Happy New Year!
[ginac.git] / ginac / inifcns.cpp
index 1fcba69..e2a09b1 100644 (file)
@@ -3,7 +3,7 @@
  *  Implementation of GiNaC's initially known functions. */
 
 /*
- *  GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
+ *  GiNaC Copyright (C) 1999-2019 Johannes Gutenberg University Mainz, Germany
  *
  *  This program is free software; you can redistribute it and/or modify
  *  it under the terms of the GNU General Public License as published by
  *
  *  You should have received a copy of the GNU General Public License
  *  along with this program; if not, write to the Free Software
- *  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
+ *  Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
  */
 
-#include <vector>
-#include <stdexcept>
-
 #include "inifcns.h"
 #include "ex.h"
 #include "constant.h"
 #include "lst.h"
+#include "fderivative.h"
 #include "matrix.h"
 #include "mul.h"
 #include "power.h"
+#include "operators.h"
 #include "relational.h"
 #include "pseries.h"
 #include "symbol.h"
+#include "symmetry.h"
 #include "utils.h"
 
+#include <stdexcept>
+#include <vector>
+
 namespace GiNaC {
 
+//////////
+// complex conjugate
+//////////
+
+static ex conjugate_evalf(const ex & arg)
+{
+       if (is_exactly_a<numeric>(arg)) {
+               return ex_to<numeric>(arg).conjugate();
+       }
+       return conjugate_function(arg).hold();
+}
+
+static ex conjugate_eval(const ex & arg)
+{
+       return arg.conjugate();
+}
+
+static void conjugate_print_latex(const ex & arg, const print_context & c)
+{
+       c.s << "\\bar{"; arg.print(c); c.s << "}";
+}
+
+static ex conjugate_conjugate(const ex & arg)
+{
+       return arg;
+}
+
+// If x is real then U.diff(x)-I*V.diff(x) represents both conjugate(U+I*V).diff(x) 
+// and conjugate((U+I*V).diff(x))
+static ex conjugate_expl_derivative(const ex & arg, const symbol & s)
+{
+       if (s.info(info_flags::real))
+               return conjugate(arg.diff(s));
+       else {
+               exvector vec_arg;
+               vec_arg.push_back(arg);
+               return fderivative(ex_to<function>(conjugate(arg)).get_serial(),0,vec_arg).hold()*arg.diff(s);
+       }
+}
+
+static ex conjugate_real_part(const ex & arg)
+{
+       return arg.real_part();
+}
+
+static ex conjugate_imag_part(const ex & arg)
+{
+       return -arg.imag_part();
+}
+
+static bool func_arg_info(const ex & arg, unsigned inf)
+{
+       // for some functions we can return the info() of its argument
+       // (think of conjugate())
+       switch (inf) {
+               case info_flags::polynomial:
+               case info_flags::integer_polynomial:
+               case info_flags::cinteger_polynomial:
+               case info_flags::rational_polynomial:
+               case info_flags::real:
+               case info_flags::rational:
+               case info_flags::integer:
+               case info_flags::crational:
+               case info_flags::cinteger:
+               case info_flags::even:
+               case info_flags::odd:
+               case info_flags::prime:
+               case info_flags::crational_polynomial:
+               case info_flags::rational_function:
+               case info_flags::positive:
+               case info_flags::negative:
+               case info_flags::nonnegative:
+               case info_flags::posint:
+               case info_flags::negint:
+               case info_flags::nonnegint:
+               case info_flags::has_indices:
+                       return arg.info(inf);
+       }
+       return false;
+}
+
+static bool conjugate_info(const ex & arg, unsigned inf)
+{
+       return func_arg_info(arg, inf);
+}
+
+REGISTER_FUNCTION(conjugate_function, eval_func(conjugate_eval).
+                                      evalf_func(conjugate_evalf).
+                                      expl_derivative_func(conjugate_expl_derivative).
+                                      info_func(conjugate_info).
+                                      print_func<print_latex>(conjugate_print_latex).
+                                      conjugate_func(conjugate_conjugate).
+                                      real_part_func(conjugate_real_part).
+                                      imag_part_func(conjugate_imag_part).
+                                      set_name("conjugate","conjugate"));
+
+//////////
+// real part
+//////////
+
+static ex real_part_evalf(const ex & arg)
+{
+       if (is_exactly_a<numeric>(arg)) {
+               return ex_to<numeric>(arg).real();
+       }
+       return real_part_function(arg).hold();
+}
+
+static ex real_part_eval(const ex & arg)
+{
+       return arg.real_part();
+}
+
+static void real_part_print_latex(const ex & arg, const print_context & c)
+{
+       c.s << "\\Re"; arg.print(c); c.s << "";
+}
+
+static ex real_part_conjugate(const ex & arg)
+{
+       return real_part_function(arg).hold();
+}
+
+static ex real_part_real_part(const ex & arg)
+{
+       return real_part_function(arg).hold();
+}
+
+static ex real_part_imag_part(const ex & arg)
+{
+       return 0;
+}
+
+// If x is real then Re(e).diff(x) is equal to Re(e.diff(x)) 
+static ex real_part_expl_derivative(const ex & arg, const symbol & s)
+{
+       if (s.info(info_flags::real))
+               return real_part_function(arg.diff(s));
+       else {
+               exvector vec_arg;
+               vec_arg.push_back(arg);
+               return fderivative(ex_to<function>(real_part(arg)).get_serial(),0,vec_arg).hold()*arg.diff(s);
+       }
+}
+
+REGISTER_FUNCTION(real_part_function, eval_func(real_part_eval).
+                                      evalf_func(real_part_evalf).
+                                      expl_derivative_func(real_part_expl_derivative).
+                                      print_func<print_latex>(real_part_print_latex).
+                                      conjugate_func(real_part_conjugate).
+                                      real_part_func(real_part_real_part).
+                                      imag_part_func(real_part_imag_part).
+                                      set_name("real_part","real_part"));
+
+//////////
+// imag part
+//////////
+
+static ex imag_part_evalf(const ex & arg)
+{
+       if (is_exactly_a<numeric>(arg)) {
+               return ex_to<numeric>(arg).imag();
+       }
+       return imag_part_function(arg).hold();
+}
+
+static ex imag_part_eval(const ex & arg)
+{
+       return arg.imag_part();
+}
+
+static void imag_part_print_latex(const ex & arg, const print_context & c)
+{
+       c.s << "\\Im"; arg.print(c); c.s << "";
+}
+
+static ex imag_part_conjugate(const ex & arg)
+{
+       return imag_part_function(arg).hold();
+}
+
+static ex imag_part_real_part(const ex & arg)
+{
+       return imag_part_function(arg).hold();
+}
+
+static ex imag_part_imag_part(const ex & arg)
+{
+       return 0;
+}
+
+// If x is real then Im(e).diff(x) is equal to Im(e.diff(x)) 
+static ex imag_part_expl_derivative(const ex & arg, const symbol & s)
+{
+       if (s.info(info_flags::real))
+               return imag_part_function(arg.diff(s));
+       else {
+               exvector vec_arg;
+               vec_arg.push_back(arg);
+               return fderivative(ex_to<function>(imag_part(arg)).get_serial(),0,vec_arg).hold()*arg.diff(s);
+       }
+}
+
+REGISTER_FUNCTION(imag_part_function, eval_func(imag_part_eval).
+                                      evalf_func(imag_part_evalf).
+                                      expl_derivative_func(imag_part_expl_derivative).
+                                      print_func<print_latex>(imag_part_print_latex).
+                                      conjugate_func(imag_part_conjugate).
+                                      real_part_func(imag_part_real_part).
+                                      imag_part_func(imag_part_imag_part).
+                                      set_name("imag_part","imag_part"));
+
 //////////
 // absolute value
 //////////
 
