/** @file inifcns_trans.cpp
*
* Implementation of transcendental (and trigonometric and hyperbolic)
- * functions.
- *
- * GiNaC Copyright (C) 1999 Johannes Gutenberg University Mainz, Germany
+ * functions. */
+
+/*
+ * GiNaC Copyright (C) 1999-2004 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
#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
//////////
-ex exp_evalf(ex const & x)
-{
- BEGIN_TYPECHECK
- TYPECHECK(x,numeric)
- END_TYPECHECK(exp(x))
-
- return exp(ex_to_numeric(x)); // -> numeric exp(numeric)
-}
-
-ex exp_eval(ex const & x)
-{
- // exp(0) -> 1
- if (x.is_zero()) {
- return exONE();
- }
- // exp(n*Pi*I/2) -> {+1|+I|-1|-I}
- ex TwoExOverPiI=(2*x)/(Pi*I);
- if (TwoExOverPiI.info(info_flags::integer)) {
- numeric z=mod(ex_to_numeric(TwoExOverPiI),numeric(4));
- if (z.is_equal(numZERO()))
- return exONE();
- if (z.is_equal(numONE()))
- return ex(I);
- if (z.is_equal(numTWO()))
- return exMINUSONE();
- if (z.is_equal(numTHREE()))
- 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::rational))
- return exp_evalf(x);
-
- return exp(x).hold();
-}
+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);
+ 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(ex_to<numeric>(x));
+
+ return exp(x).hold();
+}
-ex exp_diff(ex const & x, unsigned diff_param)
+static ex exp_deriv(const ex & x, unsigned deriv_param)
{
- ASSERT(diff_param==0);
+ GINAC_ASSERT(deriv_param==0);
- return exp(x);
+ // d/dx exp(x) -> exp(x)
+ return exp(x);
}
-REGISTER_FUNCTION(exp, exp_eval, exp_evalf, exp_diff, NULL);
+REGISTER_FUNCTION(exp, eval_func(exp_eval).
+ evalf_func(exp_evalf).
+ derivative_func(exp_deriv).
+ latex_name("\\exp"));
//////////
// natural logarithm
//////////
-ex log_evalf(ex const & x)
-{
- BEGIN_TYPECHECK
- TYPECHECK(x,numeric)
- END_TYPECHECK(log(x))
-
- return log(ex_to_numeric(x)); // -> numeric log(numeric)
-}
-
-ex log_eval(ex const & x)
-{
- if (x.info(info_flags::numeric)) {
- // log(1) -> 0
- if (x.is_equal(exONE()))
- return exZERO();
- // log(-1) -> I*Pi
- if (x.is_equal(exMINUSONE()))
- return (I*Pi);
- // log(I) -> Pi*I/2
- if (x.is_equal(I))
- return (I*Pi*numeric(1,2));
- // log(-I) -> -Pi*I/2
- if (x.is_equal(-I))
- return (I*Pi*numeric(-1,2));
- // log(0) -> throw singularity
- if (x.is_equal(exZERO()))
- throw(std::domain_error("log_eval(): log(0)"));
- // log(float)
- if (!x.info(info_flags::rational))
- return log_evalf(x);
- }
-
- return log(x).hold();
-}
+static ex log_evalf(const ex & x)
+{
+ if (is_exactly_a<numeric>(x))
+ return log(ex_to<numeric>(x));
+
+ return log(x).hold();
+}
-ex log_diff(ex const & x, unsigned diff_param)
+static ex log_eval(const ex & x)
{
- ASSERT(diff_param==0);
+ 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::real) && 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*_num1_2);
+ if (x.is_equal(-I)) // log(-I) -> -Pi*I/2
+ return (Pi*I*_num_1_2);
+ // log(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();
+}
- return power(x, -1);
+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);
}
-REGISTER_FUNCTION(log, log_eval, log_evalf, log_diff, NULL);
+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)
+ 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)
//////////
-ex sin_evalf(ex const & x)
-{
- BEGIN_TYPECHECK
- TYPECHECK(x,numeric)
- END_TYPECHECK(sin(x))
-
- return sin(ex_to_numeric(x)); // -> numeric sin(numeric)
-}
-
-ex sin_eval(ex const & x)
-{
- // sin(n*Pi) -> 0
- ex xOverPi=x/Pi;
- if (xOverPi.info(info_flags::integer))
- return exZERO();
-
- // sin((2n+1)*Pi/2) -> {+|-}1
- ex xOverPiMinusHalf=xOverPi-exHALF();
- if (xOverPiMinusHalf.info(info_flags::even))
