return psi(x)*gamma(x);
}
-static ex gamma_series(const ex & x, const symbol & s, const ex & point, int order)
+static ex gamma_series(const ex & x, const symbol & s, const ex & pt, int order)
{
// method:
// Taylor series where there is no pole falls back to psi function
// gamma(x) == gamma(x+1) / x
// from which follows
// series(gamma(x),x,-m,order) ==
- // series(gamma(x+m+1)/(x*(x+1)...*(x+m)),x,-m,order+1);
- ex xpoint = x.subs(s==point);
- if (!xpoint.info(info_flags::integer) || xpoint.info(info_flags::positive))
+ // series(gamma(x+m+1)/(x*(x+1)*...*(x+m)),x,-m,order+1);
+ const ex x_pt = x.subs(s==pt);
+ 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:
- numeric m = -ex_to_numeric(xpoint);
+ numeric m = -ex_to_numeric(x_pt);
ex ser_numer = gamma(x+m+_ex1());
ex ser_denom = _ex1();
for (numeric p; p<=m; ++p)
ser_denom *= x+p;
- return (ser_numer/ser_denom).series(s, point, order+1);
+ return (ser_numer/ser_denom).series(s, pt, order+1);
}
REGISTER_FUNCTION(gamma, gamma_eval, gamma_evalf, gamma_diff, gamma_series);
static ex beta_eval(const ex & x, const ex & y)
{
if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) {
- numeric nx(ex_to_numeric(x));
- numeric ny(ex_to_numeric(y));
// treat all problematic x and y that may not be passed into gamma,
// because they would throw there although beta(x,y) is well-defined
// using the formula beta(x,y) == (-1)^y * beta(1-x-y, y)
+ numeric nx(ex_to_numeric(x));
+ numeric ny(ex_to_numeric(y));
if (nx.is_real() && nx.is_integer() &&
ny.is_real() && ny.is_integer()) {
if (nx.is_negative()) {
return retval;
}
-static ex beta_series(const ex & x, const ex & y, const symbol & s, const ex & point, int order)
+static ex beta_series(const ex & x, const ex & y, const symbol & s, const ex & pt, int order)
{
// method:
- // Taylor series where there is no pole falls back to beta function
- // evaluation.
- // On a pole at -m use the recurrence relation
- // gamma(x) == gamma(x+1) / x
- // from which follows
- // series(gamma(x),x,-m,order) ==
- // series(gamma(x+m+1)/(x*(x+1)...*(x+m)),x,-m,order+1);
- ex xpoint = x.subs(s==point);
- ex ypoint = y.subs(s==point);
- if ((!xpoint.info(info_flags::integer) || xpoint.info(info_flags::positive)) &&
- (!ypoint.info(info_flags::integer) || ypoint.info(info_flags::positive)))
+ // Taylor series where there is no pole of one of the gamma functions
+ // falls back to beta function evaluation. Otherwise, fall back to
+ // gamma series directly.
+ // FIXME: this could need some testing, maybe it's wrong in some cases?
+ const ex x_pt = x.subs(s==pt);
+ const ex y_pt = y.subs(s==pt);
+ 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()
- // if we got here we have to care for a simple pole at -m:
- throw (std::domain_error("beta_series(): Mama, please code me!"));
+ // trap the case where x is on a pole directly:
+ if (x.info(info_flags::integer) && !x.info(info_flags::positive))
+ x_ser = gamma(x+s).series(s,pt,order);
+ else
+ x_ser = gamma(x).series(s,pt,order);
+ // trap the case where y is on a pole directly:
+ if (y.info(info_flags::integer) && !y.info(info_flags::positive))
+ y_ser = gamma(y+s).series(s,pt,order);
+ else
+ y_ser = gamma(y).series(s,pt,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 = gamma(y+x+s).series(s,pt,order);
+ else
+ xy_ser = gamma(y+x).series(s,pt,order);
+ // compose the result:
+ return (x_ser*y_ser/xy_ser).series(s,pt,order);
}
REGISTER_FUNCTION(beta, beta_eval, beta_evalf, beta_diff, beta_series);
return psi(_ex1(), x);
}
-static ex psi1_series(const ex & x, const symbol & s, const ex & point, int order)
+static ex psi1_series(const ex & x, const symbol & s, const ex & pt, int order)
{
// method:
// Taylor series where there is no pole falls back to polygamma function
// 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);
- ex xpoint = x.subs(s==point);
- if (!xpoint.info(info_flags::integer) || xpoint.info(info_flags::positive))
+ const ex x_pt = x.subs(s==pt);
+ 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:
- numeric m = -ex_to_numeric(xpoint);
+ numeric m = -ex_to_numeric(x_pt);
ex recur;
for (numeric p; p<=m; ++p)
recur += power(x+p,_ex_1());
- return (psi(x+m+_ex1())-recur).series(s, point, order);
+ return (psi(x+m+_ex1())-recur).series(s, pt, order);
}
const unsigned function_index_psi1 = function::register_new("psi", psi1_eval, psi1_evalf, psi1_diff, psi1_series);
return psi(n+_ex1(), x);
}
-static ex psi2_series(const ex & n, const ex & x, const symbol & s, const ex & point, int order)
+static ex psi2_series(const ex & n, const ex & x, const symbol & s, const ex & pt, int order)
{
// method:
// Taylor series where there is no pole falls back to polygamma function
// 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);
- ex xpoint = x.subs(s==point);
- if (!xpoint.info(info_flags::integer) || xpoint.info(info_flags::positive))
+ const ex x_pt = x.subs(s==pt);
+ 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:
- numeric m = -ex_to_numeric(xpoint);
+ numeric m = -ex_to_numeric(x_pt);
ex recur;
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(s, point, order);
+ return (psi(n, x+m+_ex1())-recur).series(s, pt, order);
}
const unsigned function_index_psi2 = function::register_new("psi", psi2_eval, psi2_evalf, psi2_diff, psi2_series);