 static ex abs_evalf(const ex & arg)
 {
-       BEGIN_TYPECHECK
-               TYPECHECK(arg,numeric)
-       END_TYPECHECK(abs(arg))
+       if (is_exactly_a<numeric>(arg))
+               return abs(ex_to<numeric>(arg));
        
-       return abs(ex_to<numeric>(arg));
+       return abs(arg).hold();
 }
 
 static ex abs_eval(const ex & arg)
 {
-       if (is_ex_exactly_of_type(arg, numeric))
+       if (is_exactly_a<numeric>(arg))
                return abs(ex_to<numeric>(arg));
+
+       if (arg.info(info_flags::nonnegative))
+               return arg;
+
+       if (arg.info(info_flags::negative) || (-arg).info(info_flags::nonnegative))
+               return -arg;
+
+       if (is_ex_the_function(arg, abs))
+               return arg;
+
+       if (is_ex_the_function(arg, exp))
+               return exp(arg.op(0).real_part());
+
+       if (is_exactly_a<power>(arg)) {
+               const ex& base = arg.op(0);
+               const ex& exponent = arg.op(1);
+               if (base.info(info_flags::positive) || exponent.info(info_flags::real))
+                       return pow(abs(base), exponent.real_part());
+       }
+
+       if (is_ex_the_function(arg, conjugate_function))
+               return abs(arg.op(0));
+
+       if (is_ex_the_function(arg, step))
+               return arg;
+
+       return abs(arg).hold();
+}
+
+static ex abs_expand(const ex & arg, unsigned options)
+{
+       if ((options & expand_options::expand_transcendental)
+               && is_exactly_a<mul>(arg)) {
+               exvector prodseq;
+               prodseq.reserve(arg.nops());
+               for (const_iterator i = arg.begin(); i != arg.end(); ++i) {
+                       if (options & expand_options::expand_function_args)
+                               prodseq.push_back(abs(i->expand(options)));
+                       else
+                               prodseq.push_back(abs(*i));
+               }
+               return dynallocate<mul>(prodseq).setflag(status_flags::expanded);
+       }
+
+       if (options & expand_options::expand_function_args)
+               return abs(arg.expand(options)).hold();
        else
                return abs(arg).hold();
 }
 
+static ex abs_expl_derivative(const ex & arg, const symbol & s)
+{
+       ex diff_arg = arg.diff(s);
+       return (diff_arg*arg.conjugate()+arg*diff_arg.conjugate())/2/abs(arg);
+}
+
+static void abs_print_latex(const ex & arg, const print_context & c)
+{
+       c.s << "{|"; arg.print(c); c.s << "|}";
+}
+
+static void abs_print_csrc_float(const ex & arg, const print_context & c)
+{
+       c.s << "fabs("; arg.print(c); c.s << ")";
+}
+
+static ex abs_conjugate(const ex & arg)
+{
+       return abs(arg).hold();
+}
+
+static ex abs_real_part(const ex & arg)
+{
+       return abs(arg).hold();
+}
+
+static ex abs_imag_part(const ex& arg)
+{
+       return 0;
+}
+
+static ex abs_power(const ex & arg, const ex & exp)
+{
+       if ((is_a<numeric>(exp) && ex_to<numeric>(exp).is_even()) || exp.info(info_flags::even)) {
+               if (arg.info(info_flags::real) || arg.is_equal(arg.conjugate()))
+                       return pow(arg, exp);
+               else
+                       return pow(arg, exp/2) * pow(arg.conjugate(), exp/2);
+       } else
+               return power(abs(arg), exp).hold();
+}
+
+bool abs_info(const ex & arg, unsigned inf)
+{
+       switch (inf) {
+               case info_flags::integer:
+               case info_flags::even:
+               case info_flags::odd:
+               case info_flags::prime:
+                       return arg.info(inf);
+               case info_flags::nonnegint:
+                       return arg.info(info_flags::integer);
+               case info_flags::nonnegative:
+               case info_flags::real:
+                       return true;
+               case info_flags::negative:
+                       return false;
+               case info_flags::positive:
+                       return arg.info(info_flags::positive) || arg.info(info_flags::negative);
+               case info_flags::has_indices: {
+                       if (arg.info(info_flags::has_indices))
+                               return true;
+                       else
+                               return false;
+               }
+       }
+       return false;
+}
+
 REGISTER_FUNCTION(abs, eval_func(abs_eval).
-                       evalf_func(abs_evalf));
+                       evalf_func(abs_evalf).
+                       expand_func(abs_expand).
+                       expl_derivative_func(abs_expl_derivative).
+                       info_func(abs_info).
+                       print_func<print_latex>(abs_print_latex).
+                       print_func<print_csrc_float>(abs_print_csrc_float).
+                       print_func<print_csrc_double>(abs_print_csrc_float).
+                       conjugate_func(abs_conjugate).
+                       real_part_func(abs_real_part).
+                       imag_part_func(abs_imag_part).
+                       power_func(abs_power));
+
+//////////
+// Step function
+//////////
+
+static ex step_evalf(const ex & arg)
+{
+       if (is_exactly_a<numeric>(arg))
+               return step(ex_to<numeric>(arg));
+       
+       return step(arg).hold();
+}
+
+static ex step_eval(const ex & arg)
+{
+       if (is_exactly_a<numeric>(arg))
+               return step(ex_to<numeric>(arg));
+       
+       else if (is_exactly_a<mul>(arg) &&
+                is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
+               numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
+               if (oc.is_real()) {
+                       if (oc > 0)
+                               // step(42*x) -> step(x)
+                               return step(arg/oc).hold();
+                       else
+                               // step(-42*x) -> step(-x)
+                               return step(-arg/oc).hold();
+               }
+               if (oc.real().is_zero()) {
+                       if (oc.imag() > 0)
+                               // step(42*I*x) -> step(I*x)
+                               return step(I*arg/oc).hold();
+                       else
+                               // step(-42*I*x) -> step(-I*x)
+                               return step(-I*arg/oc).hold();
+               }
+       }
+       
+       return step(arg).hold();
+}
+
+static ex step_series(const ex & arg,
+                      const relational & rel,
+                      int order,
+                      unsigned options)
+{
+       const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
+       if (arg_pt.info(info_flags::numeric)
+           && ex_to<numeric>(arg_pt).real().is_zero()
+           && !(options & series_options::suppress_branchcut))
+               throw (std::domain_error("step_series(): on imaginary axis"));
+       
+       epvector seq { expair(step(arg_pt), _ex0) };
+       return pseries(rel, std::move(seq));
+}
 