- return exONE();
- else if (xOverPiMinusHalf.info(info_flags::odd))
- return exMINUSONE();
-
- 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)) -> (1-x^2)^(1/2)
- if (is_ex_the_function(x, acos))
- return power(exONE()-power(t,exTWO()),exHALF());
- // sin(atan(x)) -> x*(1+x^2)^(-1/2)
- if (is_ex_the_function(x, atan))
- return t*power(exONE()+power(t,exTWO()),exMINUSHALF());
- }
-
- // sin(float) -> float
- if (x.info(info_flags::numeric) && !x.info(info_flags::rational))
- return sin_evalf(x);
-
- return sin(x).hold();
-}
-
-ex sin_diff(ex const & x, unsigned diff_param)
-{
- ASSERT(diff_param==0);
-
- return cos(x);
-}
-
-REGISTER_FUNCTION(sin, sin_eval, sin_evalf, sin_diff, NULL);
+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);
+ 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*sqrt(_ex6)*(_ex1+_ex_1_3*sqrt(_ex3));
+ if (z.is_equal(_num6)) // sin(Pi/10) -> sqrt(5)/4-1/4
+ return sign*(_ex1_4*sqrt(_ex5)+_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*sqrt(_ex2);
+ if (z.is_equal(_num18)) // sin(3/10*Pi) -> sqrt(5)/4+1/4
+ return sign*(_ex1_4*sqrt(_ex5)+_ex1_4);
+ if (z.is_equal(_num20)) // sin(Pi/3) -> sqrt(3)/2
+ return sign*_ex1_2*sqrt(_ex3);
+ if (z.is_equal(_num25)) // 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)) // 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));
+
+ 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)
//////////
-ex cos_evalf(ex const & x)
-{
- BEGIN_TYPECHECK
- TYPECHECK(x,numeric)
- END_TYPECHECK(cos(x))
-
- return cos(ex_to_numeric(x)); // -> numeric cos(numeric)
-}
-
-ex cos_eval(ex const & x)
-{
- // cos(n*Pi) -> {+|-}1
- ex xOverPi=x/Pi;
- if (xOverPi.info(info_flags::even))
- return exONE();
- else if (xOverPi.info(info_flags::odd))
- return exMINUSONE();
-
- // cos((2n+1)*Pi/2) -> 0
- ex xOverPiMinusHalf=xOverPi-exHALF();
- if (xOverPiMinusHalf.info(info_flags::integer))
- return exZERO();
-
- 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(exONE()-power(t,exTWO()),exHALF());
- // cos(atan(x)) -> (1+x^2)^(-1/2)
- if (is_ex_the_function(x, atan))
- return power(exONE()+power(t,exTWO()),exMINUSHALF());
- }
-
- // cos(float) -> float
- if (x.info(info_flags::numeric) && !x.info(info_flags::rational))
- return cos_evalf(x);
-
- return cos(x).hold();
-}
-
-ex cos_diff(ex const & x, unsigned diff_param)
-{
- ASSERT(diff_param==0);
-
- return numMINUSONE()*sin(x);
-}
-
-REGISTER_FUNCTION(cos, cos_eval, cos_evalf, cos_diff, NULL);
+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);
+ 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;
+ if (z.is_equal(_num5)) // 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)) // cos(Pi/6) -> sqrt(3)/2
+ return sign*_ex1_2*sqrt(_ex3);
+ if (z.is_equal(_num12)) // cos(Pi/5) -> sqrt(5)/4+1/4
+ return sign*(_ex1_4*sqrt(_ex5)+_ex1_4);
+ if (z.is_equal(_num15)) // cos(Pi/4) -> sqrt(2)/2
+ return sign*_ex1_2*sqrt(_ex2);
+ 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*sqrt(_ex5)+_ex_1_4);
+ if (z.is_equal(_num25)) // 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)) // 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));
+
+ 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)
//////////
-ex tan_evalf(ex const & x)
-{
- BEGIN_TYPECHECK
- TYPECHECK(x,numeric)
- END_TYPECHECK(tan(x)) // -> numeric tan(numeric)
-
- return tan(ex_to_numeric(x));
-}
-
-ex tan_eval(ex const & x)
-{
- // tan(n*Pi/3) -> {0|3^(1/2)|-(3^(1/2))}
- ex ThreeExOverPi=numTHREE()*x/Pi;
- if (ThreeExOverPi.info(info_flags::integer)) {
- numeric z=mod(ex_to_numeric(ThreeExOverPi),numeric(3));
- if (z.is_equal(numZERO()))
- return exZERO();
- if (z.is_equal(numONE()))
