X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Finifcns_trans.cpp;h=3bda254ab61c4665b4ef3fc906e5a33da0cb0e96;hp=70eccdceaff8cda1f902d812c8d95379606fdd1b;hb=b1da89e057fea6f9c9c6e32bd92ecb5980b6713a;hpb=24fe247f9ed16114a765a01c593fec5c4a2f591c diff --git a/ginac/inifcns_trans.cpp b/ginac/inifcns_trans.cpp index 70eccdce..3bda254a 100644 --- a/ginac/inifcns_trans.cpp +++ b/ginac/inifcns_trans.cpp @@ -4,7 +4,7 @@ * functions. */ /* - * GiNaC Copyright (C) 1999 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 @@ -29,6 +29,10 @@ #include "constant.h" #include "numeric.h" #include "power.h" +#include "relational.h" +#include "symbol.h" +#include "pseries.h" +#include "utils.h" namespace GiNaC { @@ -36,708 +40,1027 @@ namespace GiNaC { // exponential function ////////// -static ex exp_evalf(ex const & x) -{ - BEGIN_TYPECHECK - TYPECHECK(x,numeric) - END_TYPECHECK(exp(x)) - - return exp(ex_to_numeric(x)); // -> numeric exp(numeric) -} - -static ex exp_eval(ex const & x) -{ - // exp(0) -> 1 - if (x.is_zero()) { - return exONE(); - } - // exp(n*Pi*I/2) -> {+1|+I|-1|-I} - ex TwoExOverPiI=(2*x)/(Pi*I); - if (TwoExOverPiI.info(info_flags::integer)) { - numeric z=mod(ex_to_numeric(TwoExOverPiI),numeric(4)); - if (z.is_equal(numZERO())) - return exONE(); - if (z.is_equal(numONE())) - return ex(I); - if (z.is_equal(numTWO())) - return exMINUSONE(); - if (z.is_equal(numTHREE())) - return ex(-I); - } - // exp(log(x)) -> x - if (is_ex_the_function(x, log)) - return x.op(0); - - // exp(float) - if (x.info(info_flags::numeric) && !x.info(info_flags::rational)) - return exp_evalf(x); - - return exp(x).hold(); -} +static ex exp_evalf(const ex & x) +{ + BEGIN_TYPECHECK + TYPECHECK(x,numeric) + END_TYPECHECK(exp(x)) + + return exp(ex_to_numeric(x)); // -> numeric exp(numeric) +} + +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(); +} -static ex exp_diff(ex const & x, unsigned diff_param) +static ex exp_deriv(const ex & x, unsigned deriv_param) { - GINAC_ASSERT(diff_param==0); + GINAC_ASSERT(deriv_param==0); - return exp(x); + // d/dx exp(x) -> exp(x) + return exp(x); } -REGISTER_FUNCTION(exp, exp_eval, exp_evalf, exp_diff, NULL); +REGISTER_FUNCTION(exp, eval_func(exp_eval). + evalf_func(exp_evalf). + derivative_func(exp_deriv). + latex_name("\\exp")); ////////// // natural logarithm ////////// -static ex log_evalf(ex const & x) -{ - BEGIN_TYPECHECK - TYPECHECK(x,numeric) - END_TYPECHECK(log(x)) - - return log(ex_to_numeric(x)); // -> numeric log(numeric) -} - -static ex log_eval(ex const & x) -{ - if (x.info(info_flags::numeric)) { - // log(1) -> 0 - if (x.is_equal(exONE())) - return exZERO(); - // log(-1) -> I*Pi - if (x.is_equal(exMINUSONE())) - return (I*Pi); - // log(I) -> Pi*I/2 - if (x.is_equal(I)) - return (I*Pi*numeric(1,2)); - // log(-I) -> -Pi*I/2 - if (x.is_equal(-I)) - return (I*Pi*numeric(-1,2)); - // log(0) -> throw singularity - if (x.is_equal(exZERO())) - throw(std::domain_error("log_eval(): log(0)")); - // log(float) - if (!x.info(info_flags::rational)) - return log_evalf(x); - } - - return log(x).hold(); -} +static ex log_evalf(const ex & x) +{ + BEGIN_TYPECHECK + TYPECHECK(x,numeric) + END_TYPECHECK(log(x)) + + return log(ex_to_numeric(x)); // -> numeric log(numeric) +} -static ex log_diff(ex const & x, unsigned diff_param) +static ex log_eval(const ex & x) { - GINAC_ASSERT(diff_param==0); + if (x.info(info_flags::numeric)) { + if (x.is_equal(_ex0())) // log(0) -> infinity + throw(pole_error("log_eval(): log(0)",0)); + if (x.info(info_flags::real) && x.info(info_flags::negative)) + return (log(-x)+I*Pi); + if (x.is_equal(_ex1())) // log(1) -> 0 + return _ex0(); + if (x.is_equal(I)) // log(I) -> Pi*I/2 + return (Pi*I*_num1_2()); + if (x.is_equal(-I)) // log(-I) -> -Pi*I/2 + return (Pi*I*_num_1_2()); + // log(float) + if (!x.info(info_flags::crational)) + return log_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(); +} - return power(x, -1); +static ex log_deriv(const ex & x, unsigned deriv_param) +{ + GINAC_ASSERT(deriv_param==0); + + // d/dx log(x) -> 1/x + return power(x, _ex_1()); } -REGISTER_FUNCTION(log, log_eval, log_evalf, log_diff, NULL); +static ex log_series(const ex &arg, + const relational &rel, + int order, + unsigned options) +{ + GINAC_ASSERT(is_ex_exactly_of_type(rel.lhs(),symbol)); + 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); + } catch (pole_error) { + must_expand_arg = true; + } + // or we are at the branch point anyways + if (arg_pt.is_zero()) + must_expand_arg = true; + + if (must_expand_arg) { + // method: + // This is the branch point: Series expand the argument first, then + // trivially factorize it to isolate that part which has constant + // leading coefficient in this fashion: + // x^n + Order(x^(n+m)) -> x^n * (1 + 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); + const ex point = rel.rhs(); + const int n = argser.ldegree(*s); + epvector seq; + // construct what we carelessly called the n*log(x) term above + 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 + // (sadly, to generate them, we have to start from the beginning) + ex newarg = ex_to_pseries((arg/coeff).series(rel, order+n, options)).shift_exponents(-n).convert_to_poly(true); + return pseries(rel, seq).add_series(ex_to_pseries(log(newarg).series(rel, order, options))); + } else // it was a monomial + return pseries(rel, seq); + } + if (!(options & series_options::suppress_branchcut) && + arg_pt.info(info_flags::negative)) { + // method: + // This is the branch cut: assemble the primitive series manually and + // then add the corresponding complex step function. + const symbol *s = static_cast(rel.lhs().bp); + const ex point = rel.rhs(); + const symbol foo; + 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)); + return series(replarg - I*Pi + pseries(rel, seq), rel, order); + } + throw do_taylor(); // caught by function::series() +} + +REGISTER_FUNCTION(log, eval_func(log_eval). + evalf_func(log_evalf). + derivative_func(log_deriv). + series_func(log_series). + latex_name("\\ln")); ////////// // sine (trigonometric function) ////////// -static ex sin_evalf(ex const & x) -{ - BEGIN_TYPECHECK - TYPECHECK(x,numeric) - END_TYPECHECK(sin(x)) - - return sin(ex_to_numeric(x)); // -> numeric sin(numeric) -} - -static ex sin_eval(ex const & x) -{ - // sin(n*Pi) -> 0 - ex xOverPi=x/Pi; - if (xOverPi.info(info_flags::integer)) - return exZERO(); - - // sin((2n+1)*Pi/2) -> {+|-}1 - ex xOverPiMinusHalf=xOverPi-exHALF(); - if (xOverPiMinusHalf.info(info_flags::even)) - return exONE(); - else if (xOverPiMinusHalf.info(info_flags::odd)) - return exMINUSONE(); - - if (is_ex_exactly_of_type(x, function)) { - ex t=x.op(0); - // sin(asin(x)) -> x - if (is_ex_the_function(x, asin)) - return t; - // sin(acos(x)) -> (1-x^2)^(1/2) - if (is_ex_the_function(x, acos)) - return power(exONE()-power(t,exTWO()),exHALF()); - // sin(atan(x)) -> x*(1+x^2)^(-1/2) - if (is_ex_the_function(x, atan)) - return t*power(exONE()+power(t,exTWO()),exMINUSHALF()); - } - - // sin(float) -> float - if (x.info(info_flags::numeric) && !x.info(info_flags::rational)) - return sin_evalf(x); - - return sin(x).hold(); -} - -static ex sin_diff(ex const & x, unsigned diff_param) -{ - GINAC_ASSERT(diff_param==0); - - return cos(x); -} - -REGISTER_FUNCTION(sin, sin_eval, sin_evalf, sin_diff, NULL); +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) +} + +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(); +} + +static ex sin_deriv(const ex & x, unsigned deriv_param) +{ + GINAC_ASSERT(deriv_param==0); + + // d/dx sin(x) -> cos(x) + return