+static ex step_conjugate(const ex& arg)
+{
+       return step(arg).hold();
+}
+
+static ex step_real_part(const ex& arg)
+{
+       return step(arg).hold();
+}
+
+static ex step_imag_part(const ex& arg)
+{
+       return 0;
+}
+
+REGISTER_FUNCTION(step, eval_func(step_eval).
+                        evalf_func(step_evalf).
+                        series_func(step_series).
+                        conjugate_func(step_conjugate).
+                        real_part_func(step_real_part).
+                        imag_part_func(step_imag_part));
 
 //////////
 // Complex sign
@@ -68,20 +484,19 @@ REGISTER_FUNCTION(abs, eval_func(abs_eval).
 
 static ex csgn_evalf(const ex & arg)
 {
-       BEGIN_TYPECHECK
-               TYPECHECK(arg,numeric)
-       END_TYPECHECK(csgn(arg))
+       if (is_exactly_a<numeric>(arg))
+               return csgn(ex_to<numeric>(arg));
        
-       return csgn(ex_to<numeric>(arg));
+       return csgn(arg).hold();
 }
 
 static ex csgn_eval(const ex & arg)
 {
-       if (is_ex_exactly_of_type(arg, numeric))
+       if (is_exactly_a<numeric>(arg))
                return csgn(ex_to<numeric>(arg));
        
-       else if (is_ex_of_type(arg, mul) &&
-                is_ex_of_type(arg.op(arg.nops()-1),numeric)) {
+       else if (is_exactly_a<mul>(arg) &&
+                is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
                numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
                if (oc.is_real()) {
                        if (oc > 0)
@@ -109,75 +524,146 @@ static ex csgn_series(const ex & arg,
                       int order,
                       unsigned options)
 {
-       const ex arg_pt = arg.subs(rel);
+       const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
        if (arg_pt.info(info_flags::numeric)
            && ex_to<numeric>(arg_pt).real().is_zero()
            && !(options & series_options::suppress_branchcut))
                throw (std::domain_error("csgn_series(): on imaginary axis"));
        
-       epvector seq;
-       seq.push_back(expair(csgn(arg_pt), _ex0()));
-       return pseries(rel,seq);
+       epvector seq { expair(csgn(arg_pt), _ex0) };
+       return pseries(rel, std::move(seq));
+}
+
+static ex csgn_conjugate(const ex& arg)
+{
+       return csgn(arg).hold();
+}
+
+static ex csgn_real_part(const ex& arg)
+{
+       return csgn(arg).hold();
+}
+
+static ex csgn_imag_part(const ex& arg)
+{
+       return 0;
+}
+
+static ex csgn_power(const ex & arg, const ex & exp)
+{
+       if (is_a<numeric>(exp) && exp.info(info_flags::positive) && ex_to<numeric>(exp).is_integer()) {
+               if (ex_to<numeric>(exp).is_odd())
+                       return csgn(arg).hold();
+               else
+                       return power(csgn(arg), _ex2).hold();
+       } else
+               return power(csgn(arg), exp).hold();
 }
 
+
 REGISTER_FUNCTION(csgn, eval_func(csgn_eval).
                         evalf_func(csgn_evalf).
-                        series_func(csgn_series));
+                        series_func(csgn_series).
+                        conjugate_func(csgn_conjugate).
+                        real_part_func(csgn_real_part).
+                        imag_part_func(csgn_imag_part).
+                        power_func(csgn_power));
 
 
 //////////
-// Eta function: log(x*y) == log(x) + log(y) + eta(x,y).
+// Eta function: eta(x,y) == log(x*y) - log(x) - log(y).
+// This function is closely related to the unwinding number K, sometimes found
+// in modern literature: K(z) == (z-log(exp(z)))/(2*Pi*I).
 //////////
 
-static ex eta_evalf(const ex & x, const ex & y)
+static ex eta_evalf(const ex &x, const ex &y)
 {
-       BEGIN_TYPECHECK
-               TYPECHECK(x,numeric)
-               TYPECHECK(y,numeric)
-       END_TYPECHECK(eta(x,y))
-               
-       numeric xim = imag(ex_to<numeric>(x));
-       numeric yim = imag(ex_to<numeric>(y));
-       numeric xyim = imag(ex_to<numeric>(x*y));
-       return evalf(I/4*Pi)*((csgn(-xim)+1)*(csgn(-yim)+1)*(csgn(xyim)+1)-(csgn(xim)+1)*(csgn(yim)+1)*(csgn(-xyim)+1));
+       // It seems like we basically have to replicate the eval function here,
+       // since the expression might not be fully evaluated yet.
+       if (x.info(info_flags::positive) || y.info(info_flags::positive))
+               return _ex0;
+
+       if (x.info(info_flags::numeric) &&      y.info(info_flags::numeric)) {
+               const numeric nx = ex_to<numeric>(x);
+               const numeric ny = ex_to<numeric>(y);
+               const numeric nxy = ex_to<numeric>(x*y);
+               int cut = 0;
+               if (nx.is_real() && nx.is_negative())
+                       cut -= 4;
+               if (ny.is_real() && ny.is_negative())
+                       cut -= 4;
+               if (nxy.is_real() && nxy.is_negative())
+                       cut += 4;
+               return evalf(I/4*Pi)*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
+                                     (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
+       }
+
+       return eta(x,y).hold();
 }
 
