X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Finifcns_trans.cpp;h=239b1ea2ea1d8a3130130a9e32ee77d28458cf3b;hp=a626ce1c29332370ad13e471554e3cf4f5260ba0;hb=f78b1f296310b5f1c01b74c9fb10dd33af2a8f4a;hpb=6bff4f32e5cbe9f7e51837f392a2d0bf0d5b721d diff --git a/ginac/inifcns_trans.cpp b/ginac/inifcns_trans.cpp index a626ce1c..239b1ea2 100644 --- a/ginac/inifcns_trans.cpp +++ b/ginac/inifcns_trans.cpp @@ -4,7 +4,7 @@ * functions. */ /* - * GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany + * GiNaC Copyright (C) 1999-2003 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 +29,7 @@ #include "constant.h" #include "numeric.h" #include "power.h" +#include "operators.h" #include "relational.h" #include "symbol.h" #include "pseries.h" @@ -52,19 +53,19 @@ static ex exp_eval(const ex & x) { // exp(0) -> 1 if (x.is_zero()) { - return _ex1(); + return _ex1; } // exp(n*Pi*I/2) -> {+1|+I|-1|-I} - const ex TwoExOverPiI=(_ex2()*x)/(Pi*I); + const ex TwoExOverPiI=(_ex2*x)/(Pi*I); if (TwoExOverPiI.info(info_flags::integer)) { - numeric z=mod(ex_to(TwoExOverPiI),_num4()); - if (z.is_equal(_num0())) - return _ex1(); - if (z.is_equal(_num1())) + const numeric z = mod(ex_to(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())) + if (z.is_equal(_num2)) + return _ex_1; + if (z.is_equal(_num3)) return ex(-I); } // exp(log(x)) -> x @@ -110,21 +111,21 @@ static ex log_eval(const ex & x) 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(_ex1)) // log(1) -> 0 + return _ex0; if (x.is_equal(I)) // log(I) -> Pi*I/2 - return (Pi*I*_num1_2()); + return (Pi*I*_num1_2); if (x.is_equal(-I)) // log(-I) -> -Pi*I/2 - return (Pi*I*_num_1_2()); + return (Pi*I*_num_1_2); // log(float) if (!x.info(info_flags::crational)) return log(ex_to(x)); } // log(exp(t)) -> t (if -Pi < t.imag() <= Pi): if (is_ex_the_function(x, exp)) { - ex t = x.op(0); + const ex &t = x.op(0); if (t.info(info_flags::numeric)) { - numeric nt = ex_to(t); + const numeric &nt = ex_to(t); if (nt.is_real()) return t; } @@ -138,7 +139,7 @@ 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()); + return power(x, _ex_1); } static ex log_series(const ex &arg, @@ -146,12 +147,12 @@ static ex log_series(const ex &arg, int order, unsigned options) { - GINAC_ASSERT(is_ex_exactly_of_type(rel.lhs(),symbol)); + GINAC_ASSERT(is_a(rel.lhs())); ex arg_pt; bool must_expand_arg = false; // maybe substitution of rel into arg fails because of a pole try { - arg_pt = arg.subs(rel); + arg_pt = arg.subs(rel, subs_options::no_pattern); } catch (pole_error) { must_expand_arg = true; } @@ -164,27 +165,35 @@ 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(arg.series(rel, order, options)); - const symbol *s = static_cast(rel.lhs().bp); - const ex point = rel.rhs(); - const int n = argser.ldegree(*s); + pseries argser; // series expansion of log's argument + unsigned extra_ord = 0; // extra expansion order + do { + // oops, the argument expanded to a pure Order(x^something)... + argser = ex_to(arg.series(rel, order+extra_ord, options)); + ++extra_ord; + } while (!argser.is_terminating() && argser.nops()==1); + + const symbol &s = ex_to(rel.lhs()); + const ex &point = rel.rhs(); + const int n = argser.ldegree(s); epvector seq; // construct what we carelessly called the n*log(x) term above - ex coeff = argser.coeff(*s, n); + 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())); + seq.push_back(expair(n*log(s-point)+log(coeff), _ex0)); else - seq.push_back(expair(log(coeff*pow(*s-point, n)), _ex0())); + 