X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Finifcns_trans.cpp;h=682a981b772f3bda2854bf187698980b995a7084;hp=f737a9ec144f97052cd50a44774149e2393b3e85;hb=9e80b0d339d1ce83f51e0eb5fb101c41f23f6a71;hpb=da64e515abf7243bc4c84ca3631470931c4e6691 diff --git a/ginac/inifcns_trans.cpp b/ginac/inifcns_trans.cpp index f737a9ec..682a981b 100644 --- a/ginac/inifcns_trans.cpp +++ b/ginac/inifcns_trans.cpp @@ -4,7 +4,7 @@ * functions. */ /* - * GiNaC Copyright (C) 1999-2005 Johannes Gutenberg University Mainz, Germany + * GiNaC Copyright (C) 1999-2011 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 @@ -21,12 +21,11 @@ * 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" @@ -35,6 +34,9 @@ #include "pseries.h" #include "utils.h" +#include +#include + namespace GiNaC { ////////// @@ -59,14 +61,14 @@ static ex exp_eval(const ex & x) // 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); - if (z.is_equal(_num0)) + const numeric z = mod(ex_to(TwoExOverPiI),*_num4_p); + if (z.is_equal(*_num0_p)) return _ex1; - if (z.is_equal(_num1)) + if (z.is_equal(*_num1_p)) return ex(I); - if (z.is_equal(_num2)) + if (z.is_equal(*_num2_p)) return _ex_1; - if (z.is_equal(_num3)) + if (z.is_equal(*_num3_p)) return ex(-I); } @@ -81,6 +83,27 @@ static ex exp_eval(const ex & 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 (new mul(prodseq))->setflag(status_flags::dynallocated | status_flags::expanded); + } + + return exp(exp_arg).hold(); +} + static ex exp_deriv(const ex & x, unsigned deriv_param) { GINAC_ASSERT(deriv_param==0); @@ -89,9 +112,29 @@ static ex exp_deriv(const ex & x, unsigned deriv_param) return exp(x); } +static ex exp_real_part(const ex & x) +{ + return exp(GiNaC::real_part(x))*cos(GiNaC::imag_part(x)); +} + +static ex exp_imag_part(const ex & x) +{ + return exp(GiNaC::real_part(x))*sin(GiNaC::imag_part(x)); +} + +static ex exp_conjugate(const ex & x) +{ + // conjugate(exp(x))==exp(conjugate(x)) + return exp(x.conjugate()); +} + REGISTER_FUNCTION(exp, eval_func(exp_eval). evalf_func(exp_evalf). + expand_func(exp_expand). derivative_func(exp_deriv). + real_part_func(exp_real_part). + imag_part_func(exp_imag_part). + conjugate_func(exp_conjugate). latex_name("\\exp")); ////////// @@ -116,9 +159,9 @@ static ex log_eval(const ex & x) 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); + return (Pi*I*_ex1_2); if (x.is_equal(-I)) // log(-I) -> -Pi*I/2 - return (Pi*I*_num_1_2); + return (Pi*I*_ex_1_2); // log(float) -> float if (!x.info(info_flags::crational)) @@ -128,16 +171,10 @@ static ex log_eval(const ex & 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(t) && t.info(info_flags::real)) { + if (t.info(info_flags::real)) return t; - } - if (t.info(info_flags::numeric)) { - const numeric &nt = ex_to(t); - if (nt.is_real()) - return t; - } } - + return log(x).hold(); } @@ -167,6 +204,10 @@ static ex log_series(const ex &arg, 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 @@ -238,10 +279,89 @@ static ex log_series(const ex &arg, throw do_taylor(); // caught by function::series() } +static ex log_real_part(const ex & x) +{ + if (x.info(info_flags::nonnegative)) + return log(x).hold(); + return