cos(x); +} + +REGISTER_FUNCTION(sin, eval_func(sin_eval). + evalf_func(sin_evalf). + derivative_func(sin_deriv). + latex_name("\\sin")); ////////// // cosine (trigonometric function) ////////// -static ex cos_evalf(ex const & x) -{ - BEGIN_TYPECHECK - TYPECHECK(x,numeric) - END_TYPECHECK(cos(x)) - - return cos(ex_to_numeric(x)); // -> numeric cos(numeric) -} - -static ex cos_eval(ex const & x) -{ - // cos(n*Pi) -> {+|-}1 - ex xOverPi=x/Pi; - if (xOverPi.info(info_flags::even)) - return exONE(); - else if (xOverPi.info(info_flags::odd)) - return exMINUSONE(); - - // cos((2n+1)*Pi/2) -> 0 - ex xOverPiMinusHalf=xOverPi-exHALF(); - if (xOverPiMinusHalf.info(info_flags::integer)) - return exZERO(); - - if (is_ex_exactly_of_type(x, function)) { - ex t=x.op(0); - // cos(acos(x)) -> x - if (is_ex_the_function(x, acos)) - return t; - // cos(asin(x)) -> (1-x^2)^(1/2) - if (is_ex_the_function(x, asin)) - return power(exONE()-power(t,exTWO()),exHALF()); - // cos(atan(x)) -> (1+x^2)^(-1/2) - if (is_ex_the_function(x, atan)) - return power(exONE()+power(t,exTWO()),exMINUSHALF()); - } - - // cos(float) -> float - if (x.info(info_flags::numeric) && !x.info(info_flags::rational)) - return cos_evalf(x); - - return cos(x).hold(); -} - -static ex cos_diff(ex const & x, unsigned diff_param) -{ - GINAC_ASSERT(diff_param==0); - - return numMINUSONE()*sin(x); -} - -REGISTER_FUNCTION(cos, cos_eval, cos_evalf, cos_diff, NULL); +static ex cos_evalf(const ex & x) +{ + BEGIN_TYPECHECK + TYPECHECK(x,numeric) + END_TYPECHECK(cos(x)) + + return cos(ex_to_numeric(x)); // -> numeric cos(numeric) +} + +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(); +} + +static ex cos_deriv(const ex & x, unsigned deriv_param) +{ + GINAC_ASSERT(deriv_param==0); + + // d/dx cos(x) -> -sin(x) + return _ex_1()*sin(x); +} + +REGISTER_FUNCTION(cos, eval_func(cos_eval). + evalf_func(cos_evalf). + derivative_func(cos_deriv). + latex_name("\\cos")); ////////// // tangent (trigonometric function) ////////// -static ex tan_evalf(ex const & x) -{ - BEGIN_TYPECHECK - TYPECHECK(x,numeric) - END_TYPECHECK(tan(x)) // -> numeric tan(numeric) - - return tan(ex_to_numeric(x)); -} - -static ex tan_eval(ex const & x) -{ - // tan(n*Pi/3) -> {0|3^(1/2)|-(3^(1/2))} - ex ThreeExOverPi=numTHREE()*x/Pi; - if (ThreeExOverPi.info(info_flags::integer)) { - numeric z=mod(ex_to_numeric(ThreeExOverPi),numeric(3)); - if (z.is_equal(numZERO())) - return exZERO(); - if (z.is_equal(numONE())) - return power(exTHREE(),exHALF()); - if (z.is_equal(numTWO())) - return -power(exTHREE(),exHALF()); - } - - // tan((2n+1)*Pi/2) -> throw - ex ExOverPiMinusHalf=x/Pi-exHALF(); - if (ExOverPiMinusHalf.info(info_flags::integer)) - throw (std::domain_error("tan_eval(): infinity")); - - if (is_ex_exactly_of_type(x, function)) { - ex t=x.op(0); - // tan(atan(x)) -> x - if (is_ex_the_function(x, atan)) - return t; - // tan(asin(x)) -> x*(1+x^2)^(-1/2) - if (is_ex_the_function(x, asin)) - return t*power(exONE()-power(t,exTWO()),exMINUSHALF()); - // tan(acos(x)) -> (1-x^2)^(1/2)/x - if (is_ex_the_function(x, acos)) - return power(t,exMINUSONE())*power(exONE()-power(t,exTWO()),exHALF()); - } - - // tan(float) -> float - if (x.info(info_flags::numeric) && !x.info(info_flags::rational)) { - return tan_evalf(x); - } - - return tan(x).hold(); -} - -static ex tan_diff(ex const & x, unsigned diff_param) -{ - GINAC_ASSERT(diff_param==0); - - return (1+power(tan(x),exTWO())); -} - -REGISTER_FUNCTION(tan, tan_eval, tan_evalf, tan_diff, NULL); +static ex tan_evalf(const ex & x) +{ + BEGIN_TYPECHECK + TYPECHECK(x,numeric) + END_TYPECHECK(tan(x)) // -> numeric tan(numeric) + + return tan(ex_to_numeric(x)); +} + +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 (pole_error("tan_eval(): simple pole",1)); + } + + 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(); +} + +static ex tan_deriv(const ex & x, unsigned deriv_param) +{ + GINAC_ASSERT(deriv_param==0); + + // d/dx tan(x) -> 1+tan(x)^2; + return (_ex1()+power(tan(x),_ex2())); +} + +static ex tan_series(const ex &x, + const relational &rel, + int order, + unsigned options) +{ + GINAC_ASSERT(is_ex_exactly_of_type(rel.lhs(),symbol)); + // 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); + if (!