X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Fpseries.cpp;h=c290fe0a1f7b09b0c73fa2cebdf4e9cb1f609c38;hp=7fbf73113a74f293c3cca3b151083ff29dfa2bb9;hb=079c558d4f9758cd2777a2808a02d64cb1f70c8e;hpb=97af29c12bb3074cfb4e674d71000f0712c51ba2

diff --git a/ginac/pseries.cpp b/ginac/pseries.cpp
index 7fbf7311..c290fe0a 100644
--- a/ginac/pseries.cpp
+++ b/ginac/pseries.cpp
@@ -4,7 +4,7 @@
  *  methods for series expansion. */
 
 /*
- *  GiNaC Copyright (C) 1999-2000 Johannes Gutenberg University Mainz, Germany
+ *  GiNaC Copyright (C) 1999-2010 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
@@ -18,94 +18,62 @@
  *
  *  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 <stdexcept>
-
 #include "pseries.h"
 #include "add.h"
-#include "inifcns.h"
+#include "inifcns.h" // for Order function
 #include "lst.h"
 #include "mul.h"
 #include "power.h"
 #include "relational.h"
+#include "operators.h"
 #include "symbol.h"
+#include "integral.h"
 #include "archive.h"
 #include "utils.h"
-#include "debugmsg.h"
 
-#ifndef NO_NAMESPACE_GINAC
+#include <limits>
+#include <numeric>
+#include <stdexcept>
+
 namespace GiNaC {
-#endif // ndef NO_NAMESPACE_GINAC
 
-GINAC_IMPLEMENT_REGISTERED_CLASS(pseries, basic)
+GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(pseries, basic,
+  print_func<print_context>(&pseries::do_print).
+  print_func<print_latex>(&pseries::do_print_latex).
+  print_func<print_tree>(&pseries::do_print_tree).
+  print_func<print_python>(&pseries::do_print_python).
+  print_func<print_python_repr>(&pseries::do_print_python_repr))
+
 
 /*
- *  Default constructor, destructor, copy constructor, assignment operator and helpers
+ *  Default constructor
  */
 
-pseries::pseries() : basic(TINFO_pseries)
-{
-    debugmsg("pseries default constructor", LOGLEVEL_CONSTRUCT);
-}
-
-pseries::~pseries()
-{
-    debugmsg("pseries destructor", LOGLEVEL_DESTRUCT);
-    destroy(false);
-}
-
-pseries::pseries(const pseries &other)
-{
-    debugmsg("pseries copy constructor", LOGLEVEL_CONSTRUCT);
-    copy(other);
-}
-
-const pseries &pseries::operator=(const pseries & other)
-{
-    debugmsg("pseries operator=", LOGLEVEL_ASSIGNMENT);
-    if (this != &other) {
-        destroy(true);
-        copy(other);
-    }
-    return *this;
-}
-
-void pseries::copy(const pseries &other)
-{
-    inherited::copy(other);
-    seq = other.seq;
-    var = other.var;
-    point = other.point;
-}
-
-void pseries::destroy(bool call_parent)
-{
-    if (call_parent)
-        inherited::destroy(call_parent);
-}
+pseries::pseries() { }
 
 
 /*
- *  Other constructors
+ *  Other ctors
  */
 
 /** Construct pseries from a vector of coefficients and powers.
  *  expair.rest holds the coefficient, expair.coeff holds the power.
  *  The powers must be integers (positive or negative) and in ascending order;
- *  the last coefficient can be Order(_ex1()) to represent a truncated,
+ *  the last coefficient can be Order(_ex1) to represent a truncated,
  *  non-terminating series.
  *
- *  @param var_  series variable (must hold a symbol)
- *  @param point_  expansion point
+ *  @param rel_  expansion variable and point (must hold a relational)
  *  @param ops_  vector of {coefficient, power} pairs (coefficient must not be zero)
  *  @return newly constructed pseries */
-pseries::pseries(const ex &var_, const ex &point_, const epvector &ops_)
-    : basic(TINFO_pseries), seq(ops_), var(var_), point(point_)
+pseries::pseries(const ex &rel_, const epvector &ops_) : seq(ops_)
 {
-    debugmsg("pseries constructor from ex,ex,epvector", LOGLEVEL_CONSTRUCT);
-    GINAC_ASSERT(is_ex_exactly_of_type(var_, symbol));
+	GINAC_ASSERT(is_a<relational>(rel_));
+	GINAC_ASSERT(is_a<symbol>(rel_.lhs()));
+	point = rel_.rhs();
+	var = rel_.lhs();
 }
 
 
@@ -113,293 +81,620 @@ pseries::pseries(const ex &var_, const ex &point_, const epvector &ops_)
  *  Archiving
  */
 
-/** Construct object from archive_node. */
-pseries::pseries(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
+void pseries::read_archive(const archive_node &n, lst &sym_lst) 
 {
-    debugmsg("pseries constructor from archive_node", LOGLEVEL_CONSTRUCT);
-    for (unsigned int i=0; true; i++) {
-        ex rest;
-        ex coeff;
-        if (n.find_ex("coeff", rest, sym_lst, i) && n.find_ex("power", coeff, sym_lst, i))
-            seq.push_back(expair(rest, coeff));
-        else
-            break;
-    }
-    n.find_ex("var", var, sym_lst);
-    n.find_ex("point", point, sym_lst);
+	inherited::read_archive(n, sym_lst);
+	archive_node::archive_node_cit first = n.find_first("coeff");
+	archive_node::archive_node_cit last = n.find_last("power");
+	++last;
+	seq.reserve((last-first)/2);
+
+	for (archive_node::archive_node_cit loc = first; loc < last;) {
+		ex rest;
+		ex coeff;
+		n.find_ex_by_loc(loc++, rest, sym_lst);
+		n.find_ex_by_loc(loc++, coeff, sym_lst);
+		seq.push_back(expair(rest, coeff));
+	}
+
+	n.find_ex("var", var, sym_lst);
+	n.find_ex("point", point, sym_lst);
 }
 