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 + // in this case n more (or less) terms are needed // (sadly, to generate them, we have to start from the beginning) - ex newarg = ex_to((arg/coeff).series(rel, order+n, options)).shift_exponents(-n).convert_to_poly(true); + 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); @@ -194,13 +203,13 @@ static ex log_series(const ex &arg, // 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 &s = ex_to(rel.lhs()); + const ex &point = rel.rhs(); const symbol foo; - const ex replarg = series(log(arg), *s==foo, order).subs(foo==point); + const ex replarg = series(log(arg), s==foo, order).subs(foo==point, subs_options::no_pattern); epvector seq; - seq.push_back(expair(-I*csgn(arg*I)*Pi, _ex0())); - seq.push_back(expair(Order(_ex1()), order)); + 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() @@ -227,50 +236,50 @@ static ex sin_evalf(const ex & x) static ex sin_eval(const ex & x) { // sin(n/d*Pi) -> { all known non-nested radicals } - const ex SixtyExOverPi = _ex60()*x/Pi; - ex sign = _ex1(); + 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); + if (z>=_num60) { // wrap to interval [0, Pi) - z -= _num60(); - sign = _ex_1(); + z -= _num60; + sign = _ex_1; } - if (z>_num30()) { + if (z>_num30) { // wrap to interval [0, Pi/2) - z = _num60()-z; + 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 (z.is_equal(_num0)) // sin(0) -> 0 + return _ex0; + if (z.is_equal(_num5)) // sin(Pi/12) -> sqrt(6)/4*(1-sqrt(3)/3) + return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex_1_3*sqrt(_ex3)); + if (z.is_equal(_num6)) // sin(Pi/10) -> sqrt(5)/4-1/4 + return sign*(_ex1_4*sqrt(_ex5)+_ex_1_4); + if (z.is_equal(_num10)) // sin(Pi/6) -> 1/2 + return sign*_ex1_2; + if (z.is_equal(_num15)) // sin(Pi/4) -> sqrt(2)/2 + return sign*_ex1_2*sqrt(_ex2); + if (z.is_equal(_num18)) // sin(3/10*Pi) -> sqrt(5)/4+1/4 + return sign*(_ex1_4*sqrt(_ex5)+_ex1_4); + if (z.is_equal(_num20)) // sin(Pi/3) -> sqrt(3)/2 + return sign*_ex1_2*sqrt(_ex3); + if (z.is_equal(_num25)) // sin(5/12*Pi) -> sqrt(6)/4*(1+sqrt(3)/3) + return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex1_3*sqrt(_ex3)); + if (z.is_equal(_num30)) // sin(Pi/2) -> 1 + return sign; } if (is_exactly_a(x)) { - ex t = x.op(0); + const ex &t = x.op(0); // sin(asin(x)) -> x if (is_ex_the_function(x, asin)) return t; // sin(acos(x)) -> sqrt(1-x^2) if (is_ex_the_function(x, acos)) - return power(_ex1()-power(t,_ex2()),_ex1_2()); - // sin(atan(x)) -> x*(1+x^2)^(-1/2) + return sqrt(_ex1-power(t,_ex2)); + // sin(atan(x)) -> x/sqrt(1+x^2) if (is_ex_the_function(x, atan)) - return t*power(_ex1()+power(t,_ex2()),_ex_1_2()); + return t*power(_ex1+power(t,_ex2),_ex_1_2); } // sin(float) -> float @@ -308,50 +317,50 @@ static ex cos_evalf(const ex & x) static ex cos_eval(const ex & x) { // cos(n/d*Pi) -> { all known non-nested radicals } - const ex SixtyExOverPi = _ex60()*x/Pi; - ex sign = _ex1(); + 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); + if (z>=_num60) { // wrap to interval [0, Pi) - z = _num120()-z; + z = _num120-z; } - if (z>=_num30()) { + if (z>=_num30) { // wrap to interval [0, Pi/2) - z = _num60()-z; - sign = _ex_1(); + 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 (z.is_equal(_num0)) // cos(0) -> 1 + return sign; + if (z.is_equal(_num5)) // cos(Pi/12) -> sqrt(6)/4*(1+sqrt(3)/3) + return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex1_3*sqrt(_ex3)); + if (z.is_equal(_num10)) // cos(Pi/6) -> sqrt(3)/2 + return sign*_ex1_2*sqrt(_ex3); + if (z.is_equal(_num12)) // cos(Pi/5) -> sqrt(5)/4+1/4 + return sign*(_ex1_4*sqrt(_ex5)+_ex1_4); + if (z.is_equal(_num15)) // cos(Pi/4) -> sqrt(2)/2 + return sign*_ex1_2*sqrt(_ex2); + if (z.is_equal(_num20)) // cos(Pi/3) -> 1/2 + return sign*_ex1_2; + if (z.is_equal(_num24)) // cos(2/5*Pi) -> sqrt(5)/4-1/4x + return sign*(_ex1_4*sqrt(_ex5)+_ex_1_4); + if (z.is_equal(_num25)) // cos(5/12*Pi) -> sqrt(6)/4*(1-sqrt(3)/3) + return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex_1_3*sqrt(_ex3)); + if (z.is_equal(_num30)) // cos(Pi/2) -> 0 + return _ex0; } if (is_exactly_a(x)) { - ex t = x.op(0); + const 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) + // cos(asin(x)) -> sqrt(1-x^2) if (is_ex_the_function(x, asin)) - return power(_ex1()-power(t,_ex2()),_ex1_2()); - // cos(atan(x)) -> (1+x^2)^(-1/2) + return sqrt(_ex1-power(t,_ex2)); + // cos(atan(x)) -> 1/sqrt(1+x^2) if (is_ex_the_function(x, atan)) - return power(_ex1()+power(t,_ex2()),_ex_1_2()); + return power(_ex1+power(t,_ex2),_ex_1_2); } // cos(float) -> float @@ -366,7 +375,7 @@ 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); + return -sin(x); } REGISTER_FUNCTION(cos, eval_func(cos_eval). @@ -389,46 +398,46 @@ static ex tan_evalf(const ex & x) static ex tan_eval(const ex & x) { // tan(n/d*Pi) -> { all known non-nested radicals } - const ex SixtyExOverPi = _ex60()*x/Pi; - ex sign = _ex1(); + 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); + if (z>=_num60) { // wrap to interval [0, Pi) - z -= _num60(); + z -= _num60; } - if (z>=_num30()) { + if (z>=_num30) { // wrap to interval [0, Pi/2) - z = _num60()-z; - sign = _ex_1(); + 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 + if (z.is_equal(_num0)) // tan(0) -> 0 + return _ex0; + if (z.is_equal(_num5)) // tan(Pi/12) -> 2-sqrt(3) + return sign*(_ex2-sqrt(_ex3)); + if (z.is_equal(_num10)) // tan(Pi/6) -> sqrt(3)/3 + return sign*_ex1_3*sqrt(_ex3); + if (z.is_equal(_num15)) // tan(Pi/4) -> 1 + return sign; + if (z.is_equal(_num20)) // tan(Pi/3) -> sqrt(3) + return sign*sqrt(_ex3); + if (z.is_equal(_num25)) // tan(5/12*Pi) -> 2+sqrt(3) + return sign*(sqrt(_ex3)+_ex2); + if (z.is_equal(_num30)) // tan(Pi/2) -> infinity throw (pole_error("tan_eval(): simple pole",1)); } if (is_exactly_a(x)) { - ex t = x.op(0); + const 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) + // tan(asin(x)) -> x/sqrt(1+x^2) if (is_ex_the_function(x, asin)) - return t*power(_ex1()-power(t,_ex2()),_ex_1_2()); - // tan(acos(x)) -> (1-x^2)^(1/2)/x + return t*power(_ex1-power(t,_ex2),_ex_1_2); + // tan(acos(x)) -> sqrt(1-x^2)/x if (is_ex_the_function(x, acos)) - return power(t,_ex_1())*power(_ex1()-power(t,_ex2()),_ex1_2()); + return power(t,_ex_1)*sqrt(_ex1-power(t,_ex2)); } // tan(float) -> float @@ -444,7 +453,7 @@ 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())); + return (_ex1+power(tan(x),_ex2)); } static ex tan_series(const ex &x, @@ -452,11 +461,11 @@ static ex tan_series(const ex &x, int order, unsigned options) { - GINAC_ASSERT(is_ex_exactly_of_type(rel.lhs(),symbol)); + GINAC_ASSERT(is_a(rel.lhs())); // method: // Taylor series where there is no pole falls back to tan_deriv. // On a pole simply expand sin(x)/cos(x). - const ex x_pt = x.subs(rel); + const ex x_pt = x.subs(rel, subs_options::no_pattern); if (!