* Knows about integer arguments and that's it. Somebody ought to provide
* some good numerical evaluation some day...
*
- * @exception std::domain_error("lgamma_eval(): simple pole") */
+ * @exception std::domain_error("lgamma_eval(): logarithmic pole") */
static ex lgamma_eval(const ex & x)
{
if (x.info(info_flags::numeric)) {
return psi(x);
}
-// need to implement lgamma_series.
+
+static ex lgamma_series(const ex & x, const relational & rel, int order)
+{
+ // method:
+ // Taylor series where there is no pole falls back to psi function
+ // evaluation.
+ // On a pole at -m we could use the recurrence relation
+ // lgamma(x) == lgamma(x+1)-log(x)
+ // from which follows
+ // series(lgamma(x),x==-m,order) ==
+ // series(lgamma(x+m+1)-log(x)...-log(x+m)),x==-m,order);
+ // This, however, seems to fail utterly because you run into branch-cut
+ // problems. Somebody ought to implement it some day using an asymptotic
+ // series for tgamma:
+ const ex x_pt = x.subs(rel);
+ if (!x_pt.info(info_flags::integer) || x_pt.info(info_flags::positive))
+ throw do_taylor(); // caught by function::series()
+ // if we got here we have to care for a simple pole of tgamma(-m):
+ throw (std::domain_error("lgamma_series: please implemnt my at the poles"));
+ return _ex0(); // not reached
+}
+
REGISTER_FUNCTION(lgamma, eval_func(lgamma_eval).
evalf_func(lgamma_evalf).
- derivative_func(lgamma_deriv));
+ derivative_func(lgamma_deriv).
+ series_func(lgamma_series));
//////////
}
-static ex tgamma_series(const ex & x, const relational & r, int order)
+static ex tgamma_series(const ex & x, const relational & rel, int order)
{
// method:
// Taylor series where there is no pole falls back to psi function
// On a pole at -m use the recurrence relation
// tgamma(x) == tgamma(x+1) / x
// from which follows
- // series(tgamma(x),x,-m,order) ==
- // series(tgamma(x+m+1)/(x*(x+1)*...*(x+m)),x,-m,order+1);
- const ex x_pt = x.subs(r);
+ // series(tgamma(x),x==-m,order) ==
+ // series(tgamma(x+m+1)/(x*(x+1)*...*(x+m)),x==-m,order+1);
+ const ex x_pt = x.subs(rel);
if (!x_pt.info(info_flags::integer) || x_pt.info(info_flags::positive))
throw do_taylor(); // caught by function::series()
// if we got here we have to care for a simple pole at -m:
ex ser_denom = _ex1();
for (numeric p; p<=m; ++p)
ser_denom *= x+p;
- return (tgamma(x+m+_ex1())/ser_denom).series(r, order+1);
+ return (tgamma(x+m+_ex1())/ser_denom).series(rel, order+1);
}
}
-static ex beta_series(const ex & x, const ex & y, const relational & r, int order)
+static ex beta_series(const ex & x, const ex & y, const relational & rel, int order)
{
// method:
// Taylor series where there is no pole of one of the tgamma functions
// falls back to beta function evaluation. Otherwise, fall back to
// tgamma series directly.
- // FIXME: this could need some testing, maybe it's wrong in some cases?
- const ex x_pt = x.subs(r);
- const ex y_pt = y.subs(r);
- GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
- const symbol *s = static_cast<symbol *>(r.lhs().bp);
+ const ex x_pt = x.subs(rel);
+ const ex y_pt = y.subs(rel);
+ GINAC_ASSERT(is_ex_exactly_of_type(rel.lhs(),symbol));
+ const symbol *s = static_cast<symbol *>(rel.lhs().bp);
ex x_ser, y_ser, xy_ser;
if ((!x_pt.info(info_flags::integer) || x_pt.info(info_flags::positive)) &&
(!y_pt.info(info_flags::integer) || y_pt.info(info_flags::positive)))
throw do_taylor(); // caught by function::series()
// trap the case where x is on a pole directly:
if (x.info(info_flags::integer) && !x.info(info_flags::positive))
- x_ser = tgamma(x+*s).series(r,order);
+ x_ser = tgamma(x+*s).series(rel,order);
else
- x_ser = tgamma(x).series(r,order);
+ x_ser = tgamma(x).series(rel,order);
// trap the case where y is on a pole directly:
if (y.info(info_flags::integer) && !y.info(info_flags::positive))