-/** Unarchive the object. */
-ex pseries::unarchive(const archive_node &n, const lst &sym_lst)
+void pseries::archive(archive_node &n) const
 {
-    return (new pseries(n, sym_lst))->setflag(status_flags::dynallocated);
+	inherited::archive(n);
+	epvector::const_iterator i = seq.begin(), iend = seq.end();
+	while (i != iend) {
+		n.add_ex("coeff", i->rest);
+		n.add_ex("power", i->coeff);
+		++i;
+	}
+	n.add_ex("var", var);
+	n.add_ex("point", point);
 }
 
-/** Archive the object. */
-void pseries::archive(archive_node &n) const
+
+//////////
+// functions overriding virtual functions from base classes
+//////////
+
+void pseries::print_series(const print_context & c, const char *openbrace, const char *closebrace, const char *mul_sym, const char *pow_sym, unsigned level) const
 {
-    inherited::archive(n);
-    epvector::const_iterator i = seq.begin(), iend = seq.end();
-    while (i != iend) {
-        n.add_ex("coeff", i->rest);
-        n.add_ex("power", i->coeff);
-        i++;
-    }
-    n.add_ex("var", var);
-    n.add_ex("point", point);
+	if (precedence() <= level)
+		c.s << '(';
+		
+	// objects of type pseries must not have any zero entries, so the
+	// trivial (zero) pseries needs a special treatment here:
+	if (seq.empty())
+		c.s << '0';
+
+	epvector::const_iterator i = seq.begin(), end = seq.end();
+	while (i != end) {
+
+		// print a sign, if needed
+		if (i != seq.begin())
+			c.s << '+';
+
+		if (!is_order_function(i->rest)) {
+
+			// print 'rest', i.e. the expansion coefficient
+			if (i->rest.info(info_flags::numeric) &&
+				i->rest.info(info_flags::positive)) {
+				i->rest.print(c);
+			} else {
+				c.s << openbrace << '(';
+				i->rest.print(c);
+				c.s << ')' << closebrace;
+			}
+
+			// print 'coeff', something like (x-1)^42
+			if (!i->coeff.is_zero()) {
+				c.s << mul_sym;
+				if (!point.is_zero()) {
+					c.s << openbrace << '(';
+					(var-point).print(c);
+					c.s << ')' << closebrace;
+				} else
+					var.print(c);
+				if (i->coeff.compare(_ex1)) {
+					c.s << pow_sym;
+					c.s << openbrace;
+					if (i->coeff.info(info_flags::negative)) {
+						c.s << '(';
+						i->coeff.print(c);
+						c.s << ')';
+					} else
+						i->coeff.print(c);
+					c.s << closebrace;
+				}
+			}
+		} else
+			Order(power(var-point,i->coeff)).print(c);
+		++i;
+	}
+
+	if (precedence() <= level)
+		c.s << ')';
 }
 
+void pseries::do_print(const print_context & c, unsigned level) const
+{
+	print_series(c, "", "", "*", "^", level);
+}
 
-/*
- *  Functions overriding virtual functions from base classes
- */
+void pseries::do_print_latex(const print_latex & c, unsigned level) const
+{
+	print_series(c, "{", "}", " ", "^", level);
+}
 
-basic *pseries::duplicate() const
+void pseries::do_print_python(const print_python & c, unsigned level) const
 {
-    debugmsg("pseries duplicate", LOGLEVEL_DUPLICATE);
-    return new pseries(*this);
+	print_series(c, "", "", "*", "**", level);
 }
 
-void pseries::print(ostream &os, unsigned upper_precedence) const
+void pseries::do_print_tree(const print_tree & c, unsigned level) const
 {
-    debugmsg("pseries print", LOGLEVEL_PRINT);
-    convert_to_poly().print(os, upper_precedence);
+	c.s << std::string(level, ' ') << class_name() << " @" << this
+	    << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
+	    << std::endl;
+	size_t num = seq.size();
+	for (size_t i=0; i<num; ++i) {
+		seq[i].rest.print(c, level + c.delta_indent);
+		seq[i].coeff.print(c, level + c.delta_indent);
+		c.s << std::string(level + c.delta_indent, ' ') << "-----" << std::endl;
+	}
+	var.print(c, level + c.delta_indent);
+	point.print(c, level + c.delta_indent);
+}
+
+void pseries::do_print_python_repr(const print_python_repr & c, unsigned level) const
+{
+	c.s << class_name() << "(relational(";
+	var.print(c);
+	c.s << ',';
+	point.print(c);
+	c.s << "),[";
+	size_t num = seq.size();
+	for (size_t i=0; i<num; ++i) {
+		if (i)
+			c.s << ',';
+		c.s << '(';
+		seq[i].rest.print(c);
+		c.s << ',';
+		seq[i].coeff.print(c);
+		c.s << ')';
+	}
+	c.s << "])";
 }
 
