X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Finifcns_trans.cpp;h=98e0caff36aaae386448293c36c4371837513f2e;hp=14921f46cccd3eec28084c7d3c82b6f413ca521c;hb=ed914545e01d60ecf2544e6141d6c5142c01327f;hpb=5184d67c0ec1056ac039419e08558632793a4e2c diff --git a/ginac/inifcns_trans.cpp b/ginac/inifcns_trans.cpp index 14921f46..98e0caff 100644 --- a/ginac/inifcns_trans.cpp +++ b/ginac/inifcns_trans.cpp @@ -4,7 +4,7 @@ * functions. */ /* - * GiNaC Copyright (C) 1999-2000 Johannes Gutenberg University Mainz, Germany + * GiNaC Copyright (C) 1999-2016 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 @@ -18,24 +18,26 @@ * * 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 -#include - #include "inifcns.h" #include "ex.h" #include "constant.h" +#include "add.h" +#include "mul.h" #include "numeric.h" #include "power.h" +#include "operators.h" #include "relational.h" #include "symbol.h" +#include "pseries.h" #include "utils.h" -#ifndef NO_NAMESPACE_GINAC +#include +#include + namespace GiNaC { -#endif // ndef NO_NAMESPACE_GINAC ////////// // exponential function @@ -43,54 +45,97 @@ namespace GiNaC { 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(x)) + return exp(ex_to(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(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(x)); + + return exp(x).hold(); +} + +static ex exp_expand(const ex & arg, unsigned options) +{ + ex exp_arg; + if (options & expand_options::expand_function_args) + exp_arg = arg.expand(options); + else + exp_arg=arg; + + if ((options & expand_options::expand_transcendental) + && is_exactly_a(exp_arg)) { + exvector prodseq; + prodseq.reserve(exp_arg.nops()); + for (const_iterator i = exp_arg.begin(); i != exp_arg.end(); ++i) + prodseq.push_back(exp(*i)); + + return dynallocate(prodseq).setflag(status_flags::expanded); + } + + return exp(exp_arg).hold(); } static ex exp_deriv(const ex & x, unsigned deriv_param) { - GINAC_ASSERT(deriv_param==0); + GINAC_ASSERT(deriv_param==0); + + // d/dx exp(x) -> exp(x) + return exp(x); +} + +static ex exp_real_part(const ex & x) +{ + return exp(GiNaC::real_part(x))*cos(GiNaC::imag_part(x)); +} + +static ex exp_imag_part(const ex & x) +{ + return exp(GiNaC::real_part(x))*sin(GiNaC::imag_part(x)); +} - // d/dx exp(x) -> exp(x) - return exp(x); +static ex exp_conjugate(const ex & x) +{ + // conjugate(exp(x))==exp(conjugate(x)) + return exp(x.conjugate()); } REGISTER_FUNCTION(exp, eval_func(exp_eval). evalf_func(exp_evalf). - derivative_func(exp_deriv)); + expand_func(exp_expand). + derivative_func(exp_deriv). + real_part_func(exp_real_part). + imag_part_func(exp_imag_part). + conjugate_func(exp_conjugate). + latex_name("\\exp")); ////////// // natural logarithm @@ -98,54 +143,223 @@ REGISTER_FUNCTION(exp, eval_func(exp_eval). 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(x)) + return log(ex_to(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(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); +} + +static ex log_series(const ex &arg, + const relational &rel, + int order, + unsigned options) +{ + GINAC_ASSERT(is_a(rel.lhs())); + ex arg_pt; + bool must_expand_arg = false; + // maybe substitution of rel into arg fails because of a pole + try { + arg_pt = arg.subs(rel, subs_options::no_pattern); + } catch (pole_error) { + must_expand_arg = true; + } + // or we are at the branch point anyways + if (arg_pt.is_zero()) + must_expand_arg = true; + + if (arg.diff(ex_to(rel.lhs())).is_zero()) { + throw do_taylor(); + } + + if (must_expand_arg) { + // method: + // This is the branch point: Series expand the argument first, then + // trivially factorize it to isolate that part which has constant + // leading coefficient in this fashion: + // x^n + x^(n+1) +...