(2*x_pt/Pi).info(info_flags::odd)) + throw do_taylor(); // caught by function::series() + // if we got here we have to care for a simple pole + return (sin(x)/cos(x)).series(rel, order+2, options); +} + +REGISTER_FUNCTION(tan, eval_func(tan_eval). + evalf_func(tan_evalf). + derivative_func(tan_deriv). + series_func(tan_series). + latex_name("\\tan")); ////////// // inverse sine (arc sine) ////////// -static ex asin_evalf(ex const & x) +static ex asin_evalf(const ex & x) { - BEGIN_TYPECHECK - TYPECHECK(x,numeric) - END_TYPECHECK(asin(x)) - - return asin(ex_to_numeric(x)); // -> numeric asin(numeric) + BEGIN_TYPECHECK + TYPECHECK(x,numeric) + END_TYPECHECK(asin(x)) + + return asin(ex_to_numeric(x)); // -> numeric asin(numeric) } -static ex asin_eval(ex const & x) +static ex asin_eval(const ex & x) { - if (x.info(info_flags::numeric)) { - // asin(0) -> 0 - if (x.is_zero()) - return x; - // asin(1/2) -> Pi/6 - if (x.is_equal(exHALF())) - return numeric(1,6)*Pi; - // asin(1) -> Pi/2 - if (x.is_equal(exONE())) - return numeric(1,2)*Pi; - // asin(-1/2) -> -Pi/6 - if (x.is_equal(exMINUSHALF())) - return numeric(-1,6)*Pi; - // asin(-1) -> -Pi/2 - if (x.is_equal(exMINUSONE())) - return numeric(-1,2)*Pi; - // asin(float) -> float - if (!x.info(info_flags::rational)) - return asin_evalf(x); - } - - return asin(x).hold(); + if (x.info(info_flags::numeric)) { + // asin(0) -> 0 + if (x.is_zero()) + return x; + // asin(1/2) -> Pi/6 + if (x.is_equal(_ex1_2())) + return numeric(1,6)*Pi; + // asin(1) -> Pi/2 + if (x.is_equal(_ex1())) + return _num1_2()*Pi; + // asin(-1/2) -> -Pi/6 + if (x.is_equal(_ex_1_2())) + return numeric(-1,6)*Pi; + // asin(-1) -> -Pi/2 + if (x.is_equal(_ex_1())) + return _num_1_2()*Pi; + // asin(float) -> float + if (!x.info(info_flags::crational)) + return asin_evalf(x); + } + + return asin(x).hold(); } -static ex asin_diff(ex const & x, unsigned diff_param) +static ex asin_deriv(const ex & x, unsigned deriv_param) { - GINAC_ASSERT(diff_param==0); - - return power(1-power(x,exTWO()),exMINUSHALF()); + GINAC_ASSERT(deriv_param==0); + + // d/dx asin(x) -> 1/sqrt(1-x^2) + return power(1-power(x,_ex2()),_ex_1_2()); } -REGISTER_FUNCTION(asin, asin_eval, asin_evalf, asin_diff, NULL); +REGISTER_FUNCTION(asin, eval_func(asin_eval). + evalf_func(asin_evalf). + derivative_func(asin_deriv). + latex_name("\\arcsin")); ////////// // inverse cosine (arc cosine) ////////// -static ex acos_evalf(ex const & x) +static ex acos_evalf(const ex & x) { - BEGIN_TYPECHECK - TYPECHECK(x,numeric) - END_TYPECHECK(acos(x)) - - return acos(ex_to_numeric(x)); // -> numeric acos(numeric) + BEGIN_TYPECHECK + TYPECHECK(x,numeric) + END_TYPECHECK(acos(x)) + + return acos(ex_to_numeric(x)); // -> numeric acos(numeric) } -static ex acos_eval(ex const & x) +static ex acos_eval(const ex & x) { - if (x.info(info_flags::numeric)) { - // acos(1) -> 0 - if (x.is_equal(exONE())) - return exZERO(); - // acos(1/2) -> Pi/3 - if (x.is_equal(exHALF())) - return numeric(1,3)*Pi; - // acos(0) -> Pi/2 - if (x.is_zero()) - return numeric(1,2)*Pi; - // acos(-1/2) -> 2/3*Pi - if (x.is_equal(exMINUSHALF())) - return numeric(2,3)*Pi; - // acos(-1) -> Pi - if (x.is_equal(exMINUSONE())) - return Pi; - // acos(float) -> float - if (!x.info(info_flags::rational)) - return acos_evalf(x); - } - - return acos(x).hold(); + if (x.info(info_flags::numeric)) { + // acos(1) -> 0 + if (x.is_equal(_ex1())) + return _ex0(); + // acos(1/2) -> Pi/3 + if (x.is_equal(_ex1_2())) + return _ex1_3()*Pi; + // acos(0) -> Pi/2 + if (x.is_zero()) + return _ex1_2()*Pi; + // acos(-1/2) -> 2/3*Pi + if (x.is_equal(_ex_1_2())) + return numeric(2,3)*Pi; + // acos(-1) -> Pi + if (x.is_equal(_ex_1())) + return Pi; + // acos(float) -> float + if (!x.info(info_flags::crational)) + return acos_evalf(x); + } + + return acos(x).hold(); } -static ex acos_diff(ex const & x, unsigned diff_param) +static ex acos_deriv(const ex & x, unsigned deriv_param) { - GINAC_ASSERT(diff_param==0); - - return numMINUSONE()*power(1-power(x,exTWO()),exMINUSHALF()); + GINAC_ASSERT(deriv_param==0); + + // d/dx acos(x) -> -1/sqrt(1-x^2) + return _ex_1()*power(1-power(x,_ex2()),_ex_1_2()); } -REGISTER_FUNCTION(acos, acos_eval, acos_evalf, acos_diff, NULL); +REGISTER_FUNCTION(acos, eval_func(acos_eval). + evalf_func(acos_evalf). + derivative_func(acos_deriv). + latex_name("\\arccos")); ////////// // inverse tangent (arc tangent) ////////// -static ex atan_evalf(ex const & x) +static ex atan_evalf(const ex & x) { - BEGIN_TYPECHECK - TYPECHECK(x,numeric) - END_TYPECHECK(atan(x)) - - return atan(ex_to_numeric(x)); // -> numeric atan(numeric) + BEGIN_TYPECHECK + TYPECHECK(x,numeric) + END_TYPECHECK(atan(x)) + + return atan(ex_to_numeric(x)); // -> numeric atan(numeric) } -static ex atan_eval(ex const & x) +static ex atan_eval(const ex & x) { - if (x.info(info_flags::numeric)) { - // atan(0) -> 0 - if (x.is_equal(exZERO())) - return exZERO(); - // atan(float) -> float - if (!x.info(info_flags::rational)) - return atan_evalf(x); - } - - return atan(x).hold(); -} + if (x.info(info_flags::numeric)) { + // atan(0) -> 0 + if (x.is_equal(_ex0())) + 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(x).hold(); +} -static ex atan_diff(ex const & x, unsigned diff_param) +static ex atan_deriv(const ex & x, unsigned deriv_param) { - GINAC_ASSERT(diff_param==0); + GINAC_ASSERT(deriv_param==0); - return power(1+x*x, -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_ex_exactly_of_type(rel.lhs(),symbol)); + // 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 = static_cast(rel.lhs().bp); + const ex point = rel.rhs(); + const symbol foo; + 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, atan_eval, atan_evalf, atan_diff, NULL); +REGISTER_FUNCTION(atan, eval_func(atan_eval). + evalf_func(atan_evalf). + derivative_func(atan_deriv). + series_func(atan_series). + latex_name("\\arctan")); ////////// // inverse tangent (atan2(y,x)) ////////// -static ex atan2_evalf(ex const & y, ex const & x) +static ex atan2_evalf(const ex & y, const ex & x) { - BEGIN_TYPECHECK - TYPECHECK(y,numeric) - TYPECHECK(x,numeric) - END_TYPECHECK(atan2(y,x)) - - return atan(ex_to_numeric(y),ex_to_numeric(x)); // -> numeric atan(numeric) + 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) } -static ex atan2_eval(ex const & y, ex const & x) +static ex atan2_eval(const ex & y, const ex & x) { - if (y.info(info_flags::numeric) && !y.info(info_flags::rational) && - x.info(info_flags::numeric) && !x.info(info_flags::rational)) { - return atan2_evalf(y,x); - } - - return atan2(y,x).hold(); + if (y.info(info_flags::numeric) && !y.info(info_flags::crational) && + x.info(info_flags::numeric) && !x.info(info_flags::crational)) { + return atan2_evalf(y,x); + } + + return atan2(y,x).hold(); } -static ex atan2_diff(ex const & y, ex const & x, unsigned diff_param) +static ex atan2_deriv(const ex & y, const ex & x, unsigned deriv_param) { - GINAC_ASSERT(diff_param<2); - - if (diff_param==0) { - // d/dy atan(y,x) - return pow(x*(1+y*y/(x*x)),-1); - } - // d/dx atan(y,x) - return -y*pow(x*x+y*y,-1); + GINAC_ASSERT(deriv_param<2); + + if (deriv_param==0) { + // d/dy atan(y,x) + return x*power(power(x,_ex2())+power(y,_ex2()),_ex_1()); + } + // d/dx atan(y,x) + return -y*power(power(x,_ex2())+power(y,_ex2()),_ex_1()); } -REGISTER_FUNCTION(atan2, atan2_eval, atan2_evalf, atan2_diff, NULL); +REGISTER_FUNCTION(atan2, eval_func(atan2_eval). + evalf_func(atan2_evalf). + derivative_func(atan2_deriv)); ////////// // hyperbolic sine (trigonometric function) ////////// -static ex sinh_evalf(ex const & x) +static ex sinh_evalf(const ex & x) { - BEGIN_TYPECHECK - TYPECHECK(x,numeric) - END_TYPECHECK(sinh(x)) - - return sinh(ex_to_numeric(x)); // -> numeric sinh(numeric) + BEGIN_TYPECHECK + TYPECHECK(x,numeric) + END_TYPECHECK(sinh(x)) + + return sinh(ex_to_numeric(x)); // -> numeric sinh(numeric) } -static ex sinh_eval(ex const & x) +static ex sinh_eval(const ex & x) { - if (x.info(info_flags::numeric)) { - // sinh(0) -> 0 - if (x.is_zero()) - return exZERO(); - // sinh(float) -> float - if (!x.info(info_flags::rational)) - return sinh_evalf(x); - } - - if (is_ex_exactly_of_type(x, function)) { - ex t=x.op(0); - // sinh(asinh(x)) -> x - if (is_ex_the_function(x, asinh)) - return t; - // sinh(acosh(x)) -> (x-1)^(1/2) * (x+1)^(1/2) - if (is_ex_the_function(x, acosh)) - return power(t-exONE(),exHALF())*power(t+exONE(),exHALF()); - // sinh(atanh(x)) -> x*(1-x^2)^(-1/2) - if (is_ex_the_function(x, atanh)) - return t*power(exONE()-power(t,exTWO()),exMINUSHALF()); - } - - return sinh(x).hold(); + if (x.info(info_flags::numeric)) { + if (x.is_zero()) // sinh(0) -> 0 + return _ex0(); + if (!x.info(info_flags::crational)) // sinh(float) -> float + return sinh_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(); } -static ex sinh_diff(ex const & x, unsigned diff_param) +static ex sinh_deriv(const ex & x, unsigned deriv_param) { - GINAC_ASSERT(diff_param==0); - - return cosh(x); + GINAC_ASSERT(deriv_param==0); + + // d/dx sinh(x) -> cosh(x) + return cosh(x); } -REGISTER_FUNCTION(sinh, sinh_eval, sinh_evalf, sinh_diff, NULL); +REGISTER_FUNCTION(sinh, eval_func(sinh_eval). + evalf_func(sinh_evalf). + derivative_func(sinh_deriv). + latex_name("\\sinh")); ////////// // hyperbolic cosine (trigonometric function) ////////// -static ex cosh_evalf(ex const & x) +static ex cosh_evalf(const ex & x) { - BEGIN_TYPECHECK - TYPECHECK(x,numeric) - END_TYPECHECK(cosh(x)) - - return cosh(ex_to_numeric(x)); // -> numeric cosh(numeric) + BEGIN_TYPECHECK + TYPECHECK(x,numeric) + END_TYPECHECK(cosh(x)) + + return cosh(ex_to_numeric(x)); // -> numeric cosh(numeric) } -static ex cosh_eval(ex const & x) +static ex cosh_eval(const ex & x) { - if (x.info(info_flags::numeric)) { - // cosh(0) -> 1 - if (x.is_zero()) - return exONE(); - // cosh(float) -> float - if (!x.info(info_flags::rational)) - return cosh_evalf(x); - } - - if (is_ex_exactly_of_type(x, function)) { - ex t=x.op(0); - // cosh(acosh(x)) -> x - if (is_ex_the_function(x, acosh)) - return t; - // cosh(asinh(x)) -> (1+x^2)^(1/2) - if (is_ex_the_function(x, asinh)) - return power(exONE()+power(t,exTWO()),exHALF()); - // cosh(atanh(x)) -> (1-x^2)^(-1/2) - if (is_ex_the_function(x, atanh)) - return power(exONE()-power(t,exTWO()),exMINUSHALF()); - } - - return cosh(x).hold(); + if (x.info(info_flags::numeric)) { + if (x.is_zero()) // cosh(0) -> 1 + return _ex1(); + if (!x.info(info_flags::crational)) // cosh(float) -> float + return cosh_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(); } -static ex cosh_diff(ex const & x, unsigned diff_param) +static ex cosh_deriv(const ex & x, unsigned deriv_param) { - GINAC_ASSERT(diff_param==0); - - return sinh(x); + GINAC_ASSERT(deriv_param==0); + + // d/dx cosh(x) -> sinh(x) + return