-void pseries::printraw(ostream &os) const
+int pseries::compare_same_type(const basic & other) const
 {
-	debugmsg("pseries printraw", LOGLEVEL_PRINT);
-	os << "pseries(" << var << ";" << point << ";";
-	for (epvector::const_iterator i=seq.begin(); i!=seq.end(); i++) {
-		os << "(" << (*i).rest << "," << (*i).coeff << "),";
+	GINAC_ASSERT(is_a<pseries>(other));
+	const pseries &o = static_cast<const pseries &>(other);
+	
+	// first compare the lengths of the series...
+	if (seq.size()>o.seq.size())
+		return 1;
+	if (seq.size()<o.seq.size())
+		return -1;
+	
+	// ...then the expansion point...
+	int cmpval = var.compare(o.var);
+	if (cmpval)
+		return cmpval;
+	cmpval = point.compare(o.point);
+	if (cmpval)
+		return cmpval;
+	
+	// ...and if that failed the individual elements
+	epvector::const_iterator it = seq.begin(), o_it = o.seq.begin();
+	while (it!=seq.end() && o_it!=o.seq.end()) {
+		cmpval = it->compare(*o_it);
+		if (cmpval)
+			return cmpval;
+		++it;
+		++o_it;
 	}
-	os << ")";
+
+	// so they are equal.
+	return 0;
 }
 
-unsigned pseries::nops(void) const
+/** Return the number of operands including a possible order term. */
+size_t pseries::nops() const
 {
-    return seq.size();
+	return seq.size();
 }
 
-ex pseries::op(int i) const
+/** Return the ith term in the series when represented as a sum. */
+ex pseries::op(size_t i) const
 {
-    if (i < 0 || unsigned(i) >= seq.size())
-        throw (std::out_of_range("op() out of range"));
-    return seq[i].rest * power(var - point, seq[i].coeff);
+	if (i >= seq.size())
+		throw (std::out_of_range("op() out of range"));
+
+	if (is_order_function(seq[i].rest))
+		return Order(power(var-point, seq[i].coeff));
+	return seq[i].rest * power(var - point, seq[i].coeff);
 }
 
-ex &pseries::let_op(int i)
+/** Return degree of highest power of the series.  This is usually the exponent
+ *  of the Order term.  If s is not the expansion variable of the series, the
+ *  series is examined termwise. */
+int pseries::degree(const ex &s) const
 {
-    throw (std::logic_error("let_op not defined for pseries"));
+	if (var.is_equal(s)) {
+		// Return last exponent
+		if (seq.size())
+			return ex_to<numeric>((seq.end()-1)->coeff).to_int();
+		else
+			return 0;
+	} else {
+		epvector::const_iterator it = seq.begin(), itend = seq.end();
+		if (it == itend)
+			return 0;
+		int max_pow = std::numeric_limits<int>::min();
+		while (it != itend) {
+			int pow = it->rest.degree(s);
+			if (pow > max_pow)
+				max_pow = pow;
+			++it;
+		}
+		return max_pow;
+	}
 }
 
-int pseries::degree(const symbol &s) const
+/** Return degree of lowest power of the series.  This is usually the exponent
+ *  of the leading term.  If s is not the expansion variable of the series, the
+ *  series is examined termwise.  If s is the expansion variable but the
+ *  expansion point is not zero the series is not expanded to find the degree.
+ *  I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
+int pseries::ldegree(const ex &s) const
 {
-    if (var.is_equal(s)) {
-        // Return last exponent
-        if (seq.size())
-            return ex_to_numeric((*(seq.end() - 1)).coeff).to_int();
-        else
-            return 0;
-    } else {
-        epvector::const_iterator it = seq.begin(), itend = seq.end();
-        if (it == itend)
-            return 0;
-        int max_pow = INT_MIN;
-        while (it != itend) {
-            int pow = it->rest.degree(s);
-            if (pow > max_pow)
-                max_pow = pow;
-            it++;
-        }
-        return max_pow;
-    }
+	if (var.is_equal(s)) {
+		// Return first exponent
+		if (seq.size())
+			return ex_to<numeric>((seq.begin())->coeff).to_int();
+		else
+			return 0;
+	} else {
+		epvector::const_iterator it = seq.begin(), itend = seq.end();
+		if (it == itend)
+			return 0;
+		int min_pow = std::numeric_limits<int>::max();
+		while (it != itend) {
+			int pow = it->rest.ldegree(s);
+			if (pow < min_pow)
+				min_pow = pow;
+			++it;
+		}
+		return min_pow;
+	}
 }
 
-int pseries::ldegree(const symbol &s) const
+/** Return coefficient of degree n in power series if s is the expansion
+ *  variable.  If the expansion point is nonzero, by definition the n=1
+ *  coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
+ *  the expansion took place in the s in the first place).
+ *  If s is not the expansion variable, an attempt is made to convert the
+ *  series to a polynomial and return the corresponding coefficient from
+ *  there. */
+ex pseries::coeff(const ex &s, int n) const
 {
-    if (var.is_equal(s)) {
-        // Return first exponent
-        if (seq.size())
-            return ex_to_numeric((*(seq.begin())).coeff).to_int();
-        else
-            return 0;
-    } else {
-        epvector::const_iterator it = seq.begin(), itend = seq.end();
-        if (it == itend)
-            return 0;
-        int min_pow = INT_MAX;
-        while (it != itend) {
-            int pow = it->rest.ldegree(s);
-            if (pow < min_pow)
-                min_pow = pow;
-            it++;
-        }
-        return min_pow;
-    }
+	if (var.is_equal(s)) {
+		if (seq.empty())
+			return _ex0;
+		
+		// Binary search in sequence for given power
+		numeric looking_for = numeric(n);
+		int lo = 0, hi = seq.size() - 1;
+		while (lo <= hi) {
+			int mid = (lo + hi) / 2;
+			GINAC_ASSERT(is_exactly_a<numeric>(seq[mid].coeff));
+			int cmp = ex_to<numeric>(seq[mid].coeff).compare(looking_for);
+			switch (cmp) {
+				case -1:
+					lo = mid + 1;
+					break;
+				case 0:
+					return seq[mid].rest;
+				case 1:
+					hi = mid - 1;
+					break;
+				default:
+					throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
+			}
+		}
+		return _ex0;
+	} else
+		return convert_to_poly().coeff(s, n);
 }
 
