* 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 GiNaC::pole_error("lgamma_eval(): logarithmic pole",0) */
static ex lgamma_eval(const ex & x)
{
if (x.info(info_flags::numeric)) {
// trap integer arguments:
if (x.info(info_flags::integer)) {
// lgamma(n) -> log((n-1)!) for postitive n
- if (x.info(info_flags::posint)) {
+ if (x.info(info_flags::posint))
return log(factorial(x.exadd(_ex_1())));
- } else {
- throw (std::domain_error("lgamma_eval(): logarithmic pole"));
- }
+ else
+ throw (pole_error("lgamma_eval(): logarithmic pole",0));
}
// lgamma_evalf should be called here once it becomes available
}
return psi(x);
}
-// need to implement lgamma_series.
+
+static ex lgamma_series(const ex & arg,
+ const relational & rel,
+ int order,
+ unsigned options)
+{
+ // 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);
+ const ex arg_pt = arg.subs(rel);
+ if (!arg_pt.info(info_flags::integer) || arg_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):
+ numeric m = -ex_to_numeric(arg_pt);
+ ex recur;
+ for (numeric p; p<=m; ++p)
+ recur += log(arg+p);
+ cout << recur << endl;
+ return (lgamma(arg+m+_ex1())-recur).series(rel, order, options);
+}
+
REGISTER_FUNCTION(lgamma, eval_func(lgamma_eval).
evalf_func(lgamma_evalf).
- derivative_func(lgamma_deriv));
+ derivative_func(lgamma_deriv).
+ series_func(lgamma_series));
//////////
* arguments, half-integer arguments and that's it. Somebody ought to provide
* some good numerical evaluation some day...
*
- * @exception std::domain_error("tgamma_eval(): simple pole") */
+ * @exception pole_error("tgamma_eval(): simple pole",0) */
static ex tgamma_eval(const ex & x)
{
if (x.info(info_flags::numeric)) {
if (x.info(info_flags::posint)) {
return factorial(ex_to_numeric(x).sub(_num1()));
} else {
- throw (std::domain_error("tgamma_eval(): simple pole"));
+ throw (pole_error("tgamma_eval(): simple pole",1));
}
}
// trap half integer arguments:
}
-static ex tgamma_series(const ex & x, const relational & r, int order)
+static ex tgamma_series(const ex & arg,
+ const relational & rel,
+ int order,
+ unsigned options)
{
// 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);
- if (!x_pt.info(info_flags::integer) || x_pt.info(info_flags::positive))
+ // series(tgamma(x),x==-m,order) ==
+ // series(tgamma(x+m+1)/(x*(x+1)*...*(x+m)),x==-m,order+1);
+ const ex arg_pt = arg.subs(rel);
+ if (!arg_pt.info(info_flags::integer) || arg_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:
- numeric m = -ex_to_numeric(x_pt);
+ numeric m = -ex_to_numeric(arg_pt);
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);
+ ser_denom *= arg+p;
+ return (tgamma(arg+m+_ex1())/ser_denom).series(rel, order+1, options);
}
if (nx<=-ny)
return pow(_num_1(), ny)*beta(1-x-y, y);
else
- throw (std::domain_error("beta_eval(): simple pole"));
+ throw (pole_error("beta_eval(): simple pole",1));
}
if (ny.is_negative()) {
if (ny<=-nx)
return pow(_num_1(), nx)*beta(1-y-x, x);
else
- throw (std::domain_error("beta_eval(): simple pole"));
+ throw (pole_error("beta_eval(): simple pole",1));
}
return tgamma(x)*tgamma(y)/tgamma(x+y);
}
}
-static ex beta_series(const ex & x, const ex & y, const relational & r, int order)
+static ex beta_series(const ex & arg1,
+ const ex & arg2,
+ const relational & rel,
+ int order,
+ unsigned options)
{
// 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);
- 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)))
+ const ex arg1_pt = arg1.subs(rel);
