static ex Order_series(const ex & x, const relational & r, int order, unsigned options)
{
// Just wrap the function into a pseries object
GINAC_ASSERT(is_a<symbol>(r.lhs()));
const symbol &s = ex_to<symbol>(r.lhs());
epvector new_seq { expair(Order(_ex1), numeric(std::min(x.ldegree(s), order))) };
return pseries(r, std::move(new_seq));
}
static ex eta_series(const ex & x, const ex & y,
const relational & rel,
int order,
unsigned options)
{
const ex x_pt = x.subs(rel, subs_options::no_pattern);
const ex y_pt = y.subs(rel, subs_options::no_pattern);
if ((x_pt.info(info_flags::numeric) && x_pt.info(info_flags::negative)) ||
(y_pt.info(info_flags::numeric) && y_pt.info(info_flags::negative)) ||
((x_pt*y_pt).info(info_flags::numeric) && (x_pt*y_pt).info(info_flags::negative)))
throw (std::domain_error("eta_series(): on discontinuity"));
epvector seq { expair(eta(x_pt,y_pt), _ex0) };
return pseries(rel, std::move(seq));
}
static ex csgn_series(const ex & arg,
const relational & rel,
int order,
unsigned options)
{
const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
if (arg_pt.info(info_flags::numeric)
&& ex_to<numeric>(arg_pt).real().is_zero()
&& !(options & series_options::suppress_branchcut))
throw (std::domain_error("csgn_series(): on imaginary axis"));
epvector seq { expair(csgn(arg_pt), _ex0) };
return pseries(rel, std::move(seq));
}
static ex atan_series(const ex &arg,
const relational &rel,
int order,
unsigned options)
{
GINAC_ASSERT(is_a<symbol>(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, 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)
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<symbol>(rel.lhs());
const ex &point = rel.rhs();
const symbol foo;
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;
epvector seq;
if (order > 0) {
seq.reserve(2);
seq.push_back(expair(Order0correction, _ex0));
}
seq.push_back(expair(Order(_ex1), order));
return series(replarg - pseries(rel, std::move(seq)), rel, order);
}
throw do_taylor();
}
double incbet( double aa, double bb, double xx )
{
double a, b, t, x, xc, w, y;
int flag;
if( aa <= 0.0 || bb <= 0.0 )
goto domerr;
if( (xx <= 0.0) || ( xx >= 1.0) )
{
if( xx == 0.0 )
return(0.0);
if( xx == 1.0 )
return( 1.0 );
domerr:
mtherr( "incbet", DOMAIN );
return( 0.0 );
}
flag = 0;
if( (bb * xx) <= 1.0 && xx <= 0.95)
{
t = pseries(aa, bb, xx);
goto done;
}
w = 1.0 - xx;
/* Reverse a and b if x is greater than the mean. */
if( xx > (aa/(aa+bb)) )
{
flag = 1;
a = bb;
b = aa;
xc = xx;
x = w;
}
else
{
a = aa;
b = bb;
xc = w;
x = xx;
}
if( flag == 1 && (b * x) <= 1.0 && x <= 0.95)
{
t = pseries(a, b, x);
goto done;
}
/* Choose expansion for better convergence. */
y = x * (a+b-2.0) - (a-1.0);
if( y < 0.0 )
w = incbcf( a, b, x );
else
w = incbd( a, b, x ) / xc;
/* Multiply w by the factor
a b _ _ _
x (1-x) | (a+b) / ( a | (a) | (b) ) . */
y = a * log(x);
t = b * log(xc);
if( (a+b) < MAXGAM && fabs(y) < MAXLOG && fabs(t) < MAXLOG )
{
t = pow(xc,b);
t *= pow(x,a);
t /= a;
t *= w;
t *= gammafn(a+b) / (gammafn(a) * gammafn(b));
goto done;
}
/* Resort to logarithms. */
y += t + lgam(a+b) - lgam(a) - lgam(b);
y += log(w/a);
if( y < MINLOG )
t = 0.0;
else
t = exp(y);
done:
if( flag == 1 )
{
if( t <= MACHEP )
t = 1.0 - MACHEP;
else
t = 1.0 - t;
}
return( t );
}
static ex log_series(const ex &arg,
const relational &rel,
int order,
unsigned options)
{
GINAC_ASSERT(is_a<symbol>(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, subs_options::no_pattern);
} 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 (arg.diff(ex_to<symbol>(rel.lhs())).is_zero()) {
throw do_taylor();
}
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 + 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.
