X-Git-Url: https://ginac.de/ginac.git//ginac.git?a=blobdiff_plain;ds=sidebyside;f=ginac%2Finifcns_trans.cpp;h=507e18f51639f1c003280064b982f81a7d8f27e9;hb=559c3bfd89b822b16708b23af1b0d9af17ededfe;hp=ac0bc323c9a6968b2e9e12df665e31aed6b0c938;hpb=703c6cebb5d3d395437e73e6935f3691aed68e0a;p=ginac.git diff --git a/ginac/inifcns_trans.cpp b/ginac/inifcns_trans.cpp index ac0bc323..507e18f5 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) 1999-2001 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 @@ -34,9 +34,7 @@ #include "pseries.h" #include "utils.h" -#ifndef NO_NAMESPACE_GINAC namespace GiNaC { -#endif // ndef NO_NAMESPACE_GINAC ////////// // exponential function @@ -44,11 +42,10 @@ namespace GiNaC { static ex exp_evalf(const ex & x) { - BEGIN_TYPECHECK - TYPECHECK(x,numeric) - END_TYPECHECK(exp(x)) + if (is_exactly_a(x)) + return exp(ex_to(x)); - return exp(ex_to_numeric(x)); // -> numeric exp(numeric) + return exp(x).hold(); } static ex exp_eval(const ex & x) @@ -58,9 +55,9 @@ static ex exp_eval(const ex & x) return _ex1(); } // exp(n*Pi*I/2) -> {+1|+I|-1|-I} - ex TwoExOverPiI=(_ex2()*x)/(Pi*I); + const ex TwoExOverPiI=(_ex2()*x)/(Pi*I); if (TwoExOverPiI.info(info_flags::integer)) { - numeric z=mod(ex_to_numeric(TwoExOverPiI),_num4()); + numeric z=mod(ex_to(TwoExOverPiI),_num4()); if (z.is_equal(_num0())) return _ex1(); if (z.is_equal(_num1())) @@ -76,7 +73,7 @@ static ex exp_eval(const ex & x) // exp(float) if (x.info(info_flags::numeric) && !x.info(info_flags::crational)) - return exp_evalf(x); + return exp(ex_to(x)); return exp(x).hold(); } @@ -90,8 +87,9 @@ static ex exp_deriv(const ex & x, unsigned deriv_param) } REGISTER_FUNCTION(exp, eval_func(exp_eval). - evalf_func(exp_evalf). - derivative_func(exp_deriv)); + evalf_func(exp_evalf). + derivative_func(exp_deriv). + latex_name("\\exp")); ////////// // natural logarithm @@ -99,17 +97,16 @@ REGISTER_FUNCTION(exp, eval_func(exp_eval). static ex log_evalf(const ex & x) { - BEGIN_TYPECHECK - TYPECHECK(x,numeric) - END_TYPECHECK(log(x)) + if (is_exactly_a(x)) + return log(ex_to(x)); - return log(ex_to_numeric(x)); // -> numeric log(numeric) + return log(x).hold(); } static ex log_eval(const ex & x) { if (x.info(info_flags::numeric)) { - if (x.is_equal(_ex0())) // log(0) -> infinity + 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); @@ -121,13 +118,13 @@ static ex log_eval(const ex & x) return (Pi*I*_num_1_2()); // log(float) if (!x.info(info_flags::crational)) - return log_evalf(x); + return log(ex_to(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); + numeric nt = ex_to(t); if (nt.is_real()) return t; } @@ -145,11 +142,11 @@ static ex log_deriv(const ex & x, unsigned deriv_param) } static ex log_series(const ex &arg, - const relational &rel, - int order, - unsigned options) + const relational &rel, + int order, + unsigned options) { - GINAC_ASSERT(is_ex_exactly_of_type(rel.lhs(),symbol)); + GINAC_ASSERT(is_exactly_a(rel.lhs())); ex arg_pt; bool must_expand_arg = false; // maybe substitution of rel into arg fails because of a pole @@ -158,7 +155,7 @@ static ex log_series(const ex &arg, } catch (pole_error) { must_expand_arg = true; } - // or we are at the branch cut anyways + // or we are at the branch point anyways if (arg_pt.is_zero()) must_expand_arg = true; @@ -167,32 +164,48 @@ static ex log_series(const