return check_series(e,2,d,5);
}
+// Series expansion of logarithms around branch points
+static unsigned exam_series10(void)
+{
+ unsigned result = 0;
+ ex e, d;
+ symbol a("a");
+
+ e = log(x);
+ d = log(x);
+ result += check_series(e,0,d,5);
+
+ e = log(3/x);
+ d = log(3)-log(x);
+ result += check_series(e,0,d,5);
+
+ e = log(3*pow(x,2));
+ d = log(3)+2*log(x);
+ result += check_series(e,0,d,5);
+
+ // These ones must not be expanded because it would result in a branch cut
+ // running in the wrong direction. (Other systems tend to get this wrong.)
+ e = log(-x);
+ d = e;
+ result += check_series(e,0,d,5);
+
+ e = log(I*(x-123));
+ d = e;
+ result += check_series(e,123,d,5);
+
+ e = log(a*x);
+ d = e; // we don't know anything about a!
+ result += check_series(e,0,d,5);
+
+ e = log((1-x)/x);
+ d = log(1-x) - (x-1) + pow(x-1,2)/2 - pow(x-1,3)/3 + Order(pow(x-1,4));
+ result += check_series(e,1,d,4);
+
+ return result;
+}
+
+// Series expansion of other functions around branch points
+static unsigned exam_series11(void)
+{
+ unsigned result = 0;
+ ex e, d;
+
+ // NB: Mma and Maple give different results, but they agree if one
+ // takes into account that by assumption |x|<1.
+ e = atan(x);
+ d = (I*log(2)/2-I*log(1+I*x)/2) + (x-I)/4 + I*pow(x-I,2)/16 + Order(pow(x-I,3));
+ result += check_series(e,I,d,3);
+
+ // NB: here, at -I, Mathematica disagrees, but it is wrong -- they
+ // pick up a complex phase by incorrectly expanding logarithms.
+ e = atan(x);
+ d = (-I*log(2)/2+I*log(1-I*x)/2) + (x+I)/4 - I*pow(x+I,2)/16 + Order(pow(x+I,3));
+ result += check_series(e,-I,d,3);
+
+ // This is basically the same as above, the branch point is at +/-1:
+ e = atanh(x);
+ d = (-log(2)/2+log(x+1)/2) + (x+1)/4 + pow(x+1,2)/16 + Order(pow(x+1,3));
+ result += check_series(e,-1,d,3);
+
+ return result;
+}
+
+
unsigned exam_pseries(void)
{
unsigned result = 0;
result += exam_series7(); cout << '.' << flush;
result += exam_series8(); cout << '.' << flush;
result += exam_series9(); cout << '.' << flush;
+ result += exam_series10(); cout << '.' << flush;
+ result += exam_series11(); cout << '.' << flush;
if (!result) {
cout << " passed " << endl;
} 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;
const ex point = rel.rhs();
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
+ 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);
+ // (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)) {
+ 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);
+ 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 power(_ex1()+power(x,_ex2()), _ex_1());
}
-static ex atan_series(const ex &x,
+static ex atan_series(const ex &arg,
const relational &rel,
int order,
unsigned options)
// 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)/(1-I*x))/(2*I)
+ // (log(1+I*x)-log(1-I*x))/(2*I)
// instead.
- // (The constant term on the cut itself could be made simpler.)
- const ex x_pt = x.subs(rel);
- if (!(I*x_pt).info(info_flags::real))
+ const ex arg_pt = arg.subs(rel);
+ if (!(I*arg_pt).info(info_flags::real))
throw do_taylor(); // Re(x) != 0
- if ((I*x_pt).info(info_flags::real) && abs(I*x_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
- // if we got here we have to care for cuts and poles
- return (log((1+I*x)/(1-I*x))/(2*I)).series(rel, order, options);
+ // 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<symbol *>(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, eval_func(atan_eval).
return power(_ex1()-power(x,_ex2()),_ex_1());
}
-static ex atanh_series(const ex &x,
+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 atan_deriv.
+ // 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)/(1-x))/(2*I)
+ // (log(1+x)-log(1-x))/2
// instead.
- // (The constant term on the cut itself could be made simpler.)
- const ex x_pt = x.subs(rel);
- if (!(x_pt).info(info_flags::real))
+ const ex arg_pt = arg.subs(rel);
+ if (!(arg_pt).info(info_flags::real))
throw do_taylor(); // Im(x) != 0
- if ((x_pt).info(info_flags::real) && abs(x_pt)<_ex1())
+ if ((arg_pt).info(info_flags::real) && abs(arg_pt)<_ex1())
throw do_taylor(); // Im(x) == 0, but abs(x)<1
- // if we got here we have to care for cuts and poles
- return (log((1+x)/(1-x))/2).series(rel, order, options);
+ // 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<symbol *>(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).