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<pseries>(arg.series(rel, order+extra_ord, options));
++extra_ord;
} while (!argser.is_terminating() && argser.nops()==1);
const symbol &s = ex_to<symbol>(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
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 (or less) terms are needed
// (sadly, to generate them, we have to start from the beginning)
if (n == 0 && coeff == 1) {
ex rest = pseries(rel, epvector{expair(-1, _ex0), expair(Order(_ex1), order)}).add_series(argser);
ex acc = dynallocate<pseries>(rel, epvector());
for (int i = order-1; i>0; --i) {
epvector cterm { expair(i%2 ? _ex1/i : _ex_1/i, _ex0) };
acc = pseries(rel, std::move(cterm)).add_series(ex_to<pseries>(acc));
acc = (ex_to<pseries>(rest)).mul_series(ex_to<pseries>(acc));
}
return acc;
}
const ex newarg = ex_to<pseries>((arg/coeff).series(rel, order+n, options)).shift_exponents(-n).convert_to_poly(true);
return pseries(rel, std::move(seq)).add_series(ex_to<pseries>(log(newarg).series(rel, order, options)));
} else // it was a monomial
return pseries(rel, std::move(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 = ex_to<symbol>(rel.lhs());
const ex &point = rel.rhs();
const symbol foo;
const ex replarg = series(log(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
epvector seq;
if (order > 0) {
seq.reserve(2);
seq.push_back(expair(-I*csgn(arg*I)*Pi, _ex0));
}
seq.push_back(expair(Order(_ex1), order));
return series(replarg - I*Pi + pseries(rel, std::move(seq)), rel, order);
}
throw do_taylor(); // caught by function::series()
}
static ex Li2_series(const ex &x, const relational &rel, int order, unsigned options)
{
const ex x_pt = x.subs(rel, subs_options::no_pattern);
if (x_pt.info(info_flags::numeric)) {
// First special case: x==0 (derivatives have poles)
if (x_pt.is_zero()) {
// method:
// The problem is that in d/dx Li2(x==0) == -log(1-x)/x we cannot
// simply substitute x==0. The limit, however, exists: it is 1.
// We also know all higher derivatives' limits:
// (d/dx)^n Li2(x) == n!/n^2.
// So the primitive series expansion is
// Li2(x==0) == x + x^2/4 + x^3/9 + ...
// and so on.
// We first construct such a primitive series expansion manually in
// a dummy symbol s and then insert the argument's series expansion
// for s. Reexpanding the resulting series returns the desired
// result.
const symbol s;
ex ser;
// manually construct the primitive expansion
for (int i=1; i<order; ++i)
ser += pow(s,i) / pow(numeric(i), *_num2_p);
// substitute the argument's series expansion
ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
// maybe that was terminating, so add a proper order term
epvector nseq { expair(Order(_ex1), order) };
ser += pseries(rel, std::move(nseq));
// reexpanding it will collapse the series again
return ser.series(rel, order);
// NB: Of course, this still does not allow us to compute anything
// like sin(Li2(x)).series(x==0,2), since then this code here is
// not reached and the derivative of sin(Li2(x)) doesn't allow the
// substitution x==0. Probably limits *are* needed for the general
// cases. In case L'Hospital's rule is implemented for limits and
// basic::series() takes care of this, this whole block is probably
// obsolete!
}
// second special case: x==1 (branch point)
if (x_pt.is_equal(_ex1)) {
// method:
// construct series manually in a dummy symbol s
const symbol s;
ex ser = zeta(_ex2);
// manually construct the primitive expansion
for (int i=1; i<order; ++i)
ser += pow(1-s,i) * (numeric(1,i)*(I*Pi+log(s-1)) - numeric(1,i*i));
// substitute the argument's series expansion
ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
// maybe that was terminating, so add a proper order term
epvector nseq { expair(Order(_ex1), order) };
ser += pseries(rel, std::move(nseq));
// reexpanding it will collapse the series again
return ser.series(rel, order);
}
// third special case: x real, >=1 (branch cut)
if (!(options & series_options::suppress_branchcut) &&
ex_to<numeric>(x_pt).is_real() && ex_to<numeric>(x_pt)>1) {
// method:
// This is the branch cut: assemble the primitive series manually
// and then add the corresponding complex step function.
const symbol &s = ex_to<symbol>(rel.lhs());
const ex point = rel.rhs();
const symbol foo;
epvector seq;
// zeroth order term:
seq.push_back(expair(Li2(x_pt), _ex0));
// compute the intermediate terms:
ex replarg = series(Li2(x), s==foo, order);
for (size_t i=1; i<replarg.nops()-1; ++i)
seq.push_back(expair((replarg.op(i)/power(s-foo,i)).series(foo==point,1,options).op(0).subs(foo==s, subs_options::no_pattern),i));
// append an order term:
seq.push_back(expair(Order(_ex1), replarg.nops()-1));
return pseries(rel, std::move(seq));
}
}
// all other cases should be safe, by now:
throw do_taylor(); // caught by function::series()
}