ex &arg, // 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 + Order(x^(n+m)) -> x^n * (1 + Order(x^m)). + // 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. - const pseries argser = ex_to_pseries(arg.series(rel, order, options)); - const symbol *s = static_cast(rel.lhs().bp); + 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); + const int n = argser.ldegree(s); epvector seq; - seq.push_back(expair(n*log(*s-point), _ex0())); + // 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 terms are needed - ex newarg = ex_to_pseries(arg.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))); + // 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((arg/coeff).series(rel, order+n, options)).shift_exponents(-n).convert_to_poly(true); + return pseries(rel, seq).add_series(ex_to(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)) { + 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 = static_cast(rel.lhs().bp); + const symbol &s = ex_to(rel.lhs()); const ex point = rel.rhs(); const symbol foo; - ex replarg = series(log(arg), *s==foo, order, false).subs(foo==point); + const ex replarg = series(log(arg), s==foo, order).subs(foo==point); epvector seq; seq.push_back(expair(-I*csgn(arg*I)*Pi, _ex0())); seq.push_back(expair(Order(_ex1()), order)); @@ -202,9 +215,10 @@ static ex log_series(const ex &arg, } REGISTER_FUNCTION(log, eval_func(log_eval). - evalf_func(log_evalf). - derivative_func(log_deriv). - series_func(log_series)); + evalf_func(log_evalf). + derivative_func(log_deriv). + series_func(log_series). + latex_name("\\ln")); ////////// // sine (trigonometric function) @@ -212,20 +226,19 @@ REGISTER_FUNCTION(log, eval_func(log_eval). static ex sin_evalf(const ex & x) { - BEGIN_TYPECHECK - TYPECHECK(x,numeric) - END_TYPECHECK(sin(x)) + if (is_exactly_a(x)) + return sin(ex_to(x)); - return sin(ex_to_numeric(x)); // -> numeric sin(numeric) + 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; + const ex SixtyExOverPi = _ex60()*x/Pi; ex sign = _ex1(); if (SixtyExOverPi.info(info_flags::integer)) { - numeric z = mod(ex_to_numeric(SixtyExOverPi),_num120()); + numeric z = mod(ex_to(SixtyExOverPi),_num120()); if (z>=_num60()) { // wrap to interval [0, Pi) z -= _num60(); @@ -255,7 +268,7 @@ static ex sin_eval(const ex & x) return sign*_ex1(); } - if (is_ex_exactly_of_type(x, function)) { + if (is_exactly_a(x)) { ex t = x.op(0); // sin(asin(x)) -> x if (is_ex_the_function(x, asin)) @@ -270,7 +283,7 @@ static ex sin_eval(const ex & x) // sin(float) -> float if (x.info(info_flags::numeric) && !x.info(info_flags::crational)) - return sin_evalf(x); + return sin(ex_to(x)); return sin(x).hold(); } @@ -284,8 +297,9 @@ static ex sin_deriv(const ex & x, unsigned deriv_param) } REGISTER_FUNCTION(sin, eval_func(sin_eval). - evalf_func(sin_evalf). - derivative_func(sin_deriv)); + evalf_func(sin_evalf). + derivative_func(sin_deriv). + latex_name("\\sin")); ////////// // cosine (trigonometric function) @@ -293,20 +307,19 @@ REGISTER_FUNCTION(sin, eval_func(sin_eval). static ex cos_evalf(const ex & x) { - BEGIN_TYPECHECK - TYPECHECK(x,numeric) - END_TYPECHECK(cos(x)) + if (is_exactly_a(x)) + return cos(ex_to(x)); - return cos(ex_to_numeric(x)); // -> numeric cos(numeric) + 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; + const ex SixtyExOverPi = _ex60()*x/Pi; ex sign = _ex1(); if (SixtyExOverPi.info(info_flags::integer)) { - numeric z = mod(ex_to_numeric(SixtyExOverPi),_num120()); + numeric z = mod(ex_to(SixtyExOverPi),_num120()); if (z>=_num60()) { // wrap to interval [0, Pi) z = _num120()-z; @@ -336,7 +349,7 @@ static ex cos_eval(const ex & x) return sign*_ex0(); } - if (is_ex_exactly_of_type(x, function)) { + if (is_exactly_a(x)) { ex t = x.op(0); // cos(acos(x)) -> x if (is_ex_the_function(x, acos)) @@ -351,7 +364,7 @@ static ex cos_eval(const ex & x) // cos(float) -> float if (x.info(info_flags::numeric) && !x.info(info_flags::crational)) - return cos_evalf(x); + return cos(ex_to(x)); return cos(x).hold(); } @@ -365,8 +378,9 @@ static ex cos_deriv(const ex & x, unsigned deriv_param) } REGISTER_FUNCTION(cos, eval_func(cos_eval). - evalf_func(cos_evalf). - derivative_func(cos_deriv)); + evalf_func(cos_evalf). + derivative_func(cos_deriv). + latex_name("\\cos")); ////////// // tangent (trigonometric function) @@ -374,20 +388,19 @@ 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) + if (is_exactly_a(x)) + return tan(ex_to(x)); - return tan(ex_to_numeric(x)); + return tan(x).hold(); } static ex tan_eval(const ex & x) { // tan(n/d*Pi) -> { all known non-nested radicals } - ex SixtyExOverPi = _ex60()*x/Pi; + const ex SixtyExOverPi = _ex60()*x/Pi; ex sign = _ex1(); if (SixtyExOverPi.info(info_flags::integer)) { - numeric z = mod(ex_to_numeric(SixtyExOverPi),_num60()); + numeric z = mod(ex_to(SixtyExOverPi),_num60()); if (z>=_num60()) { // wrap to interval [0, Pi) z -= _num60(); @@ -413,7 +426,7 @@ static ex tan_eval(const ex & x) throw (pole_error("tan_eval(): simple pole",1)); } - if (is_ex_exactly_of_type(x, function)) { + if (is_exactly_a(x)) { ex t = x.op(0); // tan(atan(x)) -> x if (is_ex_the_function(x, atan)) @@ -428,7 +441,7 @@ static ex tan_eval(const ex & x) // tan(float) -> float if (x.info(info_flags::numeric) && !x.info(info_flags::crational)) { - return tan_evalf(x); + return tan(ex_to(x)); } return tan(x).hold(); @@ -443,10 +456,11 @@ static ex tan_deriv(const ex & x, unsigned deriv_param) } static ex tan_series(const ex &x, - const relational &rel, - int order, - unsigned options) + const relational &rel, + int order, + unsigned options) { + GINAC_ASSERT(is_exactly_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). @@ -454,13 +468,14 @@ static ex tan_series(const ex &x, 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); + 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)); + evalf_func(tan_evalf). + derivative_func(tan_deriv). + series_func(tan_series). + latex_name("\\tan")); ////////// // inverse sine (arc sine) @@ -468,11 +483,10 @@ REGISTER_FUNCTION(tan, eval_func(tan_eval). static ex asin_evalf(const ex & x) { - BEGIN_TYPECHECK - TYPECHECK(x,numeric) - END_TYPECHECK(asin(x)) + if (is_exactly_a(x)) + return asin(ex_to(x)); - return asin(ex_to_numeric(x)); // -> numeric asin(numeric) + return asin(x).hold(); } static ex asin_eval(const ex & x) @@ -495,7 +509,7 @@ static ex asin_eval(const ex & x) return _num_1_2()*Pi; // asin(float) -> float if (!x.info(info_flags::crational)) - return asin_evalf(x); + return asin(ex_to(x)); } return asin(x).hold(); @@ -510,8 +524,9 @@ static ex asin_deriv(const ex & x, unsigned