- return power(exTHREE(),exHALF());
- if (z.is_equal(numTWO()))
- return -power(exTHREE(),exHALF());
- }
-
- // tan((2n+1)*Pi/2) -> throw
- ex ExOverPiMinusHalf=x/Pi-exHALF();
- if (ExOverPiMinusHalf.info(info_flags::integer))
- 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(exONE()-power(t,exTWO()),exMINUSHALF());
- // tan(acos(x)) -> (1-x^2)^(1/2)/x
- if (is_ex_the_function(x, acos))
- return power(t,exMINUSONE())*power(exONE()-power(t,exTWO()),exHALF());
- }
-
- // tan(float) -> float
- if (x.info(info_flags::numeric) && !x.info(info_flags::rational)) {
- return tan_evalf(x);
- }
-
- return tan(x).hold();
-}
-
-ex tan_diff(ex const & x, unsigned diff_param)
-{
- ASSERT(diff_param==0);
-
- return (1+power(tan(x),exTWO()));
-}
-
-REGISTER_FUNCTION(tan, tan_eval, tan_evalf, tan_diff, NULL);
+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);
+ 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-sqrt(_ex3));
+ if (z.is_equal(_num10)) // tan(Pi/6) -> sqrt(3)/3
+ return sign*_ex1_3*sqrt(_ex3);
+ if (z.is_equal(_num15)) // tan(Pi/4) -> 1
+ return sign;
+ if (z.is_equal(_num20)) // tan(Pi/3) -> sqrt(3)
+ return sign*sqrt(_ex3);
+ if (z.is_equal(_num25)) // tan(5/12*Pi) -> 2+sqrt(3)
+ return sign*(sqrt(_ex3)+_ex2);
+ if (z.is_equal(_num30)) // 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));
+ }
+
+ 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+2, 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)
//////////
-ex asin_evalf(ex const & x)
+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();
}
-ex asin_eval(ex const & x)
+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(exHALF()))
- return numeric(1,6)*Pi;
- // asin(1) -> Pi/2
- if (x.is_equal(exONE()))
- return numeric(1,2)*Pi;
- // asin(-1/2) -> -Pi/6
- if (x.is_equal(exMINUSHALF()))
- return numeric(-1,6)*Pi;
- // asin(-1) -> -Pi/2
- if (x.is_equal(exMINUSONE()))
- return numeric(-1,2)*Pi;
- // asin(float) -> float
- if (!x.info(info_flags::rational))
- 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 _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(ex_to<numeric>(x));
+ }
+
+ return asin(x).hold();
}
-ex asin_diff(ex const & x, unsigned diff_param)
+static ex asin_deriv(const ex & x, unsigned deriv_param)
{
- ASSERT(diff_param==0);
-
- return power(1-power(x,exTWO()),exMINUSHALF());
+ 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_diff, NULL);
+REGISTER_FUNCTION(asin, eval_func(asin_eval).
+ evalf_func(asin_evalf).
+ derivative_func(asin_deriv).
+ latex_name("\\arcsin"));
//////////
// inverse cosine (arc cosine)
//////////
-ex acos_evalf(ex const & x)
+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();
}
-ex acos_eval(ex const & x)
+static ex acos_eval(const ex & x)
{
- if (x.info(info_flags::numeric)) {
- // acos(1) -> 0
- if (x.is_equal(exONE()))
- return exZERO();
- // acos(1/2) -> Pi/3
- if (x.is_equal(exHALF()))
- return numeric(1,3)*Pi;
- // acos(0) -> Pi/2
- if (x.is_zero())
- return numeric(1,2)*Pi;
- // acos(-1/2) -> 2/3*Pi
- if (x.is_equal(exMINUSHALF()))
- return numeric(2,3)*Pi;
- // acos(-1) -> Pi
- if (x.is_equal(exMINUSONE()))
- return Pi;
- // acos(float) -> float
- if (!x.info(info_flags::rational))
- 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));
+ }
+
+ return acos(x).hold();
}
-ex acos_diff(ex const & x, unsigned diff_param)
+static ex acos_deriv(const ex & x, unsigned deriv_param)
{
- ASSERT(diff_param==0);
-
- return numMINUSONE()*power(1-power(x,exTWO()),exMINUSHALF());
+ 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_diff, NULL);
+REGISTER_FUNCTION(acos, eval_func(acos_eval).
+ evalf_func(acos_evalf).