+ Order(x^(n+m)) -> x^n * (1 + x +...+ Order(x^m)). + // Return a plain n*log(x) for the x^n part and series expand the + // other part. Add them together and reexpand again in order to have + // one unnested pseries object. All this also works for negative n. + pseries argser; // series expansion of log's argument + unsigned extra_ord = 0; // extra expansion order + do { + // oops, the argument expanded to a pure Order(x^something)... + argser = ex_to(arg.series(rel, order+extra_ord, options)); + ++extra_ord; + } while (!argser.is_terminating() && argser.nops()==1); + + const symbol &s = ex_to(rel.lhs()); + const ex &point = rel.rhs(); + const int n = argser.ldegree(s); + epvector seq; + // construct what we carelessly called the n*log(x) term above + const ex coeff = argser.coeff(s, n); + // expand the log, but only if coeff is real and > 0, since otherwise + // it would make the branch cut run into the wrong direction + if (coeff.info(info_flags::positive)) + seq.push_back(expair(n*log(s-point)+log(coeff), _ex0)); + else + seq.push_back(expair(log(coeff*pow(s-point, n)), _ex0)); + + if (!argser.is_terminating() || argser.nops()!=1) { + // in this case n more (or less) terms are needed + // (sadly, to generate them, we have to start from the beginning) + if (n == 0 && coeff == 1) { + ex rest = pseries(rel, epvector{expair(-1, _ex0), expair(Order(_ex1), order)}).add_series(argser); + ex acc = dynallocate(rel, epvector()); + for (int i = order-1; i>0; --i) { + epvector cterm { expair(i%2 ? _ex1/i : _ex_1/i, _ex0) }; + acc = pseries(rel, std::move(cterm)).add_series(ex_to(acc)); + acc = (ex_to(rest)).mul_series(ex_to(acc)); + } + return acc; + } + const ex newarg = ex_to((arg/coeff).series(rel, order+n, options)).shift_exponents(-n).convert_to_poly(true); + return pseries(rel, std::move(seq)).add_series(ex_to(log(newarg).series(rel, order, options))); + } else // it was a monomial + return pseries(rel, std::move(seq)); + } + if (!(options & series_options::suppress_branchcut) && + arg_pt.info(info_flags::negative)) { + // method: + // This is the branch cut: assemble the primitive series manually and + // then add the corresponding complex step function. + const symbol &s = ex_to(rel.lhs()); + const ex &point = rel.rhs(); + const symbol foo; + const ex replarg = series(log(arg), s==foo, order).subs(foo==point, subs_options::no_pattern); + epvector seq; + if (order > 0) { + seq.reserve(2); + seq.push_back(expair(-I*csgn(arg*I)*Pi, _ex0)); + } + seq.push_back(expair(Order(_ex1), order)); + return series(replarg - I*Pi + pseries(rel, std::move(seq)), rel, order); + } + throw do_taylor(); // caught by function::series() +} - // d/dx log(x) -> 1/x - return power(x, _ex_1()); +static ex log_real_part(const ex & x) +{ + if (x.info(info_flags::nonnegative)) + return log(x).hold(); + return log(abs(x)); +} + +static ex log_imag_part(const ex & x) +{ + if (x.info(info_flags::nonnegative)) + return 0; + return atan2(GiNaC::imag_part(x), GiNaC::real_part(x)); +} + +static ex log_expand(const ex & arg, unsigned options) +{ + if ((options & expand_options::expand_transcendental) + && is_exactly_a(arg) && !arg.info(info_flags::indefinite)) { + exvector sumseq; + exvector prodseq; + sumseq.reserve(arg.nops()); + prodseq.reserve(arg.nops()); + bool possign=true; + + // searching for positive/negative factors + for (const_iterator i = arg.begin(); i != arg.end(); ++i) { + ex e; + if (options & expand_options::expand_function_args) + e=i->expand(options); + else + e=*i; + if (e.info(info_flags::positive)) + sumseq.push_back(log(e)); + else if (e.info(info_flags::negative)) { + sumseq.push_back(log(-e)); + possign = !possign; + } else + prodseq.push_back(e); + } + + if (sumseq.size() > 0) { + ex newarg; + if (options & expand_options::expand_function_args) + newarg=((possign?