deriv_param) } REGISTER_FUNCTION(asin, eval_func(asin_eval). - evalf_func(asin_evalf). - derivative_func(asin_deriv)); + evalf_func(asin_evalf). + derivative_func(asin_deriv). + latex_name("\\arcsin")); ////////// // inverse cosine (arc cosine) @@ -519,11 +534,10 @@ REGISTER_FUNCTION(asin, eval_func(asin_eval). static ex acos_evalf(const ex & x) { - BEGIN_TYPECHECK - TYPECHECK(x,numeric) - END_TYPECHECK(acos(x)) + if (is_exactly_a(x)) + return acos(ex_to(x)); - return acos(ex_to_numeric(x)); // -> numeric acos(numeric) + return acos(x).hold(); } static ex acos_eval(const ex & x) @@ -546,7 +560,7 @@ static ex acos_eval(const ex & x) return Pi; // acos(float) -> float if (!x.info(info_flags::crational)) - return acos_evalf(x); + return acos(ex_to(x)); } return acos(x).hold(); @@ -561,8 +575,9 @@ static ex acos_deriv(const ex & x, unsigned deriv_param) } REGISTER_FUNCTION(acos, eval_func(acos_eval). - evalf_func(acos_evalf). - derivative_func(acos_deriv)); + evalf_func(acos_evalf). + derivative_func(acos_deriv). + latex_name("\\arccos")); ////////// // inverse tangent (arc tangent) @@ -570,26 +585,33 @@ REGISTER_FUNCTION(acos, eval_func(acos_eval). static ex atan_evalf(const ex & x) { - BEGIN_TYPECHECK - TYPECHECK(x,numeric) - END_TYPECHECK(atan(x)) + if (is_exactly_a(x)) + return atan(ex_to(x)); - return atan(ex_to_numeric(x)); // -> numeric atan(numeric) + 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())) + 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_evalf(x); + return atan(ex_to(x)); } return atan(x).hold(); -} +} static ex atan_deriv(const ex & x, unsigned deriv_param) { @@ -599,22 +621,65 @@ static ex atan_deriv(const ex & x, unsigned deriv_param) 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_exactly_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); + 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); + ex Order0correction = replarg.op(0)+csgn(arg)*Pi*_ex_1_2(); + if ((I*arg_pt)<_ex0()) + Order0correction += log((I*arg_pt+_ex_1())/(I*arg_pt+_ex1()))*I*_ex_1_2(); + else + Order0correction += log((I*arg_pt+_ex1())/(I*arg_pt+_ex_1()))*I*_ex1_2(); + epvector seq; + seq.push_back(expair(Order0correction, _ex0())); + seq.push_back(expair(Order(_ex1()), order)); + return series(replarg - pseries(rel, seq), rel, order); + } + throw do_taylor(); +} + REGISTER_FUNCTION(atan, eval_func(atan_eval). - evalf_func(atan_evalf). - derivative_func(atan_deriv)); + evalf_func(atan_evalf). + derivative_func(atan_deriv). + series_func(atan_series). + latex_name("\\arctan")); ////////// // inverse tangent (atan2(y,x)) ////////// -static ex atan2_evalf(const ex & y, const ex & x) +static ex atan2_evalf(const ex &y, const ex &x) { - BEGIN_TYPECHECK - TYPECHECK(y,numeric) - TYPECHECK(x,numeric) - END_TYPECHECK(atan2(y,x)) + if (is_exactly_a(y) && is_exactly_a(x)) + return atan2(ex_to(y), ex_to(x)); - return atan(ex_to_numeric(y),ex_to_numeric(x)); // -> numeric atan(numeric) + return atan2(y, x).hold(); } static ex atan2_eval(const ex & y, const ex & x) @@ -640,8 +705,8 @@ static ex atan2_deriv(const ex & y, const ex & x, unsigned deriv_param) } REGISTER_FUNCTION(atan2, eval_func(atan2_eval). - evalf_func(atan2_evalf). - derivative_func(atan2_deriv)); + evalf_func(atan2_evalf). + derivative_func(atan2_deriv)); ////////// // hyperbolic sine (trigonometric function) @@ -649,11 +714,10 @@ REGISTER_FUNCTION(atan2, eval_func(atan2_eval). static ex sinh_evalf(const