-static ex eta_eval(const ex & x, const ex & y)
+static ex eta_eval(const ex &x, const ex &y)
 {
-       if (is_ex_exactly_of_type(x, numeric) &&
-               is_ex_exactly_of_type(y, numeric)) {
+       // trivial:  eta(x,c) -> 0  if c is real and positive
+       if (x.info(info_flags::positive) || y.info(info_flags::positive))
+               return _ex0;
+
+       if (x.info(info_flags::numeric) &&      y.info(info_flags::numeric)) {
                // don't call eta_evalf here because it would call Pi.evalf()!
-               numeric xim = imag(ex_to<numeric>(x));
-               numeric yim = imag(ex_to<numeric>(y));
-               numeric xyim = imag(ex_to<numeric>(x*y));
-               return (I/4)*Pi*((csgn(-xim)+1)*(csgn(-yim)+1)*(csgn(xyim)+1)-(csgn(xim)+1)*(csgn(yim)+1)*(csgn(-xyim)+1));
+               const numeric nx = ex_to<numeric>(x);
+               const numeric ny = ex_to<numeric>(y);
+               const numeric nxy = ex_to<numeric>(x*y);
+               int cut = 0;
+               if (nx.is_real() && nx.is_negative())
+                       cut -= 4;
+               if (ny.is_real() && ny.is_negative())
+                       cut -= 4;
+               if (nxy.is_real() && nxy.is_negative())
+                       cut += 4;
+               return (I/4)*Pi*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
+                                (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
        }
        
        return eta(x,y).hold();
 }
 
-static ex eta_series(const ex & arg1,
-                     const ex & arg2,
+static ex eta_series(const ex & x, const ex & y,
                      const relational & rel,
                      int order,
                      unsigned options)
 {
-       const ex arg1_pt = arg1.subs(rel);
-       const ex arg2_pt = arg2.subs(rel);
-       if (ex_to<numeric>(arg1_pt).imag().is_zero() ||
-               ex_to<numeric>(arg2_pt).imag().is_zero() ||
-               ex_to<numeric>(arg1_pt*arg2_pt).imag().is_zero()) {
-               throw (std::domain_error("eta_series(): on discontinuity"));
-       }
-       epvector seq;
-       seq.push_back(expair(eta(arg1_pt,arg2_pt), _ex0()));
-       return pseries(rel,seq);
+       const ex x_pt = x.subs(rel, subs_options::no_pattern);
+       const ex y_pt = y.subs(rel, subs_options::no_pattern);
+       if ((x_pt.info(info_flags::numeric) && x_pt.info(info_flags::negative)) ||
+           (y_pt.info(info_flags::numeric) && y_pt.info(info_flags::negative)) ||
+           ((x_pt*y_pt).info(info_flags::numeric) && (x_pt*y_pt).info(info_flags::negative)))
+                       throw (std::domain_error("eta_series(): on discontinuity"));
+       epvector seq { expair(eta(x_pt,y_pt), _ex0) };
+       return pseries(rel, std::move(seq));
+}
+
+static ex eta_conjugate(const ex & x, const ex & y)
+{
+       return -eta(x, y).hold();
+}
+
+static ex eta_real_part(const ex & x, const ex & y)
+{
+       return 0;
+}
+
+static ex eta_imag_part(const ex & x, const ex & y)
+{
+       return -I*eta(x, y).hold();
 }
 
 REGISTER_FUNCTION(eta, eval_func(eta_eval).
                        evalf_func(eta_evalf).
                        series_func(eta_series).
-                       latex_name("\\eta"));
+                       latex_name("\\eta").
+                       set_symmetry(sy_symm(0, 1)).
+                       conjugate_func(eta_conjugate).
+                       real_part_func(eta_real_part).
+                       imag_part_func(eta_imag_part));
 
 
 //////////
@@ -186,11 +672,10 @@ REGISTER_FUNCTION(eta, eval_func(eta_eval).
 
 static ex Li2_evalf(const ex & x)
 {
-       BEGIN_TYPECHECK
-               TYPECHECK(x,numeric)
-       END_TYPECHECK(Li2(x))
+       if (is_exactly_a<numeric>(x))
+               return Li2(ex_to<numeric>(x));
        
-       return Li2(ex_to<numeric>(x));  // -> numeric Li2(numeric)
+       return Li2(x).hold();
 }
 
 static ex Li2_eval(const ex & x)
@@ -198,25 +683,25 @@ static ex Li2_eval(const ex & x)
        if (x.info(info_flags::numeric)) {
                // Li2(0) -> 0
                if (x.is_zero())
-                       return _ex0();
+                       return _ex0;
                // Li2(1) -> Pi^2/6
-               if (x.is_equal(_ex1()))
-                       return power(Pi,_ex2())/_ex6();
+               if (x.is_equal(_ex1))
+                       return power(Pi,_ex2)/_ex6;
                // Li2(1/2) -> Pi^2/12 - log(2)^2/2
-               if (x.is_equal(_ex1_2()))
-                       return power(Pi,_ex2())/_ex12() + power(log(_ex2()),_ex2())*_ex_1_2();
+               if (x.is_equal(_ex1_2))
+                       return power(Pi,_ex2)/_ex12 + power(log(_ex2),_ex2)*_ex_1_2;
                // Li2(-1) -> -Pi^2/12
-               if (x.is_equal(_ex_1()))
-                       return -power(Pi,_ex2())/_ex12();
+               if (x.is_equal(_ex_1))
+                       return -power(Pi,_ex2)/_ex12;
                // Li2(I) -> -Pi^2/48+Catalan*I
                if (x.is_equal(I))
-                       return power(Pi,_ex2())/_ex_48() + Catalan*I;
+                       return power(Pi,_ex2)/_ex_48 + Catalan*I;
                // Li2(-I) -> -Pi^2/48-Catalan*I
                if (x.is_equal(-I))
-                       return power(Pi,_ex2())/_ex_48() - Catalan*I;
+                       return power(Pi,_ex2)/_ex_48 - Catalan*I;
                // Li2(float)
                if (!x.info(info_flags::crational))
-                       return Li2_evalf(x);
+                       return Li2(ex_to<numeric>(x));
        }
        
        return Li2(x).hold();
@@ -227,12 +712,12 @@ static ex Li2_deriv(const ex & x, unsigned deriv_param)
        GINAC_ASSERT(deriv_param==0);
        