_ex1:_ex_1)*mul(prodseq)).expand(options); + else { + newarg=(possign?_ex1:_ex_1)*mul(prodseq); + ex_to(newarg).setflag(status_flags::purely_indefinite); + } + return add(sumseq)+log(newarg); + } else { + if (!(options & expand_options::expand_function_args)) + ex_to(arg).setflag(status_flags::purely_indefinite); + } + } + + if (options & expand_options::expand_function_args) + return log(arg.expand(options)).hold(); + else + return log(arg).hold(); +} + +static ex log_conjugate(const ex & x) +{ + // conjugate(log(x))==log(conjugate(x)) unless on the branch cut which + // runs along the negative real axis. + if (x.info(info_flags::positive)) { + return log(x); + } + if (is_exactly_a(x) && + !x.imag_part().is_zero()) { + return log(x.conjugate()); + } + return conjugate_function(log(x)).hold(); } REGISTER_FUNCTION(log, eval_func(log_eval). evalf_func(log_evalf). - derivative_func(log_deriv)); + expand_func(log_expand). + derivative_func(log_deriv). + series_func(log_series). + real_part_func(log_real_part). + imag_part_func(log_imag_part). + conjugate_func(log_conjugate). + latex_name("\\ln")); ////////// // sine (trigonometric function) @@ -153,80 +367,106 @@ REGISTER_FUNCTION(log, eval_func(log_eval). 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(x)) + return sin(ex_to(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(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(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(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); +} + +static ex sin_real_part(const ex & x) +{ + return cosh(GiNaC::imag_part(x))*sin(GiNaC::real_part(x)); +} + +static ex sin_imag_part(const ex & x) +{ + return sinh(GiNaC::imag_part(x))*cos(GiNaC::real_part(x)); +} + +static ex sin_conjugate(const ex & x) +{ + // conjugate(sin(x))==sin(conjugate(x)) + return sin(x.conjugate()); } REGISTER_FUNCTION(sin, eval_func(sin_eval). evalf_func(sin_evalf). - derivative_func(sin_deriv)); + derivative_func(sin_deriv). + real_part_func(sin_real_part). + imag_part_func(sin_imag_part). + conjugate_func(sin_conjugate). + latex_name("\\sin")); ////////// // cosine (trigonometric function) @@ -234,80 +474,106 @@ REGISTER_FUNCTION(sin, eval_func(sin_eval). 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(x)) + return cos(ex_to(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(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(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(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); +} + +static ex cos_real_part(const ex & x) +{ + return cosh(GiNaC::imag_part(x))*cos(GiNaC::real_part(x)); +} + +static ex cos_imag_part(const ex & x) +{ + return -sinh(GiNaC::imag_part(x))*sin(GiNaC::real_part(x)); +} + +static ex cos_conjugate(const ex & x) +{ + // conjugate(cos(x))==cos(conjugate(x)) + return cos(x.conjugate()); } REGISTER_FUNCTION(cos, eval_func(cos_eval). evalf_func(cos_evalf). - derivative_func(cos_deriv)); + derivative_func(cos_deriv). + real_part_func(cos_real_part). + imag_part_func(cos_imag_part). + conjugate_func(cos_conjugate). + latex_name("\\cos")); ////////// // tangent (trigonometric function) @@ -315,90 +581,124 @@ REGISTER_FUNCTION(cos, eval_func(cos_eval). 