log(abs(x)); +} + +static ex log_imag_part(const ex & x) +{ + if (x.info(info_flags::nonnegative)) + return 0; + return atan2(GiNaC::imag_part(x), GiNaC::real_part(x)); +} + +static ex log_expand(const ex & arg, unsigned options) +{ + if ((options & expand_options::expand_transcendental) + && is_exactly_a(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). + 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")); ////////// @@ -262,33 +382,33 @@ static ex sin_eval(const ex & x) const ex SixtyExOverPi = _ex60*x/Pi; ex sign = _ex1; if (SixtyExOverPi.info(info_flags::integer)) { - numeric z = mod(ex_to(SixtyExOverPi),_num120); - if (z>=_num60) { + numeric z = mod(ex_to(SixtyExOverPi),*_num120_p); + if (z>=*_num60_p) { // wrap to interval [0, Pi) - z -= _num60; + z -= *_num60_p; sign = _ex_1; } - if (z>_num30) { + if (z>*_num30_p) { // wrap to interval [0, Pi/2) - z = _num60-z; + z = *_num60_p-z; } - if (z.is_equal(_num0)) // sin(0) -> 0 + if (z.is_equal(*_num0_p)) // sin(0) -> 0 return _ex0; - if (z.is_equal(_num5)) // sin(Pi/12) -> sqrt(6)/4*(1-sqrt(3)/3) + 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)) // sin(Pi/10) -> sqrt(5)/4-1/4 + 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)) // sin(Pi/6) -> 1/2 + if (z.is_equal(*_num10_p)) // sin(Pi/6) -> 1/2 return sign*_ex1_2; - if (z.is_equal(_num15)) // sin(Pi/4) -> sqrt(2)/2 + if (z.is_equal(*_num15_p)) // 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 + 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)) // sin(Pi/3) -> sqrt(3)/2 + if (z.is_equal(*_num20_p)) // 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) + 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)) // sin(Pi/2) -> 1 + if (z.is_equal(*_num30_p)) // sin(Pi/2) -> 1 return sign; } @@ -327,9 +447,28 @@ static ex sin_deriv(const ex & x, unsigned deriv_param) return cos(x); } +static ex sin_real_part(const ex & x) +{ + return cosh(GiNaC::imag_part(x))*sin(GiNaC::real_part(x)); +} + +static ex sin_imag_part(const ex & x) +{ + return sinh(GiNaC::imag_part(x))*cos(GiNaC::real_part(x)); +} + +static ex sin_conjugate(const ex & x) +{ + // conjugate(sin(x))==sin(conjugate(x)) + return sin(x.conjugate()); +} + REGISTER_FUNCTION(sin, eval_func(sin_eval). evalf_func(sin_evalf). derivative_func(sin_deriv). + real_part_func(sin_real_part). + imag_part_func(sin_imag_part). + conjugate_func(sin_conjugate). latex_name("\\sin")); ////////// @@ -350,33 +489,33 @@ static ex cos_eval(const ex & x) const ex SixtyExOverPi = _ex60*x/Pi; ex sign = _ex1; if (SixtyExOverPi.info(info_flags::integer)) { - numeric z = mod(ex_to(SixtyExOverPi),_num120); - if (z>=_num60) { + numeric z = mod(ex_to(SixtyExOverPi),*_num120_p); + if (z>=*_num60_p) { // wrap to interval [0, Pi) - z = _num120-z; + z = *_num120_p-z; } - if (z>=_num30) { + if (z>=*_num30_p) { // wrap to interval [0, Pi/2) - z = _num60-z; + z = *_num60_p-z; sign = _ex_1; } - if (z.is_equal(_num0)) // cos(0) -> 1 + if (z.is_equal(*_num0_p)) // cos(0) -> 1 return sign; - if (z.is_equal(_num5)) // cos(Pi/12) -> sqrt(6)/4*(1+sqrt(3)/3) + 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)) // cos(Pi/6) -> sqrt(3)/2 + if (z.is_equal(*_num10_p)) // 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 + 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)) // cos(Pi/4) -> sqrt(2)/2 + if (z.is_equal(*_num15_p)) // cos(Pi/4) -> sqrt(2)/2 return sign*_ex1_2*sqrt(_ex2); - if (z.is_equal(_num20)) // cos(Pi/3) -> 1/2 + if (z.is_equal(*_num20_p)) // cos(Pi/3) -> 1/2 return sign*_ex1_2; - if (z.is_equal(_num24)) // cos(2/5*Pi) -> sqrt(5)/4-1/4x + 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)) // cos(5/12*Pi) -> sqrt(6)/4*(1-sqrt(3)/3) + 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)) // cos(Pi/2) -> 0 + if (z.is_equal(*_num30_p)) // cos(Pi/2) -> 0 return _ex0; } @@ -415,9 +554,28 @@ static ex cos_deriv(const ex & x, unsigned deriv_param) return -sin(x); } +static ex cos_real_part(const ex & x) +{ + return cosh(GiNaC::imag_part(x))*cos(GiNaC::real_part(x)); +} + +static ex cos_imag_part(const ex & x) +{ + return -sinh(GiNaC::imag_part(x))*sin(GiNaC::real_part(x)); +} + +static ex cos_conjugate(const ex & x) +{ + // conjugate(cos(x))==cos(conjugate(x)) + return cos(x.conjugate()); +} + REGISTER_FUNCTION(cos, eval_func(cos_eval). evalf_func(cos_evalf). derivative_func(cos_deriv). + real_part_func(cos_real_part). + imag_part_func(cos_imag_part). + conjugate_func(cos_conjugate). latex_name("\\cos")); ////////// @@ -438,29 +596,29 @@ static ex tan_eval(const ex & x) const ex SixtyExOverPi = _ex60*x/Pi; ex sign = _ex1; if (SixtyExOverPi.info(info_flags::integer)) { - numeric z = mod(ex_to(SixtyExOverPi),_num60); - if (z>=_num60) { + numeric z = mod(ex_to(SixtyExOverPi),*_num60_p); + if (z>=*_num60_p) { // wrap to interval [0, Pi) - z -= _num60; + z -= *_num60_p; } - if (z>=_num30) { + if (z>=*_num30_p) { // wrap to interval [0, Pi/2) - z = _num60-z; + z = *_num60_p-z; sign = _ex_1; } - if (z.is_equal(_num0)) // tan(0) -> 0 + if (z.is_equal(*_num0_p)) // tan(0) -> 0 return _ex0; - if (z.is_equal(_num5)) // tan(Pi/12) -> 2-sqrt(3) + if (z.is_equal(*_num5_p)) // tan(Pi/12) -> 2-sqrt(3) return sign*(_ex2-sqrt(_ex3)); - if (z.is_equal(_num10)) // tan(Pi/6) -> sqrt(3)/3 + if (z.is_equal(*_num10_p)) // tan(Pi/6) -> sqrt(3)/3 return sign*_ex1_3*sqrt(_ex3); - if (z.is_equal(_num15)) // tan(Pi/4) -> 1 + if (z.is_equal(*_num15_p)) // tan(Pi/4) -> 1 return sign; - if (z.is_equal(_num20)) // tan(Pi/3) -> sqrt(3) + if (z.is_equal(*_num20_p)) // tan(Pi/3) -> sqrt(3) return sign*sqrt(_ex3); - if (z.is_equal(_num25)) // tan(5/12*Pi) -> 2+sqrt(3) + if (z.is_equal(*_num25_p)) // tan(5/12*Pi) -> 2+sqrt(3) return sign*(sqrt(_ex3)+_ex2); - if (z.is_equal(_num30)) // tan(Pi/2) -> infinity + if (z.is_equal(*_num30_p)) // tan(Pi/2) -> infinity throw (pole_error("tan_eval(): simple pole",1)); } @@ -500,6 +658,20 @@ static ex tan_deriv(const ex & x, unsigned deriv_param) return (_ex1+power(tan(x),_ex2)); } +static ex tan_real_part(const ex & x) +{ + ex a = GiNaC::real_part(x); + ex b = GiNaC::imag_part(x); + return tan(a)/(1+power(tan(a),2)*power(tan(b),2)); +} + +static ex tan_imag_part(const ex & x) +{ + ex a = GiNaC::real_part(x); + ex b = GiNaC::imag_part(x); + return tanh(b)/(1+power(tan(a),2)*power(tan(b),2)); +} + static ex tan_series(const ex &x, const relational &rel, int order, @@ -516,10 +688,19 @@ static ex tan_series(const ex &x, return (sin(x)/cos(x)).series(rel, order, options); } +static ex tan_conjugate(const ex & x) +{ + // conjugate(tan(x))==tan(conjugate(x)) + return tan(x.conjugate()); +} + REGISTER_FUNCTION(tan, eval_func(tan_eval). evalf_func(tan_evalf). derivative_func(tan_deriv). series_func(tan_series). + real_part_func(tan_real_part). + imag_part_func(tan_imag_part). + conjugate_func(tan_conjugate). latex_name("\\tan")); ////////// @@ -548,7 +729,7 @@ static ex asin_eval(const ex & x) // asin(1) -> Pi/2 if (x.is_equal(_ex1)) - return _num1_2*Pi; + return _ex1_2*Pi; // asin(-1/2) -> -Pi/6 if (x.is_equal(_ex_1_2)) @@ -556,7 +737,7 @@ static ex asin_eval(const ex & x) // asin(-1) -> -Pi/2 if (x.is_equal(_ex_1)) - return _num_1_2*Pi; + return _ex_1_2*Pi; // asin(float) -> float if (!x.info(info_flags::crational)) @@ -578,9 +759,21 @@ static ex asin_deriv(const ex & x, unsigned deriv_param) 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). + conjugate_func(asin_conjugate). latex_name("\\arcsin")); ////////// @@ -639,9 +832,21 @@ static ex acos_deriv(const ex & x, unsigned deriv_param) 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). + conjugate_func(acos_conjugate). latex_name("\\arccos")); ////////// @@ -738,10 +943,27 @@ static ex atan_series(const ex &arg, 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). series_func(atan_series). + conjugate_func(atan_conjugate). latex_name("\\arctan")); ////////// @@ -758,67 +980,68 @@ static ex atan2_evalf(const ex &y, const ex &x) static ex atan2_eval(const ex & y, const ex & x) { - if (y.info(info_flags::numeric) && x.info(info_flags::numeric)) { + if (y.is_zero()) { - if (y.is_zero()) { + // atan2(0, 0) -> 0 + if (x.is_zero()) + return _ex0; - // atan(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; - // atan(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; + } - // atan(0, x), x real and negative -> -Pi - if (x.info(info_flags::negative)) - return _ex_1*Pi; - } + if (x.is_zero()) { - if (x.is_zero()) { + // atan2(y, 0), y real and positive -> Pi/2 + if (y.info(info_flags::positive)) + return _ex1_2*Pi; - // atan(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; + } - // atan(y, 0), y real and negative -> -Pi/2 - if (y.info(info_flags::negative)) - return _ex_1_2*Pi; - } + if (y.is_equal(x)) { - if (y.is_equal(x)) { + // atan2(y, y), y real and positive -> Pi/4 + if (y.info(info_flags::positive)) + return _ex1_4*Pi; - // atan(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; + } - // atan(y, y), y real and negative -> -3/4*Pi - if (y.info(info_flags::negative)) - return numeric(-3, 4)*Pi; - } + if (y.is_equal(-x)) { - 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; - // atan(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; + } - // atan(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)); - // atan(float, float) -> float - if (!y.info(info_flags::crational) && !