi;
- ex sign = _ex1();
- if (SixtyExOverPi.info(info_flags::integer)) {
- numeric z = mod(ex_to_numeric(SixtyExOverPi),_num120());
- if (z>=_num60()) {
- // wrap to interval [0, Pi)
- z -= _num60();
- sign = _ex_1();
- }
- if (z>_num30()) {
- // wrap to interval [0, Pi/2)
- z = _num60()-z;
- }
- if (z.is_equal(_num0())) // sin(0) -> 0
- return _ex0();
- if (z.is_equal(_num5())) // sin(Pi/12) -> sqrt(6)/4*(1-sqrt(3)/3)
- return sign*_ex1_4()*power(_ex6(),_ex1_2())*(_ex1()+_ex_1_3()*power(_ex3(),_ex1_2()));
- if (z.is_equal(_num6())) // sin(Pi/10) -> sqrt(5)/4-1/4
- return sign*(_ex1_4()*power(_ex5(),_ex1_2())+_ex_1_4());
- if (z.is_equal(_num10())) // sin(Pi/6) -> 1/2
- return sign*_ex1_2();
- if (z.is_equal(_num15())) // sin(Pi/4) -> sqrt(2)/2
- return sign*_ex1_2()*power(_ex2(),_ex1_2());
- if (z.is_equal(_num18())) // sin(3/10*Pi) -> sqrt(5)/4+1/4
- return sign*(_ex1_4()*power(_ex5(),_ex1_2())+_ex1_4());
- if (z.is_equal(_num20())) // sin(Pi/3) -> sqrt(3)/2
- return sign*_ex1_2()*power(_ex3(),_ex1_2());
- if (z.is_equal(_num25())) // sin(5/12*Pi) -> sqrt(6)/4*(1+sqrt(3)/3)
- return sign*_ex1_4()*power(_ex6(),_ex1_2())*(_ex1()+_ex1_3()*power(_ex3(),_ex1_2()));
- if (z.is_equal(_num30())) // sin(Pi/2) -> 1
- return sign*_ex1();
- }
-
- if (is_ex_exactly_of_type(x, function)) {
- 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 power(_ex1()-power(t,_ex2()),_ex1_2());
- // sin(atan(x)) -> x*(1+x^2)^(-1/2)
- 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_evalf(x);
-
- return sin(x).hold();
+ // 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);
+ GINAC_ASSERT(deriv_param==0);
+
+ // d/dx sin(x) -> cos(x)
+ return cos(x);
}
-REGISTER_FUNCTION(sin, sin_eval, sin_evalf, sin_deriv, NULL);
+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)
{
- BEGIN_TYPECHECK
- TYPECHECK(x,numeric)
- END_TYPECHECK(cos(x))
-
- return cos(ex_to_numeric(x)); // -> numeric cos(numeric)
+ 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 }
- ex SixtyExOverPi = _ex60()*x/Pi;
- ex sign = _ex1();
- if (SixtyExOverPi.info(info_flags::integer)) {
- numeric z = mod(ex_to_numeric(SixtyExOverPi),_num120());
- if (z>=_num60()) {
- // wrap to interval [0, Pi)
- z = _num120()-z;
- }
- if (z>=_num30()) {
- // wrap to interval [0, Pi/2)
- z = _num60()-z;
- sign = _ex_1();
- }
- if (z.is_equal(_num0())) // cos(0) -> 1
- return sign*_ex1();
- if (z.is_equal(_num5())) // cos(Pi/12) -> sqrt(6)/4*(1+sqrt(3)/3)
- return sign*_ex1_4()*power(_ex6(),_ex1_2())*(_ex1()+_ex1_3()*power(_ex3(),_ex1_2()));
- if (z.is_equal(_num10())) // cos(Pi/6) -> sqrt(3)/2
- return sign*_ex1_2()*power(_ex3(),_ex1_2());
- if (z.is_equal(_num12())) // cos(Pi/5) -> sqrt(5)/4+1/4
- return sign*(_ex1_4()*power(_ex5(),_ex1_2())+_ex1_4());
- if (z.is_equal(_num15())) // cos(Pi/4) -> sqrt(2)/2
- return sign*_ex1_2()*power(_ex2(),_ex1_2());
- if (z.is_equal(_num20())) // cos(Pi/3) -> 1/2
- return sign*_ex1_2();
- if (z.is_equal(_num24())) // cos(2/5*Pi) -> sqrt(5)/4-1/4x
- return sign*(_ex1_4()*power(_ex5(),_ex1_2())+_ex_1_4());
- if (z.is_equal(_num25())) // cos(5/12*Pi) -> sqrt(6)/4*(1-sqrt(3)/3)
- return sign*_ex1_4()*power(_ex6(),_ex1_2())*(_ex1()+_ex_1_3()*power(_ex3(),_ex1_2()));
- if (z.is_equal(_num30())) // cos(Pi/2) -> 0
- return sign*_ex0();
- }
-
- if (is_ex_exactly_of_type(x, function)) {
- ex t = x.op(0);
- // cos(acos(x)) -> x
- if (is_ex_the_function(x, acos))
- return t;
- // cos(asin(x)) -> (1-x^2)^(1/2)
- if (is_ex_the_function(x, asin))
