X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Finifcns.cpp;h=e2a09b161ccf95ad01e3f95f07b823b79d2b2e54;hp=5ebd3d02139b153efa647263a232b956fb7bf5cc;hb=8cffcdf13d817a47f217f1a1043317d95969e070;hpb=dc2510946d9ce577aab2bd3e5d2f62c50d3faa30

diff --git a/ginac/inifcns.cpp b/ginac/inifcns.cpp
index 5ebd3d02..e2a09b16 100644
--- a/ginac/inifcns.cpp
+++ b/ginac/inifcns.cpp
@@ -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
@@ -17,27 +17,241 @@
  *
  *  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
 //////////
@@ -52,15 +266,217 @@ static ex abs_evalf(const ex & arg)
 
 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
@@ -76,11 +492,11 @@ static ex csgn_evalf(const ex & arg)
 
 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)
@@ -108,24 +524,56 @@ 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: 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)
@@ -133,7 +581,7 @@ static ex eta_evalf(const ex &x, const ex &y)
 	// 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();
+		return _ex0;
 
 	if (x.info(info_flags::numeric) &&	y.info(info_flags::numeric)) {
 		const numeric nx = ex_to<numeric>(x);
@@ -157,7 +605,7 @@ static ex eta_eval(const ex &x, const ex &y)
 {
 	// trivial:  eta(x,c) -> 0  if c is real and positive
 	if (x.info(info_flags::positive) || y.info(info_flags::positive))
-		return _ex0();
+		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()!
@@ -183,22 +631,39 @@ static ex eta_series(const ex & x, const ex & y,
                      int order,
                      unsigned options)
 {
-	const ex x_pt = x.subs(rel);
-	const ex y_pt = y.subs(rel);
+	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;
-	seq.push_back(expair(eta(x_pt,y_pt), _ex0()));
-	return pseries(rel,seq);
+	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").
-                       set_symmetry(sy_symm(0, 1)));
+                       set_symmetry(sy_symm(0, 1)).
+                       conjugate_func(eta_conjugate).
+                       real_part_func(eta_real_part).
+                       imag_part_func(eta_imag_part));
 
 
 //////////
@@ -218,22 +683,22 @@ 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(ex_to<numeric>(x));
@@ -247,12 +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(_ex1()-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()) {
@@ -272,13 +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
@@ -290,20 +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.is_equal(_ex1())) {
+		if (x_pt.is_equal(_ex1)) {
 			// method:
 			// construct series manually in a dummy symbol s
 			const symbol s;
-			ex ser = zeta(_ex2());
+			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);
 		}
@@ -313,30 +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
@@ -350,7 +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
@@ -363,14 +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
@@ -381,16 +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)
@@ -398,17 +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();
 }
@@ -416,52 +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));
+}
+
+static ex Order_conjugate(const ex & x)
+{
+	return Order(x).hold();
 }
 
-// Differentiation is handled in function::derivative because of its special requirements
+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));
 
 //////////
 // Solve linear system
 //////////
 
+static void insert_symbols(exset &es, const ex &e)
+{
+	if (is_a<symbol>(e)) {
+		es.insert(e);
+	} else {
+		for (const ex &sube : e) {
+			insert_symbols(es, sube);
+		}
+	}
+}
+
+static exset symbolset(const ex &e)
+{
+	exset s;
+	insert_symbols(s, e);
+	return s;
+}
+
 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"));
-		const 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"));
 		}
 	}
 	
@@ -470,10 +1093,12 @@ ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
 	matrix rhs(eqns.nops(),1);
 	matrix vars(symbols.nops(),1);
 	
-	for (unsigned r=0; r<eqns.nops(); r++) {
+	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++) {
+		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;
@@ -483,11 +1108,13 @@ ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
 	}
 	
 	// 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"));
 	}
 	
@@ -497,22 +1124,148 @@ ex lsolve(const ex &eqns, const ex &symbols, unsigned 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