sinh(x); } -REGISTER_FUNCTION(cosh, cosh_eval, cosh_evalf, cosh_diff, NULL); +REGISTER_FUNCTION(cosh, eval_func(cosh_eval). + evalf_func(cosh_evalf). + derivative_func(cosh_deriv). + latex_name("\\cosh")); ////////// // hyperbolic tangent (trigonometric function) ////////// -static ex tanh_evalf(ex const & x) +static ex tanh_evalf(const ex & x) { - BEGIN_TYPECHECK - TYPECHECK(x,numeric) - END_TYPECHECK(tanh(x)) - - return tanh(ex_to_numeric(x)); // -> numeric tanh(numeric) + BEGIN_TYPECHECK + TYPECHECK(x,numeric) + END_TYPECHECK(tanh(x)) + + return tanh(ex_to_numeric(x)); // -> numeric tanh(numeric) } -static ex tanh_eval(ex const & x) +static ex tanh_eval(const ex & x) { - if (x.info(info_flags::numeric)) { - // tanh(0) -> 0 - if (x.is_zero()) - return exZERO(); - // tanh(float) -> float - if (!x.info(info_flags::rational)) - return tanh_evalf(x); - } - - if (is_ex_exactly_of_type(x, function)) { - ex t=x.op(0); - // tanh(atanh(x)) -> x - if (is_ex_the_function(x, atanh)) - return t; - // tanh(asinh(x)) -> x*(1+x^2)^(-1/2) - if (is_ex_the_function(x, asinh)) - return t*power(exONE()+power(t,exTWO()),exMINUSHALF()); - // tanh(acosh(x)) -> (x-1)^(1/2)*(x+1)^(1/2)/x - if (is_ex_the_function(x, acosh)) - return power(t-exONE(),exHALF())*power(t+exONE(),exHALF())*power(t,exMINUSONE()); - } - - return tanh(x).hold(); + if (x.info(info_flags::numeric)) { + if (x.is_zero()) // tanh(0) -> 0 + return _ex0(); + if (!x.info(info_flags::crational)) // tanh(float) -> float + return tanh_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(); } -static ex tanh_diff(ex const & x, unsigned diff_param) +static ex tanh_deriv(const ex & x, unsigned deriv_param) { - GINAC_ASSERT(diff_param==0); - - return exONE()-power(tanh(x),exTWO()); + GINAC_ASSERT(deriv_param==0); + + // d/dx tanh(x) -> 1-tanh(x)^2 + return _ex1()-power(tanh(x),_ex2()); } -REGISTER_FUNCTION(tanh, tanh_eval, tanh_evalf, tanh_diff, NULL); +static ex tanh_series(const ex &x, + const relational &rel, + int order, + unsigned options) +{ + GINAC_ASSERT(is_ex_exactly_of_type(rel.lhs(),symbol)); + // 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); + if (!(2*I*x_pt/Pi).info(info_flags::odd)) + throw do_taylor(); // caught by function::series() + // if we got here we have to care for a simple pole + return (sinh(x)/cosh(x)).series(rel, order+2, options); +} + +REGISTER_FUNCTION(tanh, eval_func(tanh_eval). + evalf_func(tanh_evalf). + derivative_func(tanh_deriv). + series_func(tanh_series). + latex_name("\\tanh")); ////////// // inverse hyperbolic sine (trigonometric function) ////////// -static ex asinh_evalf(ex const & x) +static ex asinh_evalf(const ex & x) { - BEGIN_TYPECHECK - TYPECHECK(x,numeric) - END_TYPECHECK(asinh(x)) - - return asinh(ex_to_numeric(x)); // -> numeric asinh(numeric) + BEGIN_TYPECHECK + TYPECHECK(x,numeric) + END_TYPECHECK(asinh(x)) + + return asinh(ex_to_numeric(x)); // -> numeric asinh(numeric) } -static ex asinh_eval(ex const & x) +static ex asinh_eval(const ex & x) { - if (x.info(info_flags::numeric)) { - // asinh(0) -> 0 - if (x.is_zero()) - return exZERO(); - // asinh(float) -> float - if (!x.info(info_flags::rational)) - return asinh_evalf(x); - } - - return asinh(x).hold(); + if (x.info(info_flags::numeric)) { + // asinh(0) -> 0 + if (x.is_zero()) + return _ex0(); + // asinh(float) -> float + if (!x.info(info_flags::crational)) + return asinh_evalf(x); + } + + return asinh(x).hold(); } -static ex asinh_diff(ex const & x, unsigned diff_param) +static ex asinh_deriv(const ex & x, unsigned deriv_param) { - GINAC_ASSERT(diff_param==0); - - return power(1+power(x,exTWO()),exMINUSHALF()); + GINAC_ASSERT(deriv_param==0); + + // d/dx asinh(x) -> 1/sqrt(1+x^2) + return power(_ex1()+power(x,_ex2()),_ex_1_2()); } -REGISTER_FUNCTION(asinh, asinh_eval, asinh_evalf, asinh_diff, NULL); +REGISTER_FUNCTION(asinh, eval_func(asinh_eval). + evalf_func(asinh_evalf). + derivative_func(asinh_deriv)); ////////// // inverse