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(x)) + return tan(ex_to(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(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(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(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_real_part(const ex & x) +{ + ex a = GiNaC::real_part(x); + ex b = GiNaC::imag_part(x); + return tan(a)/(1+power(tan(a),2)*power(tan(b),2)); +} + +static ex tan_imag_part(const ex & x) +{ + ex a = GiNaC::real_part(x); + ex b = GiNaC::imag_part(x); + return tanh(b)/(1+power(tan(a),2)*power(tan(b),2)); } -static ex tan_series(const ex & x, const 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(rel.lhs())); + // method: + // Taylor series where there is no pole falls back to tan_deriv. + // On a pole simply expand sin(x)/cos(x). + const ex x_pt = x.subs(rel, subs_options::no_pattern); + if (!(2*x_pt/Pi).info(info_flags::odd)) + throw do_taylor(); // caught by function::series() + // if we got here we have to care for a simple pole + return (sin(x)/cos(x)).series(rel, order, options); +} + +static ex tan_conjugate(const ex & x) +{ + // conjugate(tan(x))==tan(conjugate(x)) + return tan(x.conjugate()); } REGISTER_FUNCTION(tan, eval_func(tan_eval). evalf_func(tan_evalf). derivative_func(tan_deriv). - series_func(tan_series)); + series_func(tan_series). + real_part_func(tan_real_part). + imag_part_func(tan_imag_part). + conjugate_func(tan_conjugate). + latex_name("\\tan")); ////////// // inverse sine (arc sine) @@ -406,50 +706,72 @@ REGISTER_FUNCTION(tan, eval_func(tan_eval). 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(x)) + return asin(ex_to(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(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); +} + +static ex asin_conjugate(const ex & x) +{ + // conjugate(asin(x))==asin(conjugate(x)) unless on the branch cuts which + // run along the real axis outside the interval [-1, +1]. + if (is_exactly_a(x) && + (!x.imag_part().is_zero() || (x > *_num_1_p && x < *_num1_p))) { + return asin(x.conjugate()); + } + return conjugate_function(asin(x)).hold(); } REGISTER_FUNCTION(asin, eval_func(asin_eval). evalf_func(asin_evalf). - derivative_func(asin_deriv)); + derivative_func(asin_deriv). + conjugate_func(asin_conjugate). + latex_name("\\arcsin")); ////////// // inverse cosine (arc cosine) @@ -457,50 +779,72 @@ REGISTER_FUNCTION(asin, eval_func(asin_eval). 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(x)) + return acos(ex_to(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(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); +} + +static ex acos_conjugate(const ex & x) +{ + // conjugate(acos(x))==acos(conjugate(x)) unless on the branch cuts which + // run along the real axis outside the interval [-1, +1]. + if (is_exactly_a(x) && + (!x.imag_part().is_zero() || (x > *_num_1_p && x < *_num1_p))) { + return acos(x.conjugate()); + } + return conjugate_function(acos(x)).hold(); } REGISTER_FUNCTION(acos, eval_func(acos_eval). evalf_func(acos_evalf). - derivative_func(acos_deriv)); + derivative_func(acos_deriv). + conjugate_func(acos_conjugate). + latex_name("\\arccos")); ////////// // inverse tangent (arc tangent) @@ -508,73 +852,213 @@ REGISTER_FUNCTION(acos, eval_func(acos_eval). 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(x)) + return atan(ex_to(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(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(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(rel.lhs()); + const ex &point = rel.rhs(); + const symbol foo; + const ex replarg = series(atan(arg), s==foo, order).subs(foo==point, subs_options::no_pattern); + ex Order0correction = replarg.op(0)+csgn(arg)*Pi*_ex_1_2; + if ((I*arg_pt)<_ex0) + Order0correction += log((I*arg_pt+_ex_1)/(I*arg_pt+_ex1))*I*_ex_1_2; + else + Order0correction += log((I*arg_pt+_ex1)/(I*arg_pt+_ex_1))*I*_ex1_2; + epvector seq; + if (order > 0) { + seq.reserve(2); + seq.push_back(expair(Order0correction, _ex0)); + } + seq.push_back(expair(Order(_ex1), order)); + return series(replarg - pseries(rel, std::move(seq)), rel, order); + } + throw do_taylor(); +} + +static ex atan_conjugate(const ex & x) +{ + // conjugate(atan(x))==atan(conjugate(x)) unless on the branch cuts which + // run along the imaginary axis outside the interval [-I, +I]. + if (x.info(info_flags::real)) + return atan(x); + if (is_exactly_a(x)) { + const numeric x_re = ex_to(x.real_part()); + const numeric x_im = ex_to(x.imag_part()); + if (!x_re.is_zero() || + (x_im > *_num_1_p && x_im < *_num1_p)) + return atan(x.conjugate()); + } + return conjugate_function(atan(x)).hold(); } REGISTER_FUNCTION(atan, eval_func(atan_eval). evalf_func(atan_evalf). - derivative_func(atan_deriv)); + derivative_func(atan_deriv). + series_func(atan_series). + conjugate_func(atan_conjugate). + latex_name("\\arctan")); ////////// // inverse tangent (atan2(y,x)) ////////// -static ex atan2_evalf(const ex & y, const ex & x) +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(y) && is_exactly_a(x)) + return atan(ex_to(y), ex_to(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.is_zero()) { + + // atan2(0, 0) -> 0 + if (x.is_zero()) + return _ex0; + + // atan2(0, x), x real and positive -> 0 + if (x.info(info_flags::positive)) + return _ex0; + + // atan2(0, x), x real and negative -> Pi + if (x.info(info_flags::negative)) + return Pi; + } + + if (x.is_zero()) { + + // atan2(y, 0), y real and positive -> Pi/2 + if (y.info(info_flags::positive)) + return _ex1_2*Pi; + + // atan2(y, 0), y real and negative -> -Pi/2 + if (y.info(info_flags::negative)) + return _ex_1_2*Pi; + } + + if (y.is_equal(x)) { + + // atan2(y, y), y real and positive -> Pi/4 + if (y.info(info_flags::positive)) + return _ex1_4*Pi; + + // atan2(y, y), y real and negative -> -3/4*Pi + if (y.info(info_flags::negative)) + return numeric(-3, 4)*Pi; + } + + if (y.is_equal(-x)) { + + // atan2(y, -y), y real and positive -> 3*Pi/4 + if (y.info(info_flags::positive)) + return numeric(3, 4)*Pi; + + // atan2(y, -y), y real and negative -> -Pi/4 + if (y.info(info_flags::negative)) + return _ex_1_4*Pi; + } + + // atan2(float, float) -> float + if (is_a(y) && !y.info(info_flags::crational) && + is_a(x) && !x.info(info_flags::crational)) + return atan(ex_to(y), ex_to(x)); + + // atan2(real, real) -> atan(y/x) +/- Pi + if (y.info(info_flags::real) && x.info(info_flags::real)) { + if (x.info(info_flags::positive)) + return atan(y/x); + + if (x.info(info_flags::negative)) { + if (y.info(info_flags::positive)) + return atan(y/x)+Pi; + if (y.info(info_flags::negative)) + return atan(y/x)-Pi; + } + } + + return atan2(y, x).hold(); } static ex atan2_deriv(const ex & y, const ex & x, unsigned deriv_param) { - GINAC_ASSERT(deriv_param<2); - - if (deriv_param==0) { - // d/dy 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 atan2(y,x) + return x*power(power(x,_ex2)+power(y,_ex2),_ex_1); + } + // d/dx atan2(y,x) + return -y*power(power(x,_ex2)+power(y,_ex2),_ex_1); } REGISTER_FUNCTION(atan2, eval_func(atan2_eval). @@ -587,53 +1071,83 @@ REGISTER_FUNCTION(atan2, eval_func(atan2_eval). 