- y_ser = tgamma(y+*s).series(r,order);
+ y_ser = tgamma(y+*s).series(rel,order);
else
- y_ser = tgamma(y).series(r,order);
+ y_ser = tgamma(y).series(rel,order);
// trap the case where y is on a pole directly:
if ((x+y).info(info_flags::integer) && !(x+y).info(info_flags::positive))
- xy_ser = tgamma(y+x+*s).series(r,order);
+ xy_ser = tgamma(y+x+*s).series(rel,order);
else
- xy_ser = tgamma(y+x).series(r,order);
+ xy_ser = tgamma(y+x).series(rel,order);
// compose the result:
- return (x_ser*y_ser/xy_ser).series(r,order);
+ return (x_ser*y_ser/xy_ser).series(rel,order);
}
if (nx.is_integer()) {
// integer case
if (nx.is_positive()) {
- // psi(n) -> 1 + 1/2 +...+ 1/(n-1) - gamma
+ // psi(n) -> 1 + 1/2 +...+ 1/(n-1) - Euler
numeric rat(0);
for (numeric i(nx+_num_1()); i.is_positive(); --i)
rat += i.inverse();
- return rat-gamma;
+ return rat-Euler;
} else {
// for non-positive integers there is a pole:
throw (std::domain_error("psi_eval(): simple pole"));
if ((_num2()*nx).is_integer()) {
// half integer case
if (nx.is_positive()) {
- // psi((2m+1)/2) -> 2/(2m+1) + 2/2m +...+ 2/1 - gamma - 2log(2)
+ // psi((2m+1)/2) -> 2/(2m+1) + 2/2m +...+ 2/1 - Euler - 2log(2)
numeric rat(0);
for (numeric i((nx+_num_1())*_num2()); i.is_positive(); i-=_num2())
rat += _num2()*i.inverse();
- return rat-gamma-_ex2()*log(_ex2());
+ return rat-Euler-_ex2()*log(_ex2());
} else {
// use the recurrence relation
// psi(-m-1/2) == psi(-m-1/2+1) - 1 / (-m-1/2)
return psi(_ex1(), x);
}
-static ex psi1_series(const ex & x, const relational & r, int order)
+static ex psi1_series(const ex & x, const relational & rel, int order)
{
// method:
// Taylor series where there is no pole falls back to polygamma function
// On a pole at -m use the recurrence relation
// psi(x) == psi(x+1) - 1/z
// from which follows
- // series(psi(x),x,-m,order) ==
- // series(psi(x+m+1) - 1/x - 1/(x+1) - 1/(x+m)),x,-m,order);
- const ex x_pt = x.subs(r);
+ // series(psi(x),x==-m,order) ==
+ // series(psi(x+m+1) - 1/x - 1/(x+1) - 1/(x+m)),x==-m,order);
+ const ex x_pt = x.subs(rel);
if (!x_pt.info(info_flags::integer) || x_pt.info(info_flags::positive))
throw do_taylor(); // caught by function::series()
// if we got here we have to care for a simple pole at -m:
ex recur;
for (numeric p; p<=m; ++p)
recur += power(x+p,_ex_1());
- return (psi(x+m+_ex1())-recur).series(r, order);
+ return (psi(x+m+_ex1())-recur).series(rel, order);
}
const unsigned function_index_psi1 =
return psi(n+_ex1(), x);
}
-static ex psi2_series(const ex & n, const ex & x, const relational & r, int order)
+static ex psi2_series(const ex & n, const ex & x, const relational & rel, int order)
{
// method:
// Taylor series where there is no pole falls back to polygamma function
// On a pole at -m use the recurrence relation
// psi(n,x) == psi(n,x+1) - (-)^n * n! / x^(n+1)
// from which follows
- // series(psi(x),x,-m,order) ==
+ // series(psi(x),x==-m,order) ==
// series(psi(x+m+1) - (-1)^n * n! * ((x)^(-n-1) + (x+1)^(-n-1) + ...
- // ... + (x+m)^(-n-1))),x,-m,order);
- const ex x_pt = x.subs(r);
+ // ... + (x+m)^(-n-1))),x==-m,order);
+ const ex x_pt = x.subs(rel);
if (!x_pt.info(info_flags::integer) || x_pt.info(info_flags::positive))
throw do_taylor(); // caught by function::series()
// if we got here we have to care for a pole of order n+1 at -m:
for (numeric p; p<=m; ++p)
recur += power(x+p,-n+_ex_1());
recur *= factorial(n)*power(_ex_1(),n);
- return (psi(n, x+m+_ex1())-recur).series(r, order);
+ return (psi(n, x+m+_ex1())-recur).series(rel, order);
}
const unsigned function_index_psi2 =
function::register_new(function_options("psi").
eval_func(psi2_eval).
evalf_func(psi2_evalf).
- derivative_func(psi2_deriv).
+ derivative_func(psi2_deriv).
series_func(psi2_series).
overloaded(2));
git://www.ginac.de / ginac.git / blobdiff