- return power(_ex1()-power(t,_ex2()),_ex1_2());
- // cos(atan(x)) -> (1+x^2)^(-1/2)
- 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_evalf(x);
-
- return cos(x).hold();
+ // 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);
+ GINAC_ASSERT(deriv_param==0);
- // d/dx cos(x) -> -sin(x)
- return _ex_1()*sin(x);
+ // d/dx cos(x) -> -sin(x)
+ return -sin(x);
}
-REGISTER_FUNCTION(cos, cos_eval, cos_evalf, cos_deriv, NULL);
+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)
{
- BEGIN_TYPECHECK
- TYPECHECK(x,numeric)
- END_TYPECHECK(tan(x)) // -> numeric tan(numeric)
-
- return tan(ex_to_numeric(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 }
- ex SixtyExOverPi = _ex60()*x/Pi;
- ex sign = _ex1();
- if (SixtyExOverPi.info(info_flags::integer)) {
- numeric z = mod(ex_to_numeric(SixtyExOverPi),_num60());
- if (z>=_num60()) {
- // wrap to interval [0, Pi)
- z -= _num60();
- }
- if (z>=_num30()) {
- // wrap to interval [0, Pi/2)
- z = _num60()-z;
- sign = _ex_1();
- }
- if (z.is_equal(_num0())) // tan(0) -> 0
- return _ex0();
- if (z.is_equal(_num5())) // tan(Pi/12) -> 2-sqrt(3)
- return sign*(_ex2()-power(_ex3(),_ex1_2()));
- if (z.is_equal(_num10())) // tan(Pi/6) -> sqrt(3)/3
- return sign*_ex1_3()*power(_ex3(),_ex1_2());
- if (z.is_equal(_num15())) // tan(Pi/4) -> 1
- return sign*_ex1();
- if (z.is_equal(_num20())) // tan(Pi/3) -> sqrt(3)
- return sign*power(_ex3(),_ex1_2());
- if (z.is_equal(_num25())) // tan(5/12*Pi) -> 2+sqrt(3)
- return sign*(power(_ex3(),_ex1_2())+_ex2());
- if (z.is_equal(_num30())) // tan(Pi/2) -> infinity
- throw (std::domain_error("tan_eval(): infinity"));
- }
-
- if (is_ex_exactly_of_type(x, function)) {
- ex t = x.op(0);
- // tan(atan(x)) -> x
- if (is_ex_the_function(x, atan))
- return t;
- // tan(asin(x)) -> x*(1+x^2)^(-1/2)
- if (is_ex_the_function(x, asin))
- return t*power(_ex1()-power(t,_ex2()),_ex_1_2());
- // tan(acos(x)) -> (1-x^2)^(1/2)/x
- if (is_ex_the_function(x, acos))
- return power(t,_ex_1())*power(_ex1()-power(t,_ex2()),_ex1_2());
- }
-
- // tan(float) -> float
- if (x.info(info_flags::numeric) && !x.info(info_flags::crational)) {
- return tan_evalf(x);
- }
-
- return tan(x).hold();
+ // 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()));
+ 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 symbol & s, const ex & pt, int order)
+static ex tan_series(const ex &x,
+ const relational &rel,
+ int order,
+ unsigned options)
{
- // 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(s==pt);
- 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(s, pt, order+2);
+ 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, tan_eval, tan_evalf, tan_deriv, tan_series);
+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)
{
- BEGIN_TYPECHECK
- TYPECHECK(x,numeric)
- END_TYPECHECK(asin(x))
-
- return asin(ex_to_numeric(x)); // -> numeric asin(numeric)
+ 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 _num1_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 _num_1_2()*Pi;
- // asin(float) -> float
- if (!x.info(info_flags::crational))
- return asin_evalf(x);
- }
-
- return asin(x).hold();
+ 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());
+ 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, asin_eval, asin_evalf, asin_deriv, NULL);
+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)
{
- BEGIN_TYPECHECK
- TYPECHECK(x,numeric)
- END_TYPECHECK(acos(x))
-
- return acos(ex_to_numeric(x)); // -> numeric acos(numeric)
