#endif // ndef NO_NAMESPACE_GINAC
//////////
-// Gamma-function
+// Logarithm of Gamma function
//////////
-static ex gamma_evalf(const ex & x)
+static ex lgamma_evalf(const ex & x)
{
BEGIN_TYPECHECK
TYPECHECK(x,numeric)
- END_TYPECHECK(gamma(x))
+ END_TYPECHECK(lgamma(x))
- return gamma(ex_to_numeric(x));
+ return lgamma(ex_to_numeric(x));
}
-/** Evaluation of gamma(x). Knows about integer arguments, half-integer
- * arguments and that's it. Somebody ought to provide some good numerical
- * evaluation some day...
+/** Evaluation of lgamma(x), the natural logarithm of the Gamma function.
+ * Knows about integer arguments and that's it. Somebody ought to provide
+ * some good numerical evaluation some day...
*
- * @exception std::domain_error("gamma_eval(): simple pole") */
-static ex gamma_eval(const ex & x)
+ * @exception std::domain_error("lgamma_eval(): logarithmic pole") */
+static ex lgamma_eval(const ex & x)
{
if (x.info(info_flags::numeric)) {
// trap integer arguments:
if (x.info(info_flags::integer)) {
- // gamma(n+1) -> n! for postitive n
+ // lgamma(n) -> log((n-1)!) for postitive n
+ if (x.info(info_flags::posint)) {
+ return log(factorial(x.exadd(_ex_1())));
+ } else {
+ throw (std::domain_error("lgamma_eval(): logarithmic pole"));
+ }
+ }
+ // lgamma_evalf should be called here once it becomes available
+ }
+
+ return lgamma(x).hold();
+}
+
+
+static ex lgamma_deriv(const ex & x, unsigned deriv_param)
+{
+ GINAC_ASSERT(deriv_param==0);
+
+ // d/dx lgamma(x) -> psi(x)
+ return psi(x);
+}
+
+
+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).
+ series_func(lgamma_series));
+
+
+//////////
+// true Gamma function
+//////////
+
+static ex tgamma_evalf(const ex & x)
+{
+ BEGIN_TYPECHECK
+ TYPECHECK(x,numeric)
+ END_TYPECHECK(tgamma(x))
+
+ return tgamma(ex_to_numeric(x));
+}
+
+
+/** Evaluation of tgamma(x), the true Gamma function. Knows about integer
+ * 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") */
+static ex tgamma_eval(const ex & x)
+{
+ if (x.info(info_flags::numeric)) {
+ // trap integer arguments:
+ if (x.info(info_flags::integer)) {
+ // tgamma(n) -> (n-1)! for postitive n
if (x.info(info_flags::posint)) {
return factorial(ex_to_numeric(x).sub(_num1()));
} else {
- throw (std::domain_error("gamma_eval(): simple pole"));
+ throw (std::domain_error("tgamma_eval(): simple pole"));
}
}
// trap half integer arguments:
if ((x*2).info(info_flags::integer)) {
// trap positive x==(n+1/2)
- // gamma(n+1/2) -> Pi^(1/2)*(1*3*..*(2*n-1))/(2^n)
+ // tgamma(n+1/2) -> Pi^(1/2)*(1*3*..*(2*n-1))/(2^n)
if ((x*_ex2()).info(info_flags::posint)) {
numeric n = ex_to_numeric(x).sub(_num1_2());
numeric coefficient = doublefactorial(n.mul(_num2()).sub(_num1()));
return coefficient * pow(Pi,_ex1_2());
} else {
// trap negative x==(-n+1/2)
- // gamma(-n+1/2) -> Pi^(1/2)*(-2)^n/(1*3*..*(2*n-1))
+ // tgamma(-n+1/2) -> Pi^(1/2)*(-2)^n/(1*3*..*(2*n-1))
numeric n = abs(ex_to_numeric(x).sub(_num1_2()));
numeric coefficient = pow(_num_2(), n);
coefficient = coefficient.div(doublefactorial(n.mul(_num2()).sub(_num1())));;
return coefficient*power(Pi,_ex1_2());
}
}
- // gamma_evalf should be called here once it becomes available
+ // tgamma_evalf should be called here once it becomes available
}
- return gamma(x).hold();
-}
+ return tgamma(x).hold();
+}
-static ex gamma_deriv(const ex & x, unsigned deriv_param)
+static ex tgamma_deriv(const ex & x, unsigned deriv_param)
{
GINAC_ASSERT(deriv_param==0);
- // d/dx log(gamma(x)) -> psi(x)
- // d/dx gamma(x) -> psi(x)*gamma(x)
- return psi(x)*gamma(x);
+ // d/dx tgamma(x) -> psi(x)*tgamma(x)
+ return psi(x)*tgamma(x);
}
-static ex gamma_series(const ex & x, const symbol & s, const ex & pt, 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
// evaluation.