        // d/dx Li2(x) -> -log(1-x)/x
-       return -log(1-x)/x;
+       return -log(_ex1-x)/x;
 }
 
 static ex Li2_series(const ex &x, const relational &rel, int order, unsigned options)
 {
-       const ex x_pt = x.subs(rel);
+       const ex x_pt = x.subs(rel, subs_options::no_pattern);
        if (x_pt.info(info_flags::numeric)) {
                // First special case: x==0 (derivatives have poles)
                if (x_pt.is_zero()) {
@@ -252,13 +737,12 @@ static ex Li2_series(const ex &x, const relational &rel, int order, unsigned opt
                        ex ser;
                        // manually construct the primitive expansion
                        for (int i=1; i<order; ++i)
-                               ser += pow(s,i) / pow(numeric(i), _num2());
+                               ser += pow(s,i) / pow(numeric(i), *_num2_p);
                        // substitute the argument's series expansion
-                       ser = ser.subs(s==x.series(rel, order));
+                       ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
                        // maybe that was terminating, so add a proper order term
-                       epvector nseq;
-                       nseq.push_back(expair(Order(_ex1()), order));
-                       ser += pseries(rel, nseq);
+                       epvector nseq { expair(Order(_ex1), order) };
+                       ser += pseries(rel, std::move(nseq));
                        // reexpanding it will collapse the series again
                        return ser.series(rel, order);
                        // NB: Of course, this still does not allow us to compute anything
@@ -270,20 +754,19 @@ static ex Li2_series(const ex &x, const relational &rel, int order, unsigned opt
                        // obsolete!
                }
                // second special case: x==1 (branch point)
-               if (x_pt == _ex1()) {
+               if (x_pt.is_equal(_ex1)) {
                        // method:
                        // construct series manually in a dummy symbol s
                        const symbol s;
-                       ex ser = zeta(2);
+                       ex ser = zeta(_ex2);
                        // manually construct the primitive expansion
                        for (int i=1; i<order; ++i)
                                ser += pow(1-s,i) * (numeric(1,i)*(I*Pi+log(s-1)) - numeric(1,i*i));
                        // substitute the argument's series expansion
-                       ser = ser.subs(s==x.series(rel, order));
+                       ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
                        // maybe that was terminating, so add a proper order term
-                       epvector nseq;
-                       nseq.push_back(expair(Order(_ex1()), order));
-                       ser += pseries(rel, nseq);
+                       epvector nseq { expair(Order(_ex1), order) };
+                       ser += pseries(rel, std::move(nseq));
                        // reexpanding it will collapse the series again
                        return ser.series(rel, order);
                }
@@ -293,30 +776,45 @@ static ex Li2_series(const ex &x, const relational &rel, int order, unsigned opt
                        // method:
                        // This is the branch cut: assemble the primitive series manually
                        // and then add the corresponding complex step function.
-                       const symbol *s = static_cast<symbol *>(rel.lhs().bp);
+                       const symbol &s = ex_to<symbol>(rel.lhs());
                        const ex point = rel.rhs();
                        const symbol foo;
                        epvector seq;
                        // zeroth order term:
-                       seq.push_back(expair(Li2(x_pt), _ex0()));
+                       seq.push_back(expair(Li2(x_pt), _ex0));
                        // compute the intermediate terms:
-                       ex replarg = series(Li2(x), *s==foo, order);
-                       for (unsigned i=1; i<replarg.nops()-1; ++i)
-                               seq.push_back(expair((replarg.op(i)/power(*s-foo,i)).series(foo==point,1,options).op(0).subs(foo==*s),i));
+                       ex replarg = series(Li2(x), s==foo, order);
+                       for (size_t i=1; i<replarg.nops()-1; ++i)
+                               seq.push_back(expair((replarg.op(i)/power(s-foo,i)).series(foo==point,1,options).op(0).subs(foo==s, subs_options::no_pattern),i));
                        // append an order term:
-                       seq.push_back(expair(Order(_ex1()), replarg.nops()-1));
-                       return pseries(rel, seq);
+                       seq.push_back(expair(Order(_ex1), replarg.nops()-1));
+                       return pseries(rel, std::move(seq));
                }
        }
        // all other cases should be safe, by now:
        throw do_taylor();  // caught by function::series()
 }
 
+static ex Li2_conjugate(const ex & x)
+{
+       // conjugate(Li2(x))==Li2(conjugate(x)) unless on the branch cuts which
+       // run along the positive real axis beginning at 1.
+       if (x.info(info_flags::negative)) {
+               return Li2(x).hold();
+       }
+       if (is_exactly_a<numeric>(x) &&
+           (!x.imag_part().is_zero() || x < *_num1_p)) {
+               return Li2(x.conjugate());
+       }
+       return conjugate_function(Li2(x)).hold();
+}
+
 REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
                        evalf_func(Li2_evalf).
                        derivative_func(Li2_deriv).
                        series_func(Li2_series).
-                       latex_name("\\mbox{Li}_2"));
+                       conjugate_func(Li2_conjugate).
+                       latex_name("\\mathrm{Li}_2"));
 
 //////////
 // trilogarithm
@@ -330,7 +828,38 @@ static ex Li3_eval(const ex & x)
 }
 
 REGISTER_FUNCTION(Li3, eval_func(Li3_eval).
-                       latex_name("\\mbox{Li}_3"));
+                       latex_name("\\mathrm{Li}_3"));
+
+//////////
+// Derivatives of Riemann's Zeta-function  zetaderiv(0,x)==zeta(x)
+//////////
+
+static ex zetaderiv_eval(const ex & n, const ex & x)
+{
+       if (n.info(info_flags::numeric)) {
+               // zetaderiv(0,x) -> zeta(x)
+               if (n.is_zero())
+                       return zeta(x).hold();
+       }
+       
+       return zetaderiv(n, x).hold();
+}
+
+static ex zetaderiv_deriv(const ex & n, const ex & x, unsigned deriv_param)
+{
+       GINAC_ASSERT(deriv_param<2);
+       
+       if (deriv_param==0) {
+               // d/dn zeta(n,x)
+               throw(std::logic_error("cannot diff zetaderiv(n,x) with respect to n"));
+       }
+       // d/dx psi(n,x)
+       return zetaderiv(n+1,x);
+}
+
+REGISTER_FUNCTION(zetaderiv, eval_func(zetaderiv_eval).
+                                derivative_func(zetaderiv_deriv).
+                                latex_name("\\zeta^\\prime"));
 
