static ex log_eval(const ex & x)
{
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(_ex_1())) // log(-1) -> I*Pi
- return (I*Pi);
if (x.is_equal(I)) // log(I) -> Pi*I/2
return (Pi*I*_num1_2());
if (x.is_equal(-I)) // log(-I) -> -Pi*I/2
return (Pi*I*_num_1_2());
- if (x.is_equal(_ex0())) // log(0) -> infinity
- throw(std::domain_error("log_eval(): log(0)"));
// log(float)
if (!x.info(info_flags::crational))
return log_evalf(x);
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());
}
-static ex log_series(const ex & x, const symbol & s, const ex & pt, int order)
+static ex log_series(const ex &arg,
+ const relational &rel,
+ int order,
+ bool branchcut)
{
- if (x.subs(s == pt).is_zero()) {
- epvector seq;
- seq.push_back(expair(log(x), _ex0()));
- return pseries(s, pt, seq);
- } else
- throw do_taylor();
+ 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 cut 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, branchcut));
+ const symbol *s = static_cast<symbol *>(rel.lhs().bp);
+ const ex point = rel.rhs();
+ const int n = argser.ldegree(*s);
+ epvector seq;
+ seq.push_back(expair(n*log(*s-point), _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, branchcut)).shift_exponents(-n).convert_to_poly(true);
+ return pseries(rel, seq).add_series(ex_to_pseries(log(newarg).series(rel, order, branchcut)));
+ } else // it was a monomial
+ return pseries(rel, seq);
+ }
+ if (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<symbol *>(rel.lhs().bp);
+ const ex point = rel.rhs();
+ const symbol foo;
+ ex replarg = series(log(arg), *s==foo, order, false).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).
if (z.is_equal(_num25())) // tan(5/12*Pi) -> 2+sqrt(3)
return sign*(power(_ex3(),_ex1_2())+_ex2());
if (z.is_equal(_num30())) // tan(Pi/2) -> infinity
- throw (std::domain_error("tan_eval(): infinity"));
+ throw (pole_error("tan_eval(): simple pole",1));
}
if (is_ex_exactly_of_type(x, function)) {
return (_ex1()+power(tan(x),_ex2()));
}
-static ex tan_series(const ex & x, const symbol & s, const ex & pt, int order)
+static ex tan_series(const ex &x,
+ const relational &rel,
+ int order,
+ bool branchcut)
{
// method:
// Taylor series where there is no pole falls back to tan_deriv.
// On a pole simply expand sin(x)/cos(x).
- const ex x_pt = x.subs(s==pt);
+ 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(s, pt, order+2);
+ return (sin(x)/cos(x)).series(rel, order+2);
}
REGISTER_FUNCTION(tan, eval_func(tan_eval).
return _ex1()-power(tanh(x),_ex2());
}
-static ex tanh_series(const ex & x, const symbol & s, const ex & pt, int order)
+static ex tanh_series(const ex &x,
+ const relational &rel,
+ int order,
+ bool branchcut)
{
// method:
// Taylor series where there is no pole falls back to tanh_deriv.
// On a pole simply expand sinh(x)/cosh(x).
- const ex x_pt = x.subs(s==pt);
+ 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(s, pt, order+2);
+ return (sinh(x)/cosh(x)).series(rel, order+2);
}
REGISTER_FUNCTION(tanh, eval_func(tanh_eval).
if (x.is_zero())
return _ex0();
// atanh({+|-}1) -> throw
- if (x.is_equal(_ex1()) || x.is_equal(_ex1()))
- throw (std::domain_error("atanh_eval(): infinity"));
+ 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);