+ 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_evalf(x);
- }
-
- return acos(x).hold();
+ 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 _ex_1()*power(1-power(x,_ex2()),_ex_1_2());
+ 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, acos_eval, acos_evalf, acos_deriv, NULL);
+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)
{
- BEGIN_TYPECHECK
- TYPECHECK(x,numeric)
- END_TYPECHECK(atan(x))
-
- return atan(ex_to_numeric(x)); // -> numeric atan(numeric)
+ 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_equal(_ex0()))
- return _ex0();
- // atan(float) -> float
- if (!x.info(info_flags::crational))
- return atan_evalf(x);
- }
-
- return atan(x).hold();
-}
+ 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);
+ GINAC_ASSERT(deriv_param==0);
+
+ // d/dx atan(x) -> 1/(1+x^2)
+ return power(_ex1+power(x,_ex2), _ex_1);
+}
- // 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, atan_eval, atan_evalf, atan_deriv, NULL);
+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)
+static ex atan2_evalf(const ex &y, const ex &x)
{
- BEGIN_TYPECHECK
- TYPECHECK(y,numeric)
- TYPECHECK(x,numeric)
- END_TYPECHECK(atan2(y,x))
-
- return atan(ex_to_numeric(y),ex_to_numeric(x)); // -> numeric atan(numeric)
+ 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) && !y.info(info_flags::crational) &&
- x.info(info_flags::numeric) && !x.info(info_flags::crational)) {
- return atan2_evalf(y,x);
- }
-
- return atan2(y,x).hold();
+ 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 _ex_1*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());
+ 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, atan2_eval, atan2_evalf, atan2_deriv, NULL);
+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)
{
- BEGIN_TYPECHECK
- TYPECHECK(x,numeric)
- END_TYPECHECK(sinh(x))
-
- return sinh(ex_to_numeric(x)); // -> numeric sinh(numeric)
+ 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)) {
- if (x.is_zero()) // sinh(0) -> 0
- return _ex0();
- if (!x.info(info_flags::crational)) // sinh(float) -> float
- return sinh_evalf(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_ex_exactly_of_type(x, function)) {
- ex t = x.op(0);
- // sinh(asinh(x)) -> x
- if (is_ex_the_function(x, asinh))
- return t;
- // sinh(acosh(x)) -> (x-1)^(1/2) * (x+1)^(1/2)
- if (is_ex_the_function(x, acosh))
- return power(t-_ex1(),_ex1_2())*power(t+_ex1(),_ex1_2());
- // sinh(atanh(x)) -> x*(1-x^2)^(-1/2)
- if (is_ex_the_function(x, atanh))
- return t*power(_ex1()-power(t,_ex2()),_ex_1_2());
- }
-
- return sinh(x).hold();
+ 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);
+ GINAC_ASSERT(deriv_param==0);
+
+ // d/dx sinh(x) -> cosh(x)
+ return cosh(x);
}
-REGISTER_FUNCTION(sinh, sinh_eval, sinh_evalf, sinh_deriv, NULL);
+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)
{
- BEGIN_TYPECHECK
- TYPECHECK(x,numeric)
- END_TYPECHECK(cosh(x))
-
- return cosh(ex_to_numeric(x)); // -> numeric cosh(numeric)
+ 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)) {
- if (x.is_zero()) // cosh(0) -> 1
- return _ex1();
- if (!x.info(info_flags::crational)) // cosh(float) -> float
- return cosh_evalf(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_ex_exactly_of_type(x, function)) {
- ex t = x.op(0);
- // cosh(acosh(x)) -> x
- if (is_ex_the_function(x, acosh))
- return t;
- // cosh(asinh(x)) -> (1+x^2)^(1/2)
- if (is_ex_the_function(x, asinh))
- return power(_ex1()+power(t,_ex2()),_ex1_2());
- // cosh(atanh(x)) -> (1-x^2)^(-1/2)