-ex pseries::coeff(const symbol &s, int n) const
+/** Does nothing. */
+ex pseries::collect(const ex &s, bool distributed) const
 {
-    if (var.is_equal(s)) {
-        epvector::const_iterator it = seq.begin(), itend = seq.end();
-        while (it != itend) {
-            int pow = ex_to_numeric(it->coeff).to_int();
-            if (pow == n)
-                return it->rest;
-            if (pow > n)
-                return _ex0();
-            it++;
-        }
-        return _ex0();
-    } else
-        return convert_to_poly().coeff(s, n);
+	return *this;
 }
 
+/** Perform coefficient-wise automatic term rewriting rules in this class. */
 ex pseries::eval(int level) const
 {
-    if (level == 1)
-        return this->hold();
-    
-    // Construct a new series with evaluated coefficients
-    epvector new_seq;
-    new_seq.reserve(seq.size());
-    epvector::const_iterator it = seq.begin(), itend = seq.end();
-    while (it != itend) {
-        new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
-        it++;
-    }
-    return (new pseries(var, point, new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
-}
-
-/** Evaluate numerically.  The order term is dropped. */
+	if (level == 1)
+		return this->hold();
+	
+	if (level == -max_recursion_level)
+		throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
+	
+	// Construct a new series with evaluated coefficients
+	epvector new_seq;
+	new_seq.reserve(seq.size());
+	epvector::const_iterator it = seq.begin(), itend = seq.end();
+	while (it != itend) {
+		new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
+		++it;
+	}
+	return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
+}
+
+/** Evaluate coefficients numerically. */
 ex pseries::evalf(int level) const
 {
-    return convert_to_poly().evalf(level);
+	if (level == 1)
+		return *this;
+	
+	if (level == -max_recursion_level)
+		throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
+	
+	// Construct a new series with evaluated coefficients
+	epvector new_seq;
+	new_seq.reserve(seq.size());
+	epvector::const_iterator it = seq.begin(), itend = seq.end();
+	while (it != itend) {
+		new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
+		++it;
+	}
+	return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
+}
+
+ex pseries::conjugate() const
+{
+	if(!var.info(info_flags::real))
+		return conjugate_function(*this).hold();
+
+	epvector * newseq = conjugateepvector(seq);
+	ex newpoint = point.conjugate();
+
+	if (!newseq	&& are_ex_trivially_equal(point, newpoint)) {
+		return *this;
+	}
+
+	ex result = (new pseries(var==newpoint, newseq ? *newseq : seq))->setflag(status_flags::dynallocated);
+	if (newseq) {
+		delete newseq;
+	}
+	return result;
+}
+
+ex pseries::real_part() const
+{
+	if(!var.info(info_flags::real))
+		return real_part_function(*this).hold();
+	ex newpoint = point.real_part();
+	if(newpoint != point)
+		return real_part_function(*this).hold();
+
+	epvector v;
+	v.reserve(seq.size());
+	for(epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i)
+		v.push_back(expair((i->rest).real_part(), i->coeff));
+	return (new pseries(var==point, v))->setflag(status_flags::dynallocated);
+}
+
+ex pseries::imag_part() const
+{
+	if(!var.info(info_flags::real))
+		return imag_part_function(*this).hold();
+	ex newpoint = point.real_part();
+	if(newpoint != point)
+		return imag_part_function(*this).hold();
+
+	epvector v;
+	v.reserve(seq.size());
+	for(epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i)
+		v.push_back(expair((i->rest).imag_part(), i->coeff));
+	return (new pseries(var==point, v))->setflag(status_flags::dynallocated);
+}
+
+ex pseries::eval_integ() const
+{
+	epvector *newseq = NULL;
+	for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
+		if (newseq) {
+			newseq->push_back(expair(i->rest.eval_integ(), i->coeff));
+			continue;
+		}
+		ex newterm = i->rest.eval_integ();
+		if (!are_ex_trivially_equal(newterm, i->rest)) {
+			newseq = new epvector;
+			newseq->reserve(seq.size());
+			for (epvector::const_iterator j=seq.begin(); j!=i; ++j)
+				newseq->push_back(*j);
+			newseq->push_back(expair(newterm, i->coeff));
+		}
+	}
+
+	ex newpoint = point.eval_integ();
+	if (newseq || !are_ex_trivially_equal(newpoint, point))
+		return (new pseries(var==newpoint, *newseq))
+		       ->setflag(status_flags::dynallocated);
+	return *this;
+}
+
+ex pseries::evalm() const
+{
+	// evalm each coefficient
+	epvector newseq;
+	bool something_changed = false;
+	for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
+		if (something_changed) {
+			ex newcoeff = i->rest.evalm();
+			if (!newcoeff.is_zero())
+				newseq.push_back(expair(newcoeff, i->coeff));
+		}
+		else {
+			ex newcoeff = i->rest.evalm();
+			if (!are_ex_trivially_equal(newcoeff, i->rest)) {
+				something_changed = true;
+				newseq.reserve(seq.size());
+				std::copy(seq.begin(), i, std::back_inserter<epvector>(newseq));
+				if (!newcoeff.is_zero())
+					newseq.push_back(expair(newcoeff, i->coeff));
+			}
+		}
+	}
+	if (something_changed)
+		return (new pseries(var==point, newseq))->setflag(status_flags::dynallocated);
+	else
+		return *this;
 }
 