 //////////
 // factorial
@@ -343,14 +872,45 @@ static ex factorial_evalf(const ex & x)
 
 static ex factorial_eval(const ex & x)
 {
-       if (is_ex_exactly_of_type(x, numeric))
+       if (is_exactly_a<numeric>(x))
                return factorial(ex_to<numeric>(x));
        else
                return factorial(x).hold();
 }
 
+static void factorial_print_dflt_latex(const ex & x, const print_context & c)
+{
+       if (is_exactly_a<symbol>(x) ||
+           is_exactly_a<constant>(x) ||
+               is_exactly_a<function>(x)) {
+               x.print(c); c.s << "!";
+       } else {
+               c.s << "("; x.print(c); c.s << ")!";
+       }
+}
+
+static ex factorial_conjugate(const ex & x)
+{
+       return factorial(x).hold();
+}
+
+static ex factorial_real_part(const ex & x)
+{
+       return factorial(x).hold();
+}
+
+static ex factorial_imag_part(const ex & x)
+{
+       return 0;
+}
+
 REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
-                             evalf_func(factorial_evalf));
+                             evalf_func(factorial_evalf).
+                             print_func<print_dflt>(factorial_print_dflt_latex).
+                             print_func<print_latex>(factorial_print_dflt_latex).
+                             conjugate_func(factorial_conjugate).
+                             real_part_func(factorial_real_part).
+                             imag_part_func(factorial_imag_part));
 
 //////////
 // binomial
@@ -361,16 +921,58 @@ static ex binomial_evalf(const ex & x, const ex & y)
        return binomial(x, y).hold();
 }
 
+static ex binomial_sym(const ex & x, const numeric & y)
+{
+       if (y.is_integer()) {
+               if (y.is_nonneg_integer()) {
+                       const unsigned N = y.to_int();
+                       if (N == 0) return _ex1;
+                       if (N == 1) return x;
+                       ex t = x.expand();
+                       for (unsigned i = 2; i <= N; ++i)
+                               t = (t * (x + i - y - 1)).expand() / i;
+                       return t;
+               } else
+                       return _ex0;
+       }
+
+       return binomial(x, y).hold();
+}
+
 static ex binomial_eval(const ex & x, const ex &y)
 {
-       if (is_ex_exactly_of_type(x, numeric) && is_ex_exactly_of_type(y, numeric))
-               return binomial(ex_to<numeric>(x), ex_to<numeric>(y));
-       else
+       if (is_exactly_a<numeric>(y)) {
+               if (is_exactly_a<numeric>(x) && ex_to<numeric>(x).is_integer())
+                       return binomial(ex_to<numeric>(x), ex_to<numeric>(y));
+               else
+                       return binomial_sym(x, ex_to<numeric>(y));
+       } else
                return binomial(x, y).hold();
 }
 
+// At the moment the numeric evaluation of a binomial function always
+// gives a real number, but if this would be implemented using the gamma
+// function, also complex conjugation should be changed (or rather, deleted).
+static ex binomial_conjugate(const ex & x, const ex & y)
+{
+       return binomial(x,y).hold();
+}
+
+static ex binomial_real_part(const ex & x, const ex & y)
+{
+       return binomial(x,y).hold();
+}
+
+static ex binomial_imag_part(const ex & x, const ex & y)
+{
+       return 0;
+}
+
 REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
-                            evalf_func(binomial_evalf));
+                            evalf_func(binomial_evalf).
+                            conjugate_func(binomial_conjugate).
+                            real_part_func(binomial_real_part).
+                            imag_part_func(binomial_imag_part));
 
 //////////
 // Order term function (for truncated power series)
@@ -378,17 +980,17 @@ REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
 
 static ex Order_eval(const ex & x)
 {
-       if (is_ex_exactly_of_type(x, numeric)) {
+       if (is_exactly_a<numeric>(x)) {
                // O(c) -> O(1) or 0
                if (!x.is_zero())
-                       return Order(_ex1()).hold();
+                       return Order(_ex1).hold();
                else
-                       return _ex0();
-       } else if (is_ex_exactly_of_type(x, mul)) {
-               mul *m = static_cast<mul *>(x.bp);
+                       return _ex0;
+       } else if (is_exactly_a<mul>(x)) {
+               const mul &m = ex_to<mul>(x);
                // O(c*expr) -> O(expr)
-               if (is_ex_exactly_of_type(m->op(m->nops() - 1), numeric))
-                       return Order(x / m->op(m->nops() - 1)).hold();
+               if (is_exactly_a<numeric>(m.op(m.nops() - 1)))
+                       return Order(x / m.op(m.nops() - 1)).hold();
        }
        return Order(x).hold();
 }
@@ -396,69 +998,93 @@ static ex Order_eval(const ex & x)
 static ex Order_series(const ex & x, const relational & r, int order, unsigned options)
 {
        // Just wrap the function into a pseries object
-       epvector new_seq;
-       GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
-       const symbol *s = static_cast<symbol *>(r.lhs().bp);
-       new_seq.push_back(expair(Order(_ex1()), numeric(std::min(x.ldegree(*s), order))));
-       return pseries(r, new_seq);
+       GINAC_ASSERT(is_a<symbol>(r.lhs()));
+       const symbol &s = ex_to<symbol>(r.lhs());
+       epvector new_seq { expair(Order(_ex1), numeric(std::min(x.ldegree(s), order))) };
+       return pseries(r, std::move(new_seq));
 }
 
-// Differentiation is handled in function::derivative because of its special requirements
+static ex Order_conjugate(const ex & x)
+{
+       return Order(x).hold();
+}
+
+static ex Order_real_part(const ex & x)
+{
+       return Order(x).hold();
+}
+
+static ex Order_imag_part(const ex & x)
+{
+       if(x.info(info_flags::real))
+               return 0;
+       return Order(x).hold();
+}
+
+static ex Order_expl_derivative(const ex & arg, const symbol & s)
+{
+       return Order(arg.diff(s));
+}
 