ex & x) { - BEGIN_TYPECHECK - TYPECHECK(x,numeric) - END_TYPECHECK(sinh(x)) + if (is_exactly_a(x)) + return sinh(ex_to(x)); - return sinh(ex_to_numeric(x)); // -> numeric sinh(numeric) + return sinh(x).hold(); } static ex sinh_eval(const ex & x) @@ -662,14 +726,14 @@ static ex sinh_eval(const ex & x) if (x.is_zero()) // sinh(0) -> 0 return _ex0(); if (!x.info(info_flags::crational)) // sinh(float) -> float - return sinh_evalf(x); + return sinh(ex_to(x)); } if ((x/Pi).info(info_flags::numeric) && - ex_to_numeric(x/Pi).real().is_zero()) // sinh(I*x) -> I*sin(x) + ex_to(x/Pi).real().is_zero()) // sinh(I*x) -> I*sin(x) return I*sin(x/I); - if (is_ex_exactly_of_type(x, function)) { + if (is_exactly_a(x)) { ex t = x.op(0); // sinh(asinh(x)) -> x if (is_ex_the_function(x, asinh)) @@ -694,8 +758,9 @@ static ex sinh_deriv(const ex & x, unsigned deriv_param) } REGISTER_FUNCTION(sinh, eval_func(sinh_eval). - evalf_func(sinh_evalf). - derivative_func(sinh_deriv)); + evalf_func(sinh_evalf). + derivative_func(sinh_deriv). + latex_name("\\sinh")); ////////// // hyperbolic cosine (trigonometric function) @@ -703,11 +768,10 @@ REGISTER_FUNCTION(sinh, eval_func(sinh_eval). static ex cosh_evalf(const ex & x) { - BEGIN_TYPECHECK - TYPECHECK(x,numeric) - END_TYPECHECK(cosh(x)) + if (is_exactly_a(x)) + return cosh(ex_to(x)); - return cosh(ex_to_numeric(x)); // -> numeric cosh(numeric) + return cosh(x).hold(); } static ex cosh_eval(const ex & x) @@ -716,14 +780,14 @@ static ex cosh_eval(const ex & x) if (x.is_zero()) // cosh(0) -> 1 return _ex1(); if (!x.info(info_flags::crational)) // cosh(float) -> float - return cosh_evalf(x); + return cosh(ex_to(x)); } if ((x/Pi).info(info_flags::numeric) && - ex_to_numeric(x/Pi).real().is_zero()) // cosh(I*x) -> cos(x) + ex_to(x/Pi).real().is_zero()) // cosh(I*x) -> cos(x) return cos(x/I); - if (is_ex_exactly_of_type(x, function)) { + if (is_exactly_a(x)) { ex t = x.op(0); // cosh(acosh(x)) -> x if (is_ex_the_function(x, acosh)) @@ -748,9 +812,9 @@ static ex cosh_deriv(const ex & x, unsigned deriv_param) } REGISTER_FUNCTION(cosh, eval_func(cosh_eval). - evalf_func(cosh_evalf). - derivative_func(cosh_deriv)); - + evalf_func(cosh_evalf). + derivative_func(cosh_deriv). + latex_name("\\cosh")); ////////// // hyperbolic tangent (trigonometric function) @@ -758,11 +822,10 @@ REGISTER_FUNCTION(cosh, eval_func(cosh_eval). static ex tanh_evalf(const ex & x) { - BEGIN_TYPECHECK - TYPECHECK(x,numeric) - END_TYPECHECK(tanh(x)) + if (is_exactly_a(x)) + return tanh(ex_to(x)); - return tanh(ex_to_numeric(x)); // -> numeric tanh(numeric) + return tanh(x).hold(); } static ex tanh_eval(const ex & x) @@ -771,14 +834,14 @@ static ex tanh_eval(const ex & x) if (x.is_zero()) // tanh(0) -> 0 return _ex0(); if (!x.info(info_flags::crational)) // tanh(float) -> float - return tanh_evalf(x); + return tanh(ex_to(x)); } if ((x/Pi).info(info_flags::numeric) && - ex_to_numeric(x/Pi).real().is_zero()) // tanh(I*x) -> I*tan(x); + ex_to(x/Pi).real().is_zero()) // tanh(I*x) -> I*tan(x); return I*tan(x/I); - if (is_ex_exactly_of_type(x, function)) { + if (is_exactly_a(x)) { ex t = x.op(0); // tanh(atanh(x)) -> x if (is_ex_the_function(x, atanh)) @@ -803,10 +866,11 @@ static ex tanh_deriv(const ex & x, unsigned deriv_param) } static ex tanh_series(const ex &x, - const relational &rel, - int