-ex pseries::subs(const lst & ls, const lst & lr) const
+ex pseries::subs(const exmap & m, unsigned options) const
 {
 	// If expansion variable is being substituted, convert the series to a
 	// polynomial and do the substitution there because the result might
 	// no longer be a power series
-	if (ls.has(var))
-		return convert_to_poly(true).subs(ls, lr);
-
+	if (m.find(var) != m.end())
+		return convert_to_poly(true).subs(m, options);
+	
 	// Otherwise construct a new series with substituted coefficients and
 	// expansion point
-	epvector new_seq;
-	new_seq.reserve(seq.size());
+	epvector newseq;
+	newseq.reserve(seq.size());
 	epvector::const_iterator it = seq.begin(), itend = seq.end();
 	while (it != itend) {
-		new_seq.push_back(expair(it->rest.subs(ls, lr), it->coeff));
-		it++;
+		newseq.push_back(expair(it->rest.subs(m, options), it->coeff));
+		++it;
+	}
+	return (new pseries(relational(var,point.subs(m, options)), newseq))->setflag(status_flags::dynallocated);
+}
+
+/** Implementation of ex::expand() for a power series.  It expands all the
+ *  terms individually and returns the resulting series as a new pseries. */
+ex pseries::expand(unsigned options) const
+{
+	epvector newseq;
+	epvector::const_iterator i = seq.begin(), end = seq.end();
+	while (i != end) {
+		ex restexp = i->rest.expand();
+		if (!restexp.is_zero())
+			newseq.push_back(expair(restexp, i->coeff));
+		++i;
 	}
-    return (new pseries(var, point.subs(ls, lr), new_seq))->setflag(status_flags::dynallocated);
+	return (new pseries(relational(var,point), newseq))
+	        ->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 }
 
-/** Implementation of ex::diff() for a power series.  It treats the series as a
- *  polynomial.
+/** Implementation of ex::diff() for a power series.
  *  @see ex::diff */
 ex pseries::derivative(const symbol & s) const
 {
-    if (s == var) {
-        epvector new_seq;
-        epvector::const_iterator it = seq.begin(), itend = seq.end();
-        
-        // FIXME: coeff might depend on var
-        while (it != itend) {
-            if (is_order_function(it->rest)) {
-                new_seq.push_back(expair(it->rest, it->coeff - 1));
-            } else {
-                ex c = it->rest * it->coeff;
-                if (!c.is_zero())
-                    new_seq.push_back(expair(c, it->coeff - 1));
-            }
-            it++;
-        }
-        return pseries(var, point, new_seq);
-    } else {
-        return *this;
-    }
-}
+	epvector new_seq;
+	epvector::const_iterator it = seq.begin(), itend = seq.end();
 
+	if (s == var) {
+		
+		// FIXME: coeff might depend on var
+		while (it != itend) {
+			if (is_order_function(it->rest)) {
+				new_seq.push_back(expair(it->rest, it->coeff - 1));
+			} else {
+				ex c = it->rest * it->coeff;
+				if (!c.is_zero())
+					new_seq.push_back(expair(c, it->coeff - 1));
+			}
+			++it;
+		}
+
+	} else {
+
+		while (it != itend) {
+			if (is_order_function(it->rest)) {
+				new_seq.push_back(*it);
+			} else {
+				ex c = it->rest.diff(s);
+				if (!c.is_zero())
+					new_seq.push_back(expair(c, it->coeff));
+			}
+			++it;
+		}
+	}
 
-/*
- *  Construct ordinary polynomial out of series
- */
+	return pseries(relational(var,point), new_seq);
+}
 
-/** Convert a pseries object to an ordinary polynomial.
- *
- *  @param no_order flag: discard higher order terms */
 ex pseries::convert_to_poly(bool no_order) const
 {
-    ex e;
-    epvector::const_iterator it = seq.begin(), itend = seq.end();
-    
-    while (it != itend) {
-        if (is_order_function(it->rest)) {
-            if (!no_order)
-                e += Order(power(var - point, it->coeff));
-        } else
-            e += it->rest * power(var - point, it->coeff);
-        it++;
-    }
-    return e;
+	ex e;
+	epvector::const_iterator it = seq.begin(), itend = seq.end();
+	
+	while (it != itend) {
+		if (is_order_function(it->rest)) {
+			if (!no_order)
+				e += Order(power(var - point, it->coeff));
+		} else
+			e += it->rest * power(var - point, it->coeff);
+		++it;
+	}
+	return e;
+}
+
+bool pseries::is_terminating() const
+{
+	return seq.empty() || !is_order_function((seq.end()-1)->rest);
+}
+
+ex pseries::coeffop(size_t i) const
+{
+	if (i >=nops())
+		throw (std::out_of_range("coeffop() out of range"));
+	return seq[i].rest;
+}
+
+ex pseries::exponop(size_t i) const
+{
+	if (i >= nops())
+		throw (std::out_of_range("exponop() out of range"));
+	return seq[i].coeff;
 }
 
 
 /*
- *  Implementation of series expansion
+ *  Implementations of series expansion
  */
 