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(x)) + return sinh(ex_to(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(x)); + + // sinh() is odd + if (x.info(info_flags::negative)) + return -sinh(-x); + } + + if ((x/Pi).info(info_flags::numeric) && + ex_to(x/Pi).real().is_zero()) // sinh(I*x) -> I*sin(x) + return I*sin(x/I); + + if (is_exactly_a(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); +} + +static ex sinh_real_part(const ex & x) +{ + return sinh(GiNaC::real_part(x))*cos(GiNaC::imag_part(x)); +} + +static ex sinh_imag_part(const ex & x) +{ + return cosh(GiNaC::real_part(x))*sin(GiNaC::imag_part(x)); +} + +static ex sinh_conjugate(const ex & x) +{ + // conjugate(sinh(x))==sinh(conjugate(x)) + return sinh(x.conjugate()); } REGISTER_FUNCTION(sinh, eval_func(sinh_eval). evalf_func(sinh_evalf). - derivative_func(sinh_deriv)); + derivative_func(sinh_deriv). + real_part_func(sinh_real_part). + imag_part_func(sinh_imag_part). + conjugate_func(sinh_conjugate). + latex_name("\\sinh")); ////////// // hyperbolic cosine (trigonometric function) @@ -641,54 +1155,83 @@ REGISTER_FUNCTION(sinh, eval_func(sinh_eval). 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(x)) + return cosh(ex_to(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(x)); + + // cosh() is even + if (x.info(info_flags::negative)) + return cosh(-x); + } + + if ((x/Pi).info(info_flags::numeric) && + ex_to(x/Pi).real().is_zero()) // cosh(I*x) -> cos(x) + return cos(x/I); + + if (is_exactly_a(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); +} + +static ex cosh_real_part(const ex & x) +{ + return cosh(GiNaC::real_part(x))*cos(GiNaC::imag_part(x)); +} + +static ex cosh_imag_part(const ex & x) +{ + return sinh(GiNaC::real_part(x))*sin(GiNaC::imag_part(x)); +} + +static ex cosh_conjugate(const ex & x) +{ + // conjugate(cosh(x))==cosh(conjugate(x)) + return cosh(x.conjugate()); } REGISTER_FUNCTION(cosh, eval_func(cosh_eval). evalf_func(cosh_evalf). - derivative_func(cosh_deriv)); - + derivative_func(cosh_deriv). + real_part_func(cosh_real_part). + imag_part_func(cosh_imag_part). + conjugate_func(cosh_conjugate). + latex_name("\\cosh")); ////////// // hyperbolic tangent (trigonometric function) @@ -696,66 +1239,104 @@ REGISTER_FUNCTION(cosh, eval_func(cosh_eval). 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(x)) + return tanh(ex_to(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(x)); + + // tanh() is odd + if (x.info(info_flags::negative)) + return -tanh(-x); + } + + if ((x/Pi).info(info_flags::numeric) && + ex_to(x/Pi).real().is_zero()) // tanh(I*x) -> I*tan(x); + return I*tan(x/I); + + if (is_exactly_a(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 relational &rel, + int order, + unsigned options) +{ + GINAC_ASSERT(is_a(rel.lhs())); + // method: + // Taylor series where there is no pole falls back to tanh_deriv. + // On a pole simply expand sinh(x)/cosh(x). + const ex x_pt = x.subs(rel, subs_options::no_pattern); + if (!(2*I*x_pt/Pi).info(info_flags::odd)) + throw do_taylor(); // caught by function::series() + // if we got here we have to care for a simple pole + return (sinh(x)/cosh(x)).series(rel, order, options); +} + +static ex tanh_real_part(const ex & x) +{ + ex a = GiNaC::real_part(x); + ex b = GiNaC::imag_part(x); + return tanh(a)/(1+power(tanh(a),2)*power(tan(b),2)); } -static ex tanh_series(const ex & x, const symbol & s, const ex & pt, int order) +static ex tanh_imag_part(const ex & x) { - // 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); + ex a = GiNaC::real_part(x); + ex b = GiNaC::imag_part(x); + return tan(b)/(1+power(tanh(a),2)*power(tan(b),2)); +} + +static ex tanh_conjugate(const ex & x) +{ + // conjugate(tanh(x))==tanh(conjugate(x)) + return tanh(x.conjugate()); } REGISTER_FUNCTION(tanh, eval_func(tanh_eval). evalf_func(tanh_evalf). derivative_func(tanh_deriv). - series_func(tanh_series)); + series_func(tanh_series). + real_part_func(tanh_real_part). + imag_part_func(tanh_imag_part). + conjugate_func(tanh_conjugate). + latex_name("\\tanh")); ////////// // inverse hyperbolic sine (trigonometric function) @@ -763,38 +1344,60 @@ REGISTER_FUNCTION(tanh, eval_func(tanh_eval). 