+ derivative_func(acos_deriv).
+ latex_name("\\arccos"));
//////////
// inverse tangent (arc tangent)
//////////
-ex atan_evalf(ex const & x)
+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();
}
-ex atan_eval(ex const & x)
+static ex atan_eval(const ex & x)
{
- if (x.info(info_flags::numeric)) {
- // atan(0) -> 0
- if (x.is_equal(exZERO()))
- return exZERO();
- // atan(float) -> float
- if (!x.info(info_flags::rational))
- 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));
+ }
+
+ return atan(x).hold();
+}
-ex atan_diff(ex const & x, unsigned diff_param)
+static ex atan_deriv(const ex & x, unsigned deriv_param)
{
- ASSERT(diff_param==0);
+ GINAC_ASSERT(deriv_param==0);
+
+ // d/dx atan(x) -> 1/(1+x^2)
+ return power(_ex1+power(x,_ex2), _ex_1);
+}
- return power(1+x*x, -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_diff, 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))
//////////
-ex atan2_evalf(ex const & y, ex const & 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 atan2(ex_to<numeric>(y), ex_to<numeric>(x));
+
+ return atan2(y, x).hold();
}
-ex atan2_eval(ex const & y, ex const & x)
+static ex atan2_eval(const ex & y, const ex & x)
{
- if (y.info(info_flags::numeric) && !y.info(info_flags::rational) &&
- x.info(info_flags::numeric) && !x.info(info_flags::rational)) {
- return atan2_evalf(y,x);
- }
-
- return atan2(y,x).hold();
+ 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();
}
-ex atan2_diff(ex const & y, ex const & x, unsigned diff_param)
+static ex atan2_deriv(const ex & y, const ex & x, unsigned deriv_param)
{
- ASSERT(diff_param<2);
-
- if (diff_param==0) {
- // d/dy atan(y,x)
- return power(x*(1+y*y/(x*x)),-1);
- }
- // d/dx atan(y,x)
- return -y*power(x*x+y*y,-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_diff, NULL);
+REGISTER_FUNCTION(atan2, eval_func(atan2_eval).
+ evalf_func(atan2_evalf).
+ derivative_func(atan2_deriv));
//////////
// hyperbolic sine (trigonometric function)
//////////
-ex sinh_evalf(ex const & x)
+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();
}
-ex sinh_eval(ex const & x)
+static ex sinh_eval(const ex & x)
{
- if (x.info(info_flags::numeric)) {
- // sinh(0) -> 0
- if (x.is_zero())
- return exZERO();
- // sinh(float) -> float
- if (!x.info(info_flags::rational))
- return sinh_evalf(x);
- }
-
- 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-exONE(),exHALF())*power(t+exONE(),exHALF());
- // sinh(atanh(x)) -> x*(1-x^2)^(-1/2)
- if (is_ex_the_function(x, atanh))
- return t*power(exONE()-power(t,exTWO()),exMINUSHALF());
- }
-
- return sinh(x).hold();
+ 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(ex_to<numeric>(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();
}
-ex sinh_diff(ex const & x, unsigned diff_param)
+static ex sinh_deriv(const ex & x, unsigned deriv_param)
{
- ASSERT(diff_param==0);
-
- 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_diff, NULL);
+REGISTER_FUNCTION(sinh, eval_func(sinh_eval).
+ evalf_func(sinh_evalf).
+ derivative_func(sinh_deriv).
+ latex_name("\\sinh"));
//////////
// hyperbolic cosine (trigonometric function)
//////////
-ex cosh_evalf(ex const & x)
+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();
}
-ex cosh_eval(ex const & x)
+static ex cosh_eval(const ex & x)
{
- if (x.info(info_flags::numeric)) {
- // cosh(0) -> 1
- if (x.is_zero())
- return exONE();
- // cosh(float) -> float
- if (!x.info(info_flags::rational))
- return cosh_evalf(x);
- }
-
- 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(exONE()+power(t,exTWO()),exHALF());
- // cosh(atanh(x)) -> (1-x^2)^(-1/2)
- if (is_ex_the_function(x, atanh))
- return power(exONE()-power(t,exTWO()),exMINUSHALF());
- }
-
- return cosh(x).hold();
+ 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(ex_to<numeric>(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();
}
-ex cosh_diff(ex const & x, unsigned diff_param)
+static ex cosh_deriv(const ex & x, unsigned deriv_param)
{
- ASSERT(diff_param==0);
-
- 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_diff, NULL);
+REGISTER_FUNCTION(cosh, eval_func(cosh_eval).