 /** Default implementation of ex::series(). This performs Taylor expansion.
  *  @see ex::series */
-ex basic::series(const symbol & s, const ex & point, int order) const
-{
-    epvector seq;
-    numeric fac(1);
-    ex deriv = *this;
-    ex coeff = deriv.subs(s == point);
-    if (!coeff.is_zero())
-        seq.push_back(expair(coeff, numeric(0)));
-    
-    int n;
-    for (n=1; n<order; n++) {
-        fac = fac.mul(numeric(n));
-        deriv = deriv.diff(s).expand();
-        if (deriv.is_zero()) {
-            // Series terminates
-            return pseries(s, point, seq);
-        }
-        coeff = fac.inverse() * deriv.subs(s == point);
-        if (!coeff.is_zero())
-            seq.push_back(expair(coeff, numeric(n)));
-    }
-    
-    // Higher-order terms, if present
-    deriv = deriv.diff(s);
-    if (!deriv.is_zero())
-        seq.push_back(expair(Order(_ex1()), numeric(n)));
-    return pseries(s, point, seq);
+ex basic::series(const relational & r, int order, unsigned options) const
+{
+	epvector seq;
+	const symbol &s = ex_to<symbol>(r.lhs());
+
+	// default for order-values that make no sense for Taylor expansion
+	if ((order <= 0) && this->has(s)) {
+		seq.push_back(expair(Order(_ex1), order));
+		return pseries(r, seq);
+	}
+
+	// do Taylor expansion
+	numeric fac = 1;
+	ex deriv = *this;
+	ex coeff = deriv.subs(r, subs_options::no_pattern);
+
+	if (!coeff.is_zero()) {
+		seq.push_back(expair(coeff, _ex0));
+	}
+
+	int n;
+	for (n=1; n<order; ++n) {
+		fac = fac.mul(n);
+		// We need to test for zero in order to see if the series terminates.
+		// The problem is that there is no such thing as a perfect test for
+		// zero.  Expanding the term occasionally helps a little...
+		deriv = deriv.diff(s).expand();
+		if (deriv.is_zero())  // Series terminates
+			return pseries(r, seq);
+
+		coeff = deriv.subs(r, subs_options::no_pattern);
+		if (!coeff.is_zero())
+			seq.push_back(expair(fac.inverse() * coeff, n));
+	}
+	
+	// Higher-order terms, if present
+	deriv = deriv.diff(s);
+	if (!deriv.expand().is_zero())
+		seq.push_back(expair(Order(_ex1), n));
+	return pseries(r, seq);
 }
 
 
 /** Implementation of ex::series() for symbols.
  *  @see ex::series */
-ex symbol::series(const symbol & s, const ex & point, int order) const
+ex symbol::series(const relational & r, int order, unsigned options) const
 {
 	epvector seq;
-	if (is_equal(s)) {
+	const ex point = r.rhs();
+	GINAC_ASSERT(is_a<symbol>(r.lhs()));
+
+	if (this->is_equal_same_type(ex_to<symbol>(r.lhs()))) {
 		if (order > 0 && !point.is_zero())
-			seq.push_back(expair(point, _ex0()));
+			seq.push_back(expair(point, _ex0));
 		if (order > 1)
-			seq.push_back(expair(_ex1(), _ex1()));
+			seq.push_back(expair(_ex1, _ex1));
 		else
-			seq.push_back(expair(Order(_ex1()), numeric(order)));
+			seq.push_back(expair(Order(_ex1), numeric(order)));
 	} else
-		seq.push_back(expair(*this, _ex0()));
-	return pseries(s, point, seq);
+		seq.push_back(expair(*this, _ex0));
+	return pseries(r, seq);
 }
 
 
@@ -410,99 +705,99 @@ ex symbol::series(const symbol & s, const ex & point, int order) const
  *  @return the sum as a pseries */
 ex pseries::add_series(const pseries &other) const
 {
-    // Adding two series with different variables or expansion points
-    // results in an empty (constant) series 
-    if (!is_compatible_to(other)) {
-        epvector nul;
-        nul.push_back(expair(Order(_ex1()), _ex0()));
-        return pseries(var, point, nul);
-    }
-    
-    // Series addition
-    epvector new_seq;
-    epvector::const_iterator a = seq.begin();
-    epvector::const_iterator b = other.seq.begin();
-    epvector::const_iterator a_end = seq.end();
-    epvector::const_iterator b_end = other.seq.end();
-    int pow_a = INT_MAX, pow_b = INT_MAX;
-    for (;;) {
-        // If a is empty, fill up with elements from b and stop
-        if (a == a_end) {
-            while (b != b_end) {
-                new_seq.push_back(*b);
-                b++;
-            }
-            break;
-        } else
-            pow_a = ex_to_numeric((*a).coeff).to_int();
-        
-        // If b is empty, fill up with elements from a and stop
-        if (b == b_end) {
-            while (a != a_end) {
-                new_seq.push_back(*a);
-                a++;
-            }
-            break;
-        } else
-            pow_b = ex_to_numeric((*b).coeff).to_int();
-        
-        // a and b are non-empty, compare powers
-        if (pow_a < pow_b) {
-            // a has lesser power, get coefficient from a
-            new_seq.push_back(*a);
-            if (is_order_function((*a).rest))
-                break;
-            a++;
-        } else if (pow_b < pow_a) {
-            // b has lesser power, get coefficient from b
-            new_seq.push_back(*b);
-            if (is_order_function((*b).rest))
-                break;
-            b++;
-        } else {
-            // Add coefficient of a and b
-            if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
-                new_seq.push_back(expair(Order(_ex1()), (*a).coeff));
-                break;  // Order term ends the sequence
-            } else {
-                ex sum = (*a).rest + (*b).rest;
-                if (!(sum.is_zero()))
-                    new_seq.push_back(expair(sum, numeric(pow_a)));
-                a++;
-                b++;
-            }
-        }
-    }
-    return pseries(var, point, new_seq);
+	// Adding two series with different variables or expansion points
+	// results in an empty (constant) series 
+	if (!is_compatible_to(other)) {
+		epvector nul;
+		nul.push_back(expair(Order(_ex1), _ex0));
+		return pseries(relational(var,point), nul);
+	}
+	
+	// Series addition
+	epvector new_seq;
+	epvector::const_iterator a = seq.begin();
+	epvector::const_iterator b = other.seq.begin();
+	epvector::const_iterator a_end = seq.end();
+	epvector::const_iterator b_end = other.seq.end();
+	int pow_a = std::numeric_limits<int>::max(), pow_b = std::numeric_limits<int>::max();
+	for (;;) {
+		// If a is empty, fill up with elements from b and stop
+		if (a == a_end) {
+			while (b != b_end) {
+				new_seq.push_back(*b);
+				++b;
+			}
+			break;
+		} else
+			pow_a = ex_to<numeric>((*a).coeff).to_int();
+		
+		// If b is empty, fill up with elements from a and stop
+		if (b == b_end) {
+			while (a != a_end) {
+				new_seq.push_back(*a);
+				++a;
+			}
+			break;
+		} else
+			pow_b = ex_to<numeric>((*b).coeff).to_int();
+		
+		// a and b are non-empty, compare powers
+		if (pow_a < pow_b) {
+			// a has lesser power, get coefficient from a
+			new_seq.push_back(*a);
+			if (is_order_function((*a).rest))
+				break;
+			++a;
+		} else if (pow_b < pow_a) {
+			// b has lesser power, get coefficient from b
+			new_seq.push_back(*b);
+			if (is_order_function((*b).rest))
+				break;
+			++b;
+		} else {
+			// Add coefficient of a and b
+			if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
+				new_seq.push_back(expair(Order(_ex1), (*a).coeff));
+				break;  // Order term ends the sequence
+			} else {
+				ex sum = (*a).rest + (*b).rest;
+				if (!(sum.is_zero()))
+					new_seq.push_back(expair(sum, numeric(pow_a)));
+				++a;
+				++b;
+			}
+		}
+	}
+	return pseries(relational(var,point), new_seq);
 }
 