 REGISTER_FUNCTION(Order, eval_func(Order_eval).
                          series_func(Order_series).
-                         latex_name("\\mathcal{O}"));
+                         latex_name("\\mathcal{O}").
+                         expl_derivative_func(Order_expl_derivative).
+                         conjugate_func(Order_conjugate).
+                         real_part_func(Order_real_part).
+                         imag_part_func(Order_imag_part));
 
 //////////
-// Inert partial differentiation operator
+// Solve linear system
 //////////
 
-static ex Derivative_eval(const ex & f, const ex & l)
+static void insert_symbols(exset &es, const ex &e)
 {
-       if (!is_ex_exactly_of_type(f, function)) {
-               throw(std::invalid_argument("Derivative(): 1st argument must be a function"));
-       }
-       if (!is_ex_exactly_of_type(l, lst)) {
-               throw(std::invalid_argument("Derivative(): 2nd argument must be a list"));
+       if (is_a<symbol>(e)) {
+               es.insert(e);
+       } else {
+               for (const ex &sube : e) {
+                       insert_symbols(es, sube);
+               }
        }
-       return Derivative(f, l).hold();
 }
 
-REGISTER_FUNCTION(Derivative, eval_func(Derivative_eval));
-
-//////////
-// Solve linear system
-//////////
+static exset symbolset(const ex &e)
+{
+       exset s;
+       insert_symbols(s, e);
+       return s;
+}
 
-ex lsolve(const ex &eqns, const ex &symbols)
+ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
 {
        // solve a system of linear equations
        if (eqns.info(info_flags::relation_equal)) {
                if (!symbols.info(info_flags::symbol))
                        throw(std::invalid_argument("lsolve(): 2nd argument must be a symbol"));
-               ex sol=lsolve(lst(eqns),lst(symbols));
+               const ex sol = lsolve(lst{eqns}, lst{symbols});
                
                GINAC_ASSERT(sol.nops()==1);
-               GINAC_ASSERT(is_ex_exactly_of_type(sol.op(0),relational));
+               GINAC_ASSERT(is_exactly_a<relational>(sol.op(0)));
                
                return sol.op(0).op(1); // return rhs of first solution
        }
        
        // syntax checks
-       if (!eqns.info(info_flags::list)) {
-               throw(std::invalid_argument("lsolve(): 1st argument must be a list"));
+       if (!(eqns.info(info_flags::list) || eqns.info(info_flags::exprseq))) {
+               throw(std::invalid_argument("lsolve(): 1st argument must be a list, a sequence, or an equation"));
        }
-       for (unsigned i=0; i<eqns.nops(); i++) {
+       for (size_t i=0; i<eqns.nops(); i++) {
                if (!eqns.op(i).info(info_flags::relation_equal)) {
                        throw(std::invalid_argument("lsolve(): 1st argument must be a list of equations"));
                }
        }
-       if (!symbols.info(info_flags::list)) {
-               throw(std::invalid_argument("lsolve(): 2nd argument must be a list"));
+       if (!(symbols.info(info_flags::list) || symbols.info(info_flags::exprseq))) {
+               throw(std::invalid_argument("lsolve(): 2nd argument must be a list, a sequence, or a symbol"));
        }
-       for (unsigned i=0; i<symbols.nops(); i++) {
+       for (size_t i=0; i<symbols.nops(); i++) {
                if (!symbols.op(i).info(info_flags::symbol)) {
-                       throw(std::invalid_argument("lsolve(): 2nd argument must be a list of symbols"));
+                       throw(std::invalid_argument("lsolve(): 2nd argument must be a list or a sequence of symbols"));
                }
        }
        
@@ -467,11 +1093,13 @@ ex lsolve(const ex &eqns, const ex &symbols)
        matrix rhs(eqns.nops(),1);
        matrix vars(symbols.nops(),1);
        
-       for (unsigned r=0; r<eqns.nops(); r++) {
-               ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
+       for (size_t r=0; r<eqns.nops(); r++) {
+               const ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
+               const exset syms = symbolset(eq);
                ex linpart = eq;
-               for (unsigned c=0; c<symbols.nops(); c++) {
-                       ex co = eq.coeff(ex_to<symbol>(symbols.op(c)),1);
+               for (size_t c=0; c<symbols.nops(); c++) {
+                       if (syms.count(symbols.op(c)) == 0) continue;
+                       const ex co = eq.coeff(ex_to<symbol>(symbols.op(c)),1);
                        linpart -= co*symbols.op(c);
                        sys(r,c) = co;
                }
@@ -480,36 +1108,164 @@ ex lsolve(const ex &eqns, const ex &symbols)
        }
        
        // test if system is linear and fill vars matrix
-       for (unsigned i=0; i<symbols.nops(); i++) {
+       const exset sys_syms = symbolset(sys);
+       const exset rhs_syms = symbolset(rhs);
+       for (size_t i=0; i<symbols.nops(); i++) {
                vars(i,0) = symbols.op(i);
-               if (sys.has(symbols.op(i)))
+               if (sys_syms.count(symbols.op(i)) != 0)
                        throw(std::logic_error("lsolve: system is not linear"));
-               if (rhs.has(symbols.op(i)))
+               if (rhs_syms.count(symbols.op(i)) != 0)
                        throw(std::logic_error("lsolve: system is not linear"));
        }
        
        matrix solution;
        try {
-               solution = sys.solve(vars,rhs);
+               solution = sys.solve(vars,rhs,options);
        } catch (const std::runtime_error & e) {
                // Probably singular matrix or otherwise overdetermined system:
                // It is consistent to return an empty list
-               return lst();
-       }    
+               return lst{};
+       }
        GINAC_ASSERT(solution.cols()==1);
        GINAC_ASSERT(solution.rows()==symbols.nops());
        