(2*x_pt/Pi).info(info_flags::odd)) throw do_taylor(); // caught by function::series() // if we got here we have to care for a simple pole @@ -488,17 +497,17 @@ static ex asin_eval(const ex & x) if (x.is_zero()) return x; // asin(1/2) -> Pi/6 - if (x.is_equal(_ex1_2())) + if (x.is_equal(_ex1_2)) return numeric(1,6)*Pi; // asin(1) -> Pi/2 - if (x.is_equal(_ex1())) - return _num1_2()*Pi; + if (x.is_equal(_ex1)) + return _num1_2*Pi; // asin(-1/2) -> -Pi/6 - if (x.is_equal(_ex_1_2())) + 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; + if (x.is_equal(_ex_1)) + return _num_1_2*Pi; // asin(float) -> float if (!x.info(info_flags::crational)) return asin(ex_to(x)); @@ -512,7 +521,7 @@ static ex asin_deriv(const ex & x, unsigned deriv_param) GINAC_ASSERT(deriv_param==0); // d/dx asin(x) -> 1/sqrt(1-x^2) - return power(1-power(x,_ex2()),_ex_1_2()); + return power(1-power(x,_ex2),_ex_1_2); } REGISTER_FUNCTION(asin, eval_func(asin_eval). @@ -536,19 +545,19 @@ static ex acos_eval(const ex & x) { if (x.info(info_flags::numeric)) { // acos(1) -> 0 - if (x.is_equal(_ex1())) - return _ex0(); + if (x.is_equal(_ex1)) + return _ex0; // acos(1/2) -> Pi/3 - if (x.is_equal(_ex1_2())) - return _ex1_3()*Pi; + if (x.is_equal(_ex1_2)) + return _ex1_3*Pi; // acos(0) -> Pi/2 if (x.is_zero()) - return _ex1_2()*Pi; + return _ex1_2*Pi; // acos(-1/2) -> 2/3*Pi - if (x.is_equal(_ex_1_2())) + if (x.is_equal(_ex_1_2)) return numeric(2,3)*Pi; // acos(-1) -> Pi - if (x.is_equal(_ex_1())) + if (x.is_equal(_ex_1)) return Pi; // acos(float) -> float if (!x.info(info_flags::crational)) @@ -563,7 +572,7 @@ static ex acos_deriv(const ex & x, unsigned deriv_param) GINAC_ASSERT(deriv_param==0); // d/dx acos(x) -> -1/sqrt(1-x^2) - return _ex_1()*power(1-power(x,_ex2()),_ex_1_2()); + return -power(1-power(x,_ex2),_ex_1_2); } REGISTER_FUNCTION(acos, eval_func(acos_eval). @@ -588,13 +597,13 @@ static ex atan_eval(const ex & x) if (x.info(info_flags::numeric)) { // atan(0) -> 0 if (x.is_zero()) - return _ex0(); + return _ex0; // atan(1) -> Pi/4 - if (x.is_equal(_ex1())) - return _ex1_4()*Pi; + 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(_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 @@ -610,7 +619,7 @@ static ex atan_deriv(const ex & x, unsigned deriv_param) GINAC_ASSERT(deriv_param==0); // d/dx atan(x) -> 1/(1+x^2) - return power(_ex1()+power(x,_ex2()), _ex_1()); + return power(_ex1+power(x,_ex2), _ex_1); } static ex atan_series(const ex &arg, @@ -618,7 +627,7 @@ static ex atan_series(const ex &arg, int order, unsigned options) { - GINAC_ASSERT(is_ex_exactly_of_type(rel.lhs(),symbol)); + GINAC_ASSERT(is_a(rel.lhs())); // method: // Taylor series where there is no pole or cut falls back to atan_deriv. // There are two branch cuts, one runnig from I up the imaginary axis and @@ -627,10 +636,10 @@ static ex atan_series(const ex &arg, // 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); + const ex arg_pt = arg.subs(rel, subs_options::no_pattern); if (!(I*arg_pt).info(info_flags::real)) throw do_taylor(); // Re(x) != 0 - if ((I*arg_pt).info(info_flags::real) && abs(I*arg_pt)<_ex1()) + 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)) @@ -639,18 +648,18 @@ static ex atan_series(const ex &arg, // 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 &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(); + const ex replarg = series(atan(arg), s==foo, order).subs(foo==point, subs_options::no_pattern); + ex Order0correction = replarg.op(0)+csgn(arg)*Pi*_ex_1_2; + if ((I*arg_pt)<_ex0) + Order0correction += log((I*arg_pt+_ex_1)/(I*arg_pt+_ex1))*I*_ex_1_2; else - Order0correction += log((I*arg_pt+_ex1())/(I*arg_pt+_ex_1()))*I*_ex1_2(); + 