 
 /** Implementation of ex::series() for sums. This performs series addition when
  *  adding pseries objects.
  *  @see ex::series */
-ex add::series(const symbol & s, const ex & point, int order) const
-{
-    ex acc; // Series accumulator
-    
-    // Get first term from overall_coeff
-    acc = overall_coeff.series(s, point, order);
-
-    // Add remaining terms
-    epvector::const_iterator it = seq.begin();
-    epvector::const_iterator itend = seq.end();
-    for (; it!=itend; it++) {
-        ex op;
-        if (is_ex_exactly_of_type(it->rest, pseries))
-            op = it->rest;
-        else
-            op = it->rest.series(s, point, order);
-        if (!it->coeff.is_equal(_ex1()))
-            op = ex_to_pseries(op).mul_const(ex_to_numeric(it->coeff));
-        
-        // Series addition
-        acc = ex_to_pseries(acc).add_series(ex_to_pseries(op));
-    }
-    return acc;
+ex add::series(const relational & r, int order, unsigned options) const
+{
+	ex acc; // Series accumulator
+	
+	// Get first term from overall_coeff
+	acc = overall_coeff.series(r, order, options);
+	
+	// Add remaining terms
+	epvector::const_iterator it = seq.begin();
+	epvector::const_iterator itend = seq.end();
+	for (; it!=itend; ++it) {
+		ex op;
+		if (is_exactly_a<pseries>(it->rest))
+			op = it->rest;
+		else
+			op = it->rest.series(r, order, options);
+		if (!it->coeff.is_equal(_ex1))
+			op = ex_to<pseries>(op).mul_const(ex_to<numeric>(it->coeff));
+		
+		// Series addition
+		acc = ex_to<pseries>(acc).add_series(ex_to<pseries>(op));
+	}
+	return acc;
 }
 
 
@@ -513,18 +808,18 @@ ex add::series(const symbol & s, const ex & point, int order) const
  *  @return the product as a pseries */
 ex pseries::mul_const(const numeric &other) const
 {
-    epvector new_seq;
-    new_seq.reserve(seq.size());
-    
-    epvector::const_iterator it = seq.begin(), itend = seq.end();
-    while (it != itend) {
-        if (!is_order_function(it->rest))
-            new_seq.push_back(expair(it->rest * other, it->coeff));
-        else
-            new_seq.push_back(*it);
-        it++;
-    }
-    return pseries(var, point, new_seq);
+	epvector new_seq;
+	new_seq.reserve(seq.size());
+	
+	epvector::const_iterator it = seq.begin(), itend = seq.end();
+	while (it != itend) {
+		if (!is_order_function(it->rest))
+			new_seq.push_back(expair(it->rest * other, it->coeff));
+		else
+			new_seq.push_back(*it);
+		++it;
+	}
+	return pseries(relational(var,point), new_seq);
 }
 