- if (is_ex_the_function(x, atanh))
- return power(_ex1()-power(t,_ex2()),_ex_1_2());
- }
-
- return cosh(x).hold();
+ 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);
+ GINAC_ASSERT(deriv_param==0);
+
+ // d/dx cosh(x) -> sinh(x)
+ return sinh(x);
}
-REGISTER_FUNCTION(cosh, cosh_eval, cosh_evalf, cosh_deriv, NULL);
+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)
{
- BEGIN_TYPECHECK
- TYPECHECK(x,numeric)
- END_TYPECHECK(tanh(x))
-
- return tanh(ex_to_numeric(x)); // -> numeric tanh(numeric)
+ 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)) {
- if (x.is_zero()) // tanh(0) -> 0
- return _ex0();
- if (!x.info(info_flags::crational)) // tanh(float) -> float
- return tanh_evalf(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_ex_exactly_of_type(x, function)) {
- ex t = x.op(0);
- // tanh(atanh(x)) -> x
- if (is_ex_the_function(x, atanh))
- return t;
- // tanh(asinh(x)) -> x*(1+x^2)^(-1/2)
- if (is_ex_the_function(x, asinh))
- return t*power(_ex1()+power(t,_ex2()),_ex_1_2());
- // tanh(acosh(x)) -> (x-1)^(1/2)*(x+1)^(1/2)/x
- if (is_ex_the_function(x, acosh))
- return power(t-_ex1(),_ex1_2())*power(t+_ex1(),_ex1_2())*power(t,_ex_1());
- }
-
- return tanh(x).hold();
+ 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());
+ 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 symbol & s, const ex & pt, int order)
+static ex tanh_series(const ex &x,
+ const relational &rel,
+ int order,
+ unsigned options)
{
- // 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(s==pt);
- 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(s, pt, order+2);
+ 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, tanh_eval, tanh_evalf, tanh_deriv, tanh_series);
+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)
{
- BEGIN_TYPECHECK
- TYPECHECK(x,numeric)
- END_TYPECHECK(asinh(x))
-
- return asinh(ex_to_numeric(x)); // -> numeric asinh(numeric)
+ 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_evalf(x);
- }
-
- return asinh(x).hold();
+ 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());
+ 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, asinh_eval, asinh_evalf, asinh_deriv, NULL);
+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)
{
- BEGIN_TYPECHECK
- TYPECHECK(x,numeric)
- END_TYPECHECK(acosh(x))
-
- return acosh(ex_to_numeric(x)); // -> numeric acosh(numeric)
+ 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_evalf(x);
- }
-
- return acosh(x).hold();
+ 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());
+ 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, acosh_eval, acosh_evalf, acosh_deriv, NULL);
+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)
{
- BEGIN_TYPECHECK
- TYPECHECK(x,numeric)
- END_TYPECHECK(atanh(x))
-
- return atanh(ex_to_numeric(x)); // -> numeric atanh(numeric)
+ 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(_ex1()))
- throw (std::domain_error("atanh_eval(): infinity"));
- // atanh(float) -> float
- if (!x.info(info_flags::crational))
- return atanh_evalf(x);
- }
-
- return atanh(x).hold();
+ 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());
+ GINAC_ASSERT(deriv_param==0);
+
+ // d/dx atanh(x) -> 1/(1-x^2)
+ return power(_ex1-power(x,_ex2),_ex_1);
}
-REGISTER_FUNCTION(atanh, atanh_eval, atanh_evalf, atanh_deriv, NULL);
+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));
+
-#ifndef NO_NAMESPACE_GINAC
} // namespace GiNaC
-#endif // ndef NO_NAMESPACE_GINAC
git://www.ginac.de / ginac.git / blobdiff