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)); + seq.push_back(expair(Order0correction, _ex0)); + seq.push_back(expair(Order(_ex1), order)); return series(replarg - pseries(rel, seq), rel, order); } throw do_taylor(); @@ -690,10 +699,10 @@ static ex atan2_deriv(const ex & y, const ex & x, unsigned deriv_param) if (deriv_param==0) { // d/dy atan(y,x) - return x*power(power(x,_ex2())+power(y,_ex2()),_ex_1()); + 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()); + return -y*power(power(x,_ex2)+power(y,_ex2),_ex_1); } REGISTER_FUNCTION(atan2, eval_func(atan2_eval). @@ -716,7 +725,7 @@ static ex sinh_eval(const ex & x) { if (x.info(info_flags::numeric)) { if (x.is_zero()) // sinh(0) -> 0 - return _ex0(); + return _ex0; if (!x.info(info_flags::crational)) // sinh(float) -> float return sinh(ex_to(x)); } @@ -726,16 +735,16 @@ static ex sinh_eval(const ex & x) return I*sin(x/I); if (is_exactly_a(x)) { - ex t = x.op(0); + const 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) + // sinh(acosh(x)) -> sqrt(x-1) * sqrt(x+1) 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) + return sqrt(t-_ex1)*sqrt(t+_ex1); + // sinh(atanh(x)) -> x/sqrt(1-x^2) if (is_ex_the_function(x, atanh)) - return t*power(_ex1()-power(t,_ex2()),_ex_1_2()); + return t*power(_ex1-power(t,_ex2),_ex_1_2); } return sinh(x).hold(); @@ -770,7 +779,7 @@ static ex cosh_eval(const ex & x) { if (x.info(info_flags::numeric)) { if (x.is_zero()) // cosh(0) -> 1 - return _ex1(); + return _ex1; if (!x.info(info_flags::crational)) // cosh(float) -> float return cosh(ex_to(x)); } @@ -780,16 +789,16 @@ static ex cosh_eval(const ex & x) return cos(x/I); if (is_exactly_a(x)) { - ex t = x.op(0); + const 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) + // cosh(asinh(x)) -> sqrt(1+x^2) if (is_ex_the_function(x, asinh)) - return power(_ex1()+power(t,_ex2()),_ex1_2()); - // cosh(atanh(x)) -> (1-x^2)^(-1/2) + return sqrt(_ex1+power(t,_ex2)); + // cosh(atanh(x)) -> 1/sqrt(1-x^2) if (is_ex_the_function(x, atanh)) - return power(_ex1()-power(t,_ex2()),_ex_1_2()); + return power(_ex1-power(t,_ex2),_ex_1_2); } return cosh(x).hold(); @@ -824,7 +833,7 @@ static ex tanh_eval(const ex & x) { if (x.info(info_flags::numeric)) { if (x.is_zero()) // tanh(0) -> 0 - return _ex0(); + return _ex0; if (!x.info(info_flags::crational)) // tanh(float) -> float return tanh(ex_to(x)); } @@ -834,16 +843,16 @@ static ex tanh_eval(const ex & x) return I*tan(x/I); if (is_exactly_a(x)) { - ex t = x.op(0); + const 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) + // tanh(asinh(x)) -> x/sqrt(1+x^2) if (is_ex_the_function(x, asinh)) - return t*power(_ex1()+power(t,_ex2()),_ex_1_2()); - // tanh(acosh(x)) -> (x-1)^(1/2)*(x+1)^(1/2)/x + return t*power(_ex1+power(t,_ex2),_ex_1_2); + // tanh(acosh(x)) -> sqrt(x-1)*sqrt(x+1)/x if (is_ex_the_function(x, acosh)) - return power(t-_ex1(),_ex1_2())*power(t+_ex1(),_ex1_2())*power(t,_ex_1()); + return sqrt(t-_ex1)*sqrt(t+_ex1)*power(t,_ex_1); } return tanh(x).hold(); @@ -854,7 +863,7 @@ static ex tanh_deriv(const ex & x, unsigned deriv_param) GINAC_ASSERT(deriv_param==0); // d/dx tanh(x) -> 1-tanh(x)^2 - return _ex1()-power(tanh(x),_ex2()); + return _ex1-power(tanh(x),_ex2); } static ex tanh_series(const ex &x, @@ -862,11 +871,11 @@ static ex tanh_series(const ex &x, int order, unsigned options) { - GINAC_ASSERT(is_ex_exactly_of_type(rel.lhs(),symbol)); + GINAC_ASSERT(is_a(rel.lhs())); // method: // Taylor series where there is no pole falls back to tanh_deriv. // On a pole simply expand sinh(x)/cosh(x). - const ex x_pt = x.subs(rel); + const ex x_pt = x.subs(rel, subs_options::no_pattern); if (!