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(x)) + return asinh(ex_to(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(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); +} + +static ex asinh_conjugate(const ex & x) +{ + // conjugate(asinh(x))==asinh(conjugate(x)) unless on the branch cuts which + // run along the imaginary axis outside the interval [-I, +I]. + if (x.info(info_flags::real)) + return asinh(x); + if (is_exactly_a(x)) { + const numeric x_re = ex_to(x.real_part()); + const numeric x_im = ex_to(x.imag_part()); + if (!x_re.is_zero() || + (x_im > *_num_1_p && x_im < *_num1_p)) + return asinh(x.conjugate()); + } + return conjugate_function(asinh(x)).hold(); } REGISTER_FUNCTION(asinh, eval_func(asinh_eval). evalf_func(asinh_evalf). - derivative_func(asinh_deriv)); + derivative_func(asinh_deriv). + conjugate_func(asinh_conjugate)); ////////// // inverse hyperbolic cosine (trigonometric function) @@ -802,44 +1405,63 @@ REGISTER_FUNCTION(asinh, eval_func(asinh_eval). 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(x)) + return acosh(ex_to(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(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); +} + +static ex acosh_conjugate(const ex & x) +{ + // conjugate(acosh(x))==acosh(conjugate(x)) unless on the branch cut + // which runs along the real axis from +1 to -inf. + if (is_exactly_a(x) && + (!x.imag_part().is_zero() || x > *_num1_p)) { + return acosh(x.conjugate()); + } + return conjugate_function(acosh(x)).hold(); } REGISTER_FUNCTION(acosh, eval_func(acosh_eval). evalf_func(acosh_evalf). - derivative_func(acosh_deriv)); + derivative_func(acosh_deriv). + conjugate_func(acosh_conjugate)); ////////// // inverse hyperbolic tangent (trigonometric function) @@ -847,42 +1469,106 @@ REGISTER_FUNCTION(acosh, eval_func(acosh_eval). 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(x)) + return atanh(ex_to(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(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); +} + +static ex atanh_series(const ex &arg, + const relational &rel, + int order, + unsigned options) +{ + GINAC_ASSERT(is_a(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(rel.lhs()); + const ex &point = rel.rhs(); + const symbol foo; + const ex replarg = series(atanh(arg), s==foo, order).subs(foo==point, subs_options::no_pattern); + ex Order0correction = replarg.op(0)+csgn(I*arg)*Pi*I*_ex1_2; + if (arg_pt<_ex0) + Order0correction += log((arg_pt+_ex_1)/(arg_pt+_ex1))*_ex1_2; + else + Order0correction += log((arg_pt+_ex1)/(arg_pt+_ex_1))*_ex_1_2; + epvector seq; + if (order > 0) { + seq.reserve(2); + seq.push_back(expair(Order0correction, _ex0)); + } + seq.push_back(expair(Order(_ex1), order)); + return series(replarg - pseries(rel, std::move(seq)), rel, order); + } + throw do_taylor(); +} + +static ex atanh_conjugate(const ex & x) +{ + // conjugate(atanh(x))==atanh(conjugate(x)) unless on the branch cuts which + // run along the real axis outside the interval [-1, +1]. + if (is_exactly_a(x) && + (!x.imag_part().is_zero() || (x > *_num_1_p && x < *_num1_p))) { + return atanh(x.conjugate()); + } + return conjugate_function(atanh(x)).hold(); } REGISTER_FUNCTION(atanh, eval_func(atanh_eval). evalf_func(atanh_evalf). - derivative_func(atanh_deriv)); + derivative_func(atanh_deriv). + series_func(atanh_series). + conjugate_func(atanh_conjugate)); + -#ifndef NO_NAMESPACE_GINAC } // namespace GiNaC -#endif // ndef NO_NAMESPACE_GINAC