+ evalf_func(cosh_evalf).
+ derivative_func(cosh_deriv).
+ latex_name("\\cosh"));
//////////
// hyperbolic tangent (trigonometric function)
//////////
-ex tanh_evalf(ex const & x)
+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();
}
-ex tanh_eval(ex const & x)
+static ex tanh_eval(const ex & x)
{
- if (x.info(info_flags::numeric)) {
- // tanh(0) -> 0
- if (x.is_zero())
- return exZERO();
- // tanh(float) -> float
- if (!x.info(info_flags::rational))
- return tanh_evalf(x);
- }
-
- 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(exONE()+power(t,exTWO()),exMINUSHALF());
- // tanh(acosh(x)) -> (x-1)^(1/2)*(x+1)^(1/2)/x
- if (is_ex_the_function(x, acosh))
- return power(t-exONE(),exHALF())*power(t+exONE(),exHALF())*power(t,exMINUSONE());
- }
-
- return tanh(x).hold();
+ 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(ex_to<numeric>(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();
}
-ex tanh_diff(ex const & x, unsigned diff_param)
+static ex tanh_deriv(const ex & x, unsigned deriv_param)
{
- ASSERT(diff_param==0);
-
- return exONE()-power(tanh(x),exTWO());
+ GINAC_ASSERT(deriv_param==0);
+
+ // d/dx tanh(x) -> 1-tanh(x)^2
+ return _ex1-power(tanh(x),_ex2);
}
-REGISTER_FUNCTION(tanh, tanh_eval, tanh_evalf, tanh_diff, NULL);
+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+2, 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)
//////////
-ex asinh_evalf(ex const & x)
+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();
}
-ex asinh_eval(ex const & x)
+static ex asinh_eval(const ex & x)
{
- if (x.info(info_flags::numeric)) {
- // asinh(0) -> 0
- if (x.is_zero())
- return exZERO();
- // asinh(float) -> float
- if (!x.info(info_flags::rational))
- 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));
+ }
+
+ return asinh(x).hold();
}
-ex asinh_diff(ex const & x, unsigned diff_param)
+static ex asinh_deriv(const ex & x, unsigned deriv_param)
{
- ASSERT(diff_param==0);
-
- return power(1+power(x,exTWO()),exMINUSHALF());
+ 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_diff, NULL);
+REGISTER_FUNCTION(asinh, eval_func(asinh_eval).
+ evalf_func(asinh_evalf).
+ derivative_func(asinh_deriv));
//////////
// inverse hyperbolic cosine (trigonometric function)
//////////
-ex acosh_evalf(ex const & x)
+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();
}
-ex acosh_eval(ex const & x)
+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(exONE()))
- return exZERO();
- // acosh(-1) -> Pi*I
- if (x.is_equal(exMINUSONE()))
- return Pi*I;
- // acosh(float) -> float
- if (!x.info(info_flags::rational))
- 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));
+ }
+
+ return acosh(x).hold();
}
-ex acosh_diff(ex const & x, unsigned diff_param)
+static ex acosh_deriv(const ex & x, unsigned deriv_param)
{
- ASSERT(diff_param==0);
-
- return power(x-1,exMINUSHALF())*power(x+1,exMINUSHALF());
+ 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_diff, NULL);
+REGISTER_FUNCTION(acosh, eval_func(acosh_eval).
+ evalf_func(acosh_evalf).
+ derivative_func(acosh_deriv));
//////////
// inverse hyperbolic tangent (trigonometric function)
//////////
-ex atanh_evalf(ex const & x)
+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();
}
-ex atanh_eval(ex const & x)
+static ex atanh_eval(const ex & x)
{
- if (x.info(info_flags::numeric)) {
- // atanh(0) -> 0
- if (x.is_zero())
- return exZERO();
- // atanh({+|-}1) -> throw
- if (x.is_equal(exONE()) || x.is_equal(exONE()))
- throw (std::domain_error("atanh_eval(): infinity"));
- // atanh(float) -> float
- if (!x.info(info_flags::rational))
- 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));
+ }
+
+ return atanh(x).hold();
}
-ex atanh_diff(ex const & x, unsigned diff_param)
+static ex atanh_deriv(const ex & x, unsigned deriv_param)
{
- ASSERT(diff_param==0);
-
- return power(exONE()-power(x,exTWO()),exMINUSONE());
+ 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_diff, 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));
+
+
+} // namespace GiNaC
git://www.ginac.de / ginac.git / blobdiff