+ const ex arg2_pt = arg2.subs(rel);
+ GINAC_ASSERT(is_ex_exactly_of_type(rel.lhs(),symbol));
+ const symbol *s = static_cast<symbol *>(rel.lhs().bp);
+ ex arg1_ser, arg2_ser, arg1arg2_ser;
+ if ((!arg1_pt.info(info_flags::integer) || arg1_pt.info(info_flags::positive)) &&
+ (!arg2_pt.info(info_flags::integer) || arg2_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);
+ // trap the case where arg1 is on a pole:
+ if (arg1.info(info_flags::integer) && !arg1.info(info_flags::positive))
+ arg1_ser = tgamma(arg1+*s).series(rel, order, options);
else
- x_ser = tgamma(x).series(r,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);
+ arg1_ser = tgamma(arg1).series(rel,order);
+ // trap the case where arg2 is on a pole:
+ if (arg2.info(info_flags::integer) && !arg2.info(info_flags::positive))
+ arg2_ser = tgamma(arg2+*s).series(rel, order, options);
else
- y_ser = tgamma(y).series(r,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);
+ arg2_ser = tgamma(arg2).series(rel,order);
+ // trap the case where arg1+arg2 is on a pole:
+ if ((arg1+arg2).info(info_flags::integer) && !(arg1+arg2).info(info_flags::positive))
+ arg1arg2_ser = tgamma(arg2+arg1+*s).series(rel, order, options);
else
- xy_ser = tgamma(y+x).series(r,order);
- // compose the result:
- return (x_ser*y_ser/xy_ser).series(r,order);
+ arg1arg2_ser = tgamma(arg2+arg1).series(rel,order);
+ // compose the result (expanding all the terms):
+ return (arg1_ser*arg2_ser/arg1arg2_ser).series(rel, order, options).expand();
}
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"));
+ throw (pole_error("psi_eval(): simple pole",1));
}
}
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 & arg,
+ const relational & rel,
+ int order,
+ unsigned options)
{
// 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);
- if (!x_pt.info(info_flags::integer) || x_pt.info(info_flags::positive))
+ // series(psi(x),x==-m,order) ==
+ // series(psi(x+m+1) - 1/x - 1/(x+1) - 1/(x+m)),x==-m,order);
+ const ex arg_pt = arg.subs(rel);
+ if (!arg_pt.info(info_flags::integer) || arg_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:
- numeric m = -ex_to_numeric(x_pt);
+ numeric m = -ex_to_numeric(arg_pt);
ex recur;
for (numeric p; p<=m; ++p)
- recur += power(x+p,_ex_1());
- return (psi(x+m+_ex1())-recur).series(r, order);
+ recur += power(arg+p,_ex_1());
+ return (psi(arg+m+_ex1())-recur).series(rel, order, options);
}
const unsigned function_index_psi1 =
return recur+psi(n,_ex1());
} else {
// for non-positive integers there is a pole:
- throw (std::domain_error("psi2_eval(): pole"));
+ throw (pole_error("psi2_eval(): pole",1));
}
}
if ((_num2()*nx).is_integer()) {
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 & arg,
+ const relational & rel,
+ int order,
+ unsigned options)
{
// 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);
- if (!x_pt.info(info_flags::integer) || x_pt.info(info_flags::positive))
+ // ... + (x+m)^(-n-1))),x==-m,order);
+ const ex arg_pt = arg.subs(rel);
+ if (!arg_pt.info(info_flags::integer) || arg_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:
- numeric m = -ex_to_numeric(x_pt);
+ numeric m = -ex_to_numeric(arg_pt);
ex recur;
for (numeric p; p<=m; ++p)
- recur += power(x+p,-n+_ex_1());
+ recur += power(arg+p,-n+_ex_1());
recur *= factorial(n)*power(_ex_1(),n);
- return (psi(n, x+m+_ex1())-recur).series(r, order);
+ return (psi(n, arg+m+_ex1())-recur).series(rel, order, options);
}
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