+ // lgamma(n) -> log((n-1)!) for postitive n
+ if (x.info(info_flags::posint)) {
+ return log(factorial(x.exadd(_ex_1())));
+ } else {
+ throw (pole_error("lgamma_eval(): logarithmic pole",0));
+ }
+ }
+ // 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 & 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);
+ // 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 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):
+ throw (std::overflow_error("lgamma_series: please implement 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 pole_error("tgamma_eval(): simple pole",0) */
+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