// On a pole at -m use the recurrence relation
- // gamma(x) == gamma(x+1) / x
+ // tgamma(x) == tgamma(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);
- const ex x_pt = x.subs(s==pt);
+ // 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 (gamma(x+m+_ex1())/ser_denom).series(s, pt, order+1);
+ return (tgamma(x+m+_ex1())/ser_denom).series(rel, order+1);
}
-REGISTER_FUNCTION(gamma, eval_func(gamma_eval).
- evalf_func(gamma_evalf).
- derivative_func(gamma_deriv).
- series_func(gamma_series));
+REGISTER_FUNCTION(tgamma, eval_func(tgamma_eval).
+ evalf_func(tgamma_evalf).
+ derivative_func(tgamma_deriv).
+ series_func(tgamma_series));
//////////
-// Beta-function
+// beta-function
//////////
static ex beta_evalf(const ex & x, const ex & y)
TYPECHECK(y,numeric)
END_TYPECHECK(beta(x,y))
- return gamma(ex_to_numeric(x))*gamma(ex_to_numeric(y))/gamma(ex_to_numeric(x+y));
+ return tgamma(ex_to_numeric(x))*tgamma(ex_to_numeric(y))/tgamma(ex_to_numeric(x+y));
}
static ex beta_eval(const ex & x, const ex & y)
{
if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) {
- // treat all problematic x and y that may not be passed into gamma,
+ // treat all problematic x and y that may not be passed into tgamma,
// 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));
else
throw (std::domain_error("beta_eval(): simple pole"));
}
- return gamma(x)*gamma(y)/gamma(x+y);
+ return tgamma(x)*tgamma(y)/tgamma(x+y);
}
// no problem in numerator, but denominator has pole:
if ((nx+ny).is_real() &&
!(nx+ny).is_positive())
return _ex0();
// everything is ok:
- return gamma(x)*gamma(y)/gamma(x+y);
+ return tgamma(x)*tgamma(y)/tgamma(x+y);
}
return beta(x,y).hold();
}
-static ex beta_series(const ex & x, const ex & y, const symbol & s, const ex & pt, 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 gamma functions
+ // Taylor series where there is no pole of one of the tgamma 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);
+ // tgamma series directly.
+ 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 = gamma(x+s).series(s,pt,order);
+ x_ser = tgamma(x+*s).series(rel,order);
else
- x_ser = gamma(x).series(s,pt,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 = gamma(y+s).series(s,pt,order);
+ y_ser = tgamma(y+*s).series(rel,order);
else
- y_ser = gamma(y).series(s,pt,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 = gamma(y+x+s).series(s,pt,order);
+ xy_ser = tgamma(y+x+*s).series(rel,order);
else
- xy_ser = gamma(y+x).series(s,pt,order);
+ xy_ser = tgamma(y+x).series(rel,order);
// compose the result:
- return (x_ser*y_ser/xy_ser).series(s,pt,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) - EulerGamma
+ // 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-EulerGamma;
+ 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 - EulerGamma - 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-EulerGamma-_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 symbol & s, const ex & pt, 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(s==pt);
+ // 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(s, pt, order);
+ return (psi(x+m+_ex1())-recur).series(rel, order);
}
const unsigned function_index_psi1 =
// psi(0,x) -> psi(x)
if (n.is_zero())
return psi(x);
- // psi(-1,x) -> log(gamma(x))
+ // psi(-1,x) -> log(tgamma(x))
if (n.is_equal(_ex_1()))
- return log(gamma(x));
+ return log(tgamma(x));
if (n.info(info_flags::numeric) && n.info(info_flags::posint) &&
x.info(info_flags::numeric)) {
numeric nn = ex_to_numeric(n);
return psi(n+_ex1(), x);
}
-static ex psi2_series(const ex & n, const ex & x, const symbol & s, const ex & pt, 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(s==pt);
+ // ... + (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(s, pt, 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