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); - // atan(real, real) -> atan(y/x) +/- Pi - if (y.info(info_flags::real) && x.info(info_flags::real)) { - if (x.info(info_flags::positive)) - return atan(y/x); - else if(y.info(info_flags::positive)) + if (x.info(info_flags::negative)) { + if (y.info(info_flags::positive)) return atan(y/x)+Pi; - else + if (y.info(info_flags::negative)) return atan(y/x)-Pi; } } @@ -831,10 +1054,10 @@ 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) + // d/dy atan2(y,x) return x*power(power(x,_ex2)+power(y,_ex2),_ex_1); } - // d/dx atan(y,x) + // d/dx atan2(y,x) return -y*power(power(x,_ex2)+power(y,_ex2),_ex_1); } @@ -902,9 +1125,28 @@ static ex sinh_deriv(const ex & x, unsigned deriv_param) return cosh(x); } +static ex sinh_real_part(const ex & x) +{ + return sinh(GiNaC::real_part(x))*cos(GiNaC::imag_part(x)); +} + +static ex sinh_imag_part(const ex & x) +{ + return cosh(GiNaC::real_part(x))*sin(GiNaC::imag_part(x)); +} + +static ex sinh_conjugate(const ex & x) +{ + // conjugate(sinh(x))==sinh(conjugate(x)) + return sinh(x.conjugate()); +} + REGISTER_FUNCTION(sinh, eval_func(sinh_eval). evalf_func(sinh_evalf). derivative_func(sinh_deriv). + real_part_func(sinh_real_part). + imag_part_func(sinh_imag_part). + conjugate_func(sinh_conjugate). latex_name("\\sinh")); ////////// @@ -967,9 +1209,28 @@ static ex cosh_deriv(const ex & x, unsigned deriv_param) return sinh(x); } +static ex cosh_real_part(const ex & x) +{ + return cosh(GiNaC::real_part(x))*cos(GiNaC::imag_part(x)); +} + +static ex cosh_imag_part(const ex & x) +{ + return sinh(GiNaC::real_part(x))*sin(GiNaC::imag_part(x)); +} + +static ex cosh_conjugate(const ex & x) +{ + // conjugate(cosh(x))==cosh(conjugate(x)) + return cosh(x.conjugate()); +} + REGISTER_FUNCTION(cosh, eval_func(cosh_eval). evalf_func(cosh_evalf). derivative_func(cosh_deriv). + real_part_func(cosh_real_part). + imag_part_func(cosh_imag_part). + conjugate_func(cosh_conjugate). latex_name("\\cosh")); ////////// @@ -1048,10 +1309,33 @@ static ex tanh_series(const ex &x, return (sinh(x)/cosh(x)).series(rel, order, options); } +static ex tanh_real_part(const ex & x) +{ + ex a = GiNaC::real_part(x); + ex b = GiNaC::imag_part(x); + return tanh(a)/(1+power(tanh(a),2)*power(tan(b),2)); +} + +static ex tanh_imag_part(const ex & x) +{ + ex a = GiNaC::real_part(x); + ex b = GiNaC::imag_part(x); + return tan(b)/(1+power(tanh(a),2)*power(tan(b),2)); +} + +static ex tanh_conjugate(const ex & x) +{ + // conjugate(tanh(x))==tanh(conjugate(x)) + return tanh(x.conjugate()); +} + REGISTER_FUNCTION(tanh, eval_func(tanh_eval). evalf_func(tanh_evalf). derivative_func(tanh_deriv). series_func(tanh_series). + real_part_func(tanh_real_part). + imag_part_func(tanh_imag_part). + conjugate_func(tanh_conjugate). latex_name("\\tanh")); ////////// @@ -1094,9 +1378,26 @@ static ex asinh_deriv(const ex & x, unsigned deriv_param) 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) @@ -1146,9 +1447,21 @@ static ex acosh_deriv(const ex & x, unsigned deriv_param) 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) @@ -1237,10 +1550,22 @@ static ex atanh_series(const ex &arg, 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). - series_func(atanh_series)); + series_func(atanh_series). + conjugate_func(atanh_conjugate)); } // namespace GiNaC