(2*I*x_pt/Pi).info(info_flags::odd)) throw do_taylor(); // caught by function::series() // if we got here we have to care for a simple pole @@ -896,7 +905,7 @@ static ex asinh_eval(const ex & x) if (x.info(info_flags::numeric)) { // asinh(0) -> 0 if (x.is_zero()) - return _ex0(); + return _ex0; // asinh(float) -> float if (!x.info(info_flags::crational)) return asinh(ex_to(x)); @@ -910,7 +919,7 @@ static ex asinh_deriv(const ex & x, unsigned deriv_param) GINAC_ASSERT(deriv_param==0); // d/dx asinh(x) -> 1/sqrt(1+x^2) - return power(_ex1()+power(x,_ex2()),_ex_1_2()); + return power(_ex1+power(x,_ex2),_ex_1_2); } REGISTER_FUNCTION(asinh, eval_func(asinh_eval). @@ -936,10 +945,10 @@ static ex acosh_eval(const ex & x) if (x.is_zero()) return Pi*I*numeric(1,2); // acosh(1) -> 0 - if (x.is_equal(_ex1())) - return _ex0(); + if (x.is_equal(_ex1)) + return _ex0; // acosh(-1) -> Pi*I - if (x.is_equal(_ex_1())) + if (x.is_equal(_ex_1)) return Pi*I; // acosh(float) -> float if (!x.info(info_flags::crational)) @@ -954,7 +963,7 @@ static ex acosh_deriv(const ex & x, unsigned deriv_param) GINAC_ASSERT(deriv_param==0); // d/dx acosh(x) -> 1/(sqrt(x-1)*sqrt(x+1)) - return power(x+_ex_1(),_ex_1_2())*power(x+_ex1(),_ex_1_2()); + return power(x+_ex_1,_ex_1_2)*power(x+_ex1,_ex_1_2); } REGISTER_FUNCTION(acosh, eval_func(acosh_eval). @@ -978,9 +987,9 @@ static ex atanh_eval(const ex & x) if (x.info(info_flags::numeric)) { // atanh(0) -> 0 if (x.is_zero()) - return _ex0(); + return _ex0; // atanh({+|-}1) -> throw - if (x.is_equal(_ex1()) || x.is_equal(_ex_1())) + 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)) @@ -995,7 +1004,7 @@ static ex atanh_deriv(const ex & x, unsigned deriv_param) GINAC_ASSERT(deriv_param==0); // d/dx atanh(x) -> 1/(1-x^2) - return power(_ex1()-power(x,_ex2()),_ex_1()); + return power(_ex1-power(x,_ex2),_ex_1); } static ex atanh_series(const ex &arg, @@ -1003,7 +1012,7 @@ static ex atanh_series(const ex &arg, int order, unsigned options) { - GINAC_ASSERT(is_ex_exactly_of_type(rel.lhs(),symbol)); + GINAC_ASSERT(is_a(rel.lhs())); // method: // Taylor series where there is no pole or cut falls back to atanh_deriv. // There are two branch cuts, one runnig from 1 up the real axis and one @@ -1011,31 +1020,31 @@ static ex atanh_series(const ex &arg, // On the branch cuts and the poles series expand // (log(1+x)-log(1-x))/2 // instead. - const ex arg_pt = arg.subs(rel); + const ex arg_pt = arg.subs(rel, subs_options::no_pattern); if (!(arg_pt).info(info_flags::real)) throw do_taylor(); // Im(x) != 0 - if ((arg_pt).info(info_flags::real) && abs(arg_pt)<_ex1()) + 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); + 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 &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(); + const ex replarg = series(atanh(arg), s==foo, order).subs(foo==point, subs_options::no_pattern); + ex Order0correction = replarg.op(0)+csgn(I*arg)*Pi*I*_ex1_2; + if (arg_pt<_ex0) + Order0correction += log((arg_pt+_ex_1)/(arg_pt+_ex1))*_ex1_2; else - Order0correction += log((arg_pt+_ex1())/(arg_pt+_ex_1()))*_ex_1_2(); + 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)); + seq.push_back(expair(Order0correction, _ex0)); + seq.push_back(expair(Order(_ex1), order)); return series(replarg - pseries(rel, seq), rel, order); } throw do_taylor();