hyperbolic cosine (trigonometric function) ////////// -static ex acosh_evalf(ex const & x) +static ex acosh_evalf(const ex & x) { - BEGIN_TYPECHECK - TYPECHECK(x,numeric) - END_TYPECHECK(acosh(x)) - - return acosh(ex_to_numeric(x)); // -> numeric acosh(numeric) + BEGIN_TYPECHECK + TYPECHECK(x,numeric) + END_TYPECHECK(acosh(x)) + + return acosh(ex_to_numeric(x)); // -> numeric acosh(numeric) } -static ex acosh_eval(ex const & x) +static ex acosh_eval(const ex & x) { - if (x.info(info_flags::numeric)) { - // acosh(0) -> Pi*I/2 - if (x.is_zero()) - return Pi*I*numeric(1,2); - // acosh(1) -> 0 - if (x.is_equal(exONE())) - return exZERO(); - // acosh(-1) -> Pi*I - if (x.is_equal(exMINUSONE())) - return Pi*I; - // acosh(float) -> float - if (!x.info(info_flags::rational)) - return acosh_evalf(x); - } - - return acosh(x).hold(); + if (x.info(info_flags::numeric)) { + // acosh(0) -> Pi*I/2 + if (x.is_zero()) + return Pi*I*numeric(1,2); + // acosh(1) -> 0 + if (x.is_equal(_ex1())) + return _ex0(); + // acosh(-1) -> Pi*I + if (x.is_equal(_ex_1())) + return Pi*I; + // acosh(float) -> float + if (!x.info(info_flags::crational)) + return acosh_evalf(x); + } + + return acosh(x).hold(); } -static ex acosh_diff(ex const & x, unsigned diff_param) +static ex acosh_deriv(const ex & x, unsigned deriv_param) { - GINAC_ASSERT(diff_param==0); - - return power(x-1,exMINUSHALF())*power(x+1,exMINUSHALF()); + GINAC_ASSERT(deriv_param==0); + + // d/dx acosh(x) -> 1/(sqrt(x-1)*sqrt(x+1)) + return power(x+_ex_1(),_ex_1_2())*power(x+_ex1(),_ex_1_2()); } -REGISTER_FUNCTION(acosh, acosh_eval, acosh_evalf, acosh_diff, NULL); +REGISTER_FUNCTION(acosh, eval_func(acosh_eval). + evalf_func(acosh_evalf). + derivative_func(acosh_deriv)); ////////// // inverse hyperbolic tangent (trigonometric function) ////////// -static ex atanh_evalf(ex const & x) +static ex atanh_evalf(const ex & x) { - BEGIN_TYPECHECK - TYPECHECK(x,numeric) - END_TYPECHECK(atanh(x)) - - return atanh(ex_to_numeric(x)); // -> numeric atanh(numeric) + BEGIN_TYPECHECK + TYPECHECK(x,numeric) + END_TYPECHECK(atanh(x)) + + return atanh(ex_to_numeric(x)); // -> numeric atanh(numeric) } -static ex atanh_eval(ex const & x) +static ex atanh_eval(const ex & x) { - if (x.info(info_flags::numeric)) { - // atanh(0) -> 0 - if (x.is_zero()) - return exZERO(); - // atanh({+|-}1) -> throw - if (x.is_equal(exONE()) || x.is_equal(exONE())) - throw (std::domain_error("atanh_eval(): infinity")); - // atanh(float) -> float - if (!x.info(info_flags::rational)) - return atanh_evalf(x); - } - - return atanh(x).hold(); + if (x.info(info_flags::numeric)) { + // atanh(0) -> 0 + if (x.is_zero()) + return _ex0(); + // atanh({+|-}1) -> throw + if (x.is_equal(_ex1()) || x.is_equal(_ex_1())) + throw (pole_error("atanh_eval(): logarithmic pole",0)); + // atanh(float) -> float + if (!x.info(info_flags::crational)) + return atanh_evalf(x); + } + + return atanh(x).hold(); } -static ex atanh_diff(ex const & x, unsigned diff_param) +static ex atanh_deriv(const ex & x, unsigned deriv_param) { - GINAC_ASSERT(diff_param==0); - - return power(exONE()-power(x,exTWO()),exMINUSONE()); + GINAC_ASSERT(deriv_param==0); + + // d/dx atanh(x) -> 1/(1-x^2) + return power(_ex1()-power(x,_ex2()),_ex_1()); } -REGISTER_FUNCTION(atanh, atanh_eval, atanh_evalf, atanh_diff, NULL); +static ex atanh_series(const ex &arg, + const relational &rel, + int order, + unsigned options) +{ + GINAC_ASSERT(is_ex_exactly_of_type(rel.lhs(),symbol)); + // 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 = static_cast(rel.lhs().bp); + const ex point = rel.rhs(); + const symbol foo; + 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). + series_func(atanh_series)); + } // namespace GiNaC