order, - unsigned options) + const relational &rel, + int order, + unsigned options) { + GINAC_ASSERT(is_exactly_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). @@ -814,13 +878,14 @@ static ex tanh_series(const ex &x, 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); + 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)); + evalf_func(tanh_evalf). + derivative_func(tanh_deriv). + series_func(tanh_series). + latex_name("\\tanh")); ////////// // inverse hyperbolic sine (trigonometric function) @@ -828,11 +893,10 @@ REGISTER_FUNCTION(tanh, eval_func(tanh_eval). static ex asinh_evalf(const ex & x) { - BEGIN_TYPECHECK - TYPECHECK(x,numeric) - END_TYPECHECK(asinh(x)) + if (is_exactly_a(x)) + return asinh(ex_to(x)); - return asinh(ex_to_numeric(x)); // -> numeric asinh(numeric) + return asinh(x).hold(); } static ex asinh_eval(const ex & x) @@ -843,7 +907,7 @@ static ex asinh_eval(const ex & x) return _ex0(); // asinh(float) -> float if (!x.info(info_flags::crational)) - return asinh_evalf(x); + return asinh(ex_to(x)); } return asinh(x).hold(); @@ -858,8 +922,8 @@ static ex asinh_deriv(const ex & x, unsigned deriv_param) } REGISTER_FUNCTION(asinh, eval_func(asinh_eval). - evalf_func(asinh_evalf). - derivative_func(asinh_deriv)); + evalf_func(asinh_evalf). + derivative_func(asinh_deriv)); ////////// // inverse hyperbolic cosine (trigonometric function) @@ -867,11 +931,10 @@ REGISTER_FUNCTION(asinh, eval_func(asinh_eval). static ex acosh_evalf(const ex & x) { - BEGIN_TYPECHECK - TYPECHECK(x,numeric) - END_TYPECHECK(acosh(x)) + if (is_exactly_a(x)) + return acosh(ex_to(x)); - return acosh(ex_to_numeric(x)); // -> numeric acosh(numeric) + return acosh(x).hold(); } static ex acosh_eval(const ex & x) @@ -888,7 +951,7 @@ static ex acosh_eval(const ex & x) return Pi*I; // acosh(float) -> float if (!x.info(info_flags::crational)) - return acosh_evalf(x); + return acosh(ex_to(x)); } return acosh(x).hold(); @@ -903,8 +966,8 @@ static ex acosh_deriv(const ex & x, unsigned deriv_param) } REGISTER_FUNCTION(acosh, eval_func(acosh_eval). - evalf_func(acosh_evalf). - derivative_func(acosh_deriv)); + evalf_func(acosh_evalf). + derivative_func(acosh_deriv)); ////////// // inverse hyperbolic tangent (trigonometric function) @@ -912,11 +975,10 @@ REGISTER_FUNCTION(acosh, eval_func(acosh_eval). static ex atanh_evalf(const ex & x) { - BEGIN_TYPECHECK - TYPECHECK(x,numeric) - END_TYPECHECK(atanh(x)) + if (is_exactly_a(x)) + return atanh(ex_to(x)); - return atanh(ex_to_numeric(x)); // -> numeric atanh(numeric) + return atanh(x).hold(); } static ex atanh_eval(const ex & x) @@ -930,7 +992,7 @@ static ex atanh_eval(const ex & x) throw (pole_error("atanh_eval(): logarithmic pole",0)); // atanh(float) -> float if (!x.info(info_flags::crational)) - return atanh_evalf(x); + return atanh(ex_to(x)); } return atanh(x).hold(); @@ -944,10 +1006,53 @@ static ex atanh_deriv(const ex & x, unsigned deriv_param) 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_exactly_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); + 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); + 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)); + evalf_func(atanh_evalf). + derivative_func(atanh_deriv). + series_func(atanh_series)); + -#ifndef NO_NAMESPACE_GINAC } // namespace GiNaC -#endif // ndef NO_NAMESPACE_GINAC