-       // return list of equations of the form lst(var1==sol1,var2==sol2,...)
+       // return list of equations of the form lst{var1==sol1,var2==sol2,...}
        lst sollist;
-       for (unsigned i=0; i<symbols.nops(); i++)
+       for (size_t i=0; i<symbols.nops(); i++)
                sollist.append(symbols.op(i)==solution(i,0));
        
        return sollist;
 }
 
+//////////
+// Find real root of f(x) numerically
+//////////
+
+const numeric
+fsolve(const ex& f_in, const symbol& x, const numeric& x1, const numeric& x2)
+{
+       if (!x1.is_real() || !x2.is_real()) {
+               throw std::runtime_error("fsolve(): interval not bounded by real numbers");
+       }
+       if (x1==x2) {
+               throw std::runtime_error("fsolve(): vanishing interval");
+       }
+       // xx[0] == left interval limit, xx[1] == right interval limit.
+       // fx[0] == f(xx[0]), fx[1] == f(xx[1]).
+       // We keep the root bracketed: xx[0]<xx[1] and fx[0]*fx[1]<0.
+       numeric xx[2] = { x1<x2 ? x1 : x2,
+                         x1<x2 ? x2 : x1 };
+       ex f;
+       if (is_a<relational>(f_in)) {
+               f = f_in.lhs()-f_in.rhs();
+       } else {
+               f = f_in;
+       }
+       const ex fx_[2] = { f.subs(x==xx[0]).evalf(),
+                           f.subs(x==xx[1]).evalf() };
+       if (!is_a<numeric>(fx_[0]) || !is_a<numeric>(fx_[1])) {
+               throw std::runtime_error("fsolve(): function does not evaluate numerically");
+       }
+       numeric fx[2] = { ex_to<numeric>(fx_[0]),
+                         ex_to<numeric>(fx_[1]) };
+       if (!fx[0].is_real() || !fx[1].is_real()) {
+               throw std::runtime_error("fsolve(): function evaluates to complex values at interval boundaries");
+       }
+       if (fx[0]*fx[1]>=0) {
+               throw std::runtime_error("fsolve(): function does not change sign at interval boundaries");
+       }
+
+       // The Newton-Raphson method has quadratic convergence!  Simply put, it
+       // replaces x with x-f(x)/f'(x) at each step.  -f/f' is the delta:
+       const ex ff = normal(-f/f.diff(x));
+       int side = 0;  // Start at left interval limit.
+       numeric xxprev;
+       numeric fxprev;
+       do {
+               xxprev = xx[side];
+               fxprev = fx[side];
+               ex dx_ = ff.subs(x == xx[side]).evalf();
+               if (!is_a<numeric>(dx_))
+                       throw std::runtime_error("fsolve(): function derivative does not evaluate numerically");
+               xx[side] += ex_to<numeric>(dx_);
+               // Now check if Newton-Raphson method shot out of the interval 
+               bool bad_shot = (side == 0 && xx[0] < xxprev) || 
+                               (side == 1 && xx[1] > xxprev) || xx[0] > xx[1];
+               if (!bad_shot) {
+                       // Compute f(x) only if new x is inside the interval.
+                       // The function might be difficult to compute numerically
+                       // or even ill defined outside the interval. Also it's
+                       // a small optimization. 
+                       ex f_x = f.subs(x == xx[side]).evalf();
+                       if (!is_a<numeric>(f_x))
+                               throw std::runtime_error("fsolve(): function does not evaluate numerically");
+                       fx[side] = ex_to<numeric>(f_x);
+               }
+               if (bad_shot) {
+                       // Oops, Newton-Raphson method shot out of the interval.
+                       // Restore, and try again with the other side instead!
+                       xx[side] = xxprev;
+                       fx[side] = fxprev;
+                       side = !side;
+                       xxprev = xx[side];
+                       fxprev = fx[side];
+
+                       ex dx_ = ff.subs(x == xx[side]).evalf();
+                       if (!is_a<numeric>(dx_))
+                               throw std::runtime_error("fsolve(): function derivative does not evaluate numerically [2]");
+                       xx[side] += ex_to<numeric>(dx_);
+
+                       ex f_x = f.subs(x==xx[side]).evalf();
+                       if (!is_a<numeric>(f_x))
+                               throw std::runtime_error("fsolve(): function does not evaluate numerically [2]");
+                       fx[side] = ex_to<numeric>(f_x);
+               }
+               if ((fx[side]<0 && fx[!side]<0) || (fx[side]>0 && fx[!side]>0)) {
+                       // Oops, the root isn't bracketed any more.
+                       // Restore, and perform a bisection!
+                       xx[side] = xxprev;
+                       fx[side] = fxprev;
+
+                       // Ah, the bisection! Bisections converge linearly. Unfortunately,
+                       // they occur pretty often when Newton-Raphson arrives at an x too
+                       // close to the result on one side of the interval and
+                       // f(x-f(x)/f'(x)) turns out to have the same sign as f(x) due to
+                       // precision errors! Recall that this function does not have a
+                       // precision goal as one of its arguments but instead relies on
+                       // x converging to a fixed point. We speed up the (safe but slow)
+                       // bisection method by mixing in a dash of the (unsafer but faster)
+                       // secant method: Instead of splitting the interval at the
+                       // arithmetic mean (bisection), we split it nearer to the root as
+                       // determined by the secant between the values xx[0] and xx[1].
+                       // Don't set the secant_weight to one because that could disturb
+                       // the convergence in some corner cases!
+                       constexpr double secant_weight = 0.984375;  // == 63/64 < 1
+                       numeric xxmid = (1-secant_weight)*0.5*(xx[0]+xx[1])
+                           + secant_weight*(xx[0]+fx[0]*(xx[0]-xx[1])/(fx[1]-fx[0]));
+                       ex fxmid_ = f.subs(x == xxmid).evalf();
+                       if (!is_a<numeric>(fxmid_))
+                               throw std::runtime_error("fsolve(): function does not evaluate numerically [3]");
+                       numeric fxmid = ex_to<numeric>(fxmid_);
+                       if (fxmid.is_zero()) {
+                               // Luck strikes...
+                               return xxmid;
+                       }
+                       if ((fxmid<0 && fx[side]>0) || (fxmid>0 && fx[side]<0)) {
+                               side = !side;
+                       }
+                       xxprev = xx[side];
+                       fxprev = fx[side];
+                       xx[side] = xxmid;
+                       fx[side] = fxmid;
+               }
+       } while (xxprev!=xx[side]);
+       return xxprev;
+}
+
+
 /* Force inclusion of functions from inifcns_gamma and inifcns_zeta
  * for static lib (so ginsh will see them). */
-unsigned force_include_tgamma = function_index_tgamma;
-unsigned force_include_zeta1 = function_index_zeta1;
+unsigned force_include_tgamma = tgamma_SERIAL::serial;
+unsigned force_include_zeta1 = zeta1_SERIAL::serial;
 
 } // namespace GiNaC