 
@@ -535,82 +830,168 @@ ex pseries::mul_const(const numeric &other) const
  *  @return the product as a pseries */
 ex pseries::mul_series(const pseries &other) const
 {
-    // Multiplying two series with different variables or expansion points
-    // results in an empty (constant) series 
-    if (!is_compatible_to(other)) {
-        epvector nul;
-        nul.push_back(expair(Order(_ex1()), _ex0()));
-        return pseries(var, point, nul);
-    }
-
-    // Series multiplication
-    epvector new_seq;
-    
-    const symbol *s = static_cast<symbol *>(var.bp);
-    int a_max = degree(*s);
-    int b_max = other.degree(*s);
-    int a_min = ldegree(*s);
-    int b_min = other.ldegree(*s);
-    int cdeg_min = a_min + b_min;
-    int cdeg_max = a_max + b_max;
-    
-    int higher_order_a = INT_MAX;
-    int higher_order_b = INT_MAX;
-    if (is_order_function(coeff(*s, a_max)))
-        higher_order_a = a_max + b_min;
-    if (is_order_function(other.coeff(*s, b_max)))
-        higher_order_b = b_max + a_min;
-    int higher_order_c = min(higher_order_a, higher_order_b);
-    if (cdeg_max >= higher_order_c)
-        cdeg_max = higher_order_c - 1;
-    
-    for (int cdeg=cdeg_min; cdeg<=cdeg_max; cdeg++) {
-        ex co = _ex0();
-        // c(i)=a(0)b(i)+...+a(i)b(0)
-        for (int i=a_min; cdeg-i>=b_min; i++) {
-            ex a_coeff = coeff(*s, i);
-            ex b_coeff = other.coeff(*s, cdeg-i);
-            if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
-                co += coeff(*s, i) * other.coeff(*s, cdeg-i);
-        }
-        if (!co.is_zero())
-            new_seq.push_back(expair(co, numeric(cdeg)));
-    }
-    if (higher_order_c < INT_MAX)
-        new_seq.push_back(expair(Order(_ex1()), numeric(higher_order_c)));
-    return pseries(var, point, new_seq);
+	// Multiplying two series with different variables or expansion points
+	// results in an empty (constant) series 
+	if (!is_compatible_to(other)) {
+		epvector nul;
+		nul.push_back(expair(Order(_ex1), _ex0));
+		return pseries(relational(var,point), nul);
+	}
+
+	if (seq.empty() || other.seq.empty()) {
+		return (new pseries(var==point, epvector()))
+		       ->setflag(status_flags::dynallocated);
+	}
+	
+	// Series multiplication
+	epvector new_seq;
+	int a_max = degree(var);
+	int b_max = other.degree(var);
+	int a_min = ldegree(var);
+	int b_min = other.ldegree(var);
+	int cdeg_min = a_min + b_min;
+	int cdeg_max = a_max + b_max;
+	
+	int higher_order_a = std::numeric_limits<int>::max();
+	int higher_order_b = std::numeric_limits<int>::max();
+	if (is_order_function(coeff(var, a_max)))
+		higher_order_a = a_max + b_min;
+	if (is_order_function(other.coeff(var, b_max)))
+		higher_order_b = b_max + a_min;
+	int higher_order_c = std::min(higher_order_a, higher_order_b);
+	if (cdeg_max >= higher_order_c)
+		cdeg_max = higher_order_c - 1;
+	
+	for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
+		ex co = _ex0;
+		// c(i)=a(0)b(i)+...+a(i)b(0)
+		for (int i=a_min; cdeg-i>=b_min; ++i) {
+			ex a_coeff = coeff(var, i);
+			ex b_coeff = other.coeff(var, cdeg-i);
+			if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
+				co += a_coeff * b_coeff;
+		}
+		if (!co.is_zero())
+			new_seq.push_back(expair(co, numeric(cdeg)));
+	}
+	if (higher_order_c < std::numeric_limits<int>::max())
+		new_seq.push_back(expair(Order(_ex1), numeric(higher_order_c)));
+	return pseries(relational(var, point), new_seq);
 }
 
 
 /** Implementation of ex::series() for product. This performs series
  *  multiplication when multiplying series.
  *  @see ex::series */
-ex mul::series(const symbol & s, const ex & point, int order) const
-{
-    ex acc; // Series accumulator
-    
-    // Get first term from overall_coeff
-    acc = overall_coeff.series(s, point, order);
-    
-    // Multiply with remaining terms
-    epvector::const_iterator it = seq.begin();
-    epvector::const_iterator itend = seq.end();
-    for (; it!=itend; it++) {
-        ex op = it->rest;
-        if (op.info(info_flags::numeric)) {
-            // series * const (special case, faster)
-            ex f = power(op, it->coeff);
-            acc = ex_to_pseries(acc).mul_const(ex_to_numeric(f));
-            continue;
-        } else if (!is_ex_exactly_of_type(op, pseries))
-            op = op.series(s, point, order);
-        if (!it->coeff.is_equal(_ex1()))
-            op = ex_to_pseries(op).power_const(ex_to_numeric(it->coeff), order);
-
-        // Series multiplication
-        acc = ex_to_pseries(acc).mul_series(ex_to_pseries(op));
-    }
-    return acc;
+ex mul::series(const relational & r, int order, unsigned options) const
+{
+	pseries acc; // Series accumulator
+
+	GINAC_ASSERT(is_a<symbol>(r.lhs()));
+	const ex& sym = r.lhs();
+		
+	// holds ldegrees of the series of individual factors
+	std::vector<int> ldegrees;
+	std::vector<bool> ldegree_redo;
+
+	// find minimal degrees
+	const epvector::const_iterator itbeg = seq.begin();
+	const epvector::const_iterator itend = seq.end();
+	// first round: obtain a bound up to which minimal degrees have to be
+	// considered
+	for (epvector::const_iterator it=itbeg; it!=itend; ++it) {
+
+		ex expon = it->coeff;
+		int factor = 1;
+		ex buf;
+		if (expon.info(info_flags::integer)) {
+			buf = it->rest;
+			factor = ex_to<numeric>(expon).to_int();
+		} else {
+			buf = recombine_pair_to_ex(*it);
+		}
+
+		int real_ldegree = 0;
+		bool flag_redo = false;
+		try {
+			real_ldegree = buf.expand().ldegree(sym-r.rhs());
+		} catch (std::runtime_error) {}
+
+		if (real_ldegree == 0) {
+			if ( factor < 0 ) {
+				// This case must terminate, otherwise we would have division by
+				// zero.
+				int orderloop = 0;
+				do {
+					orderloop++;
+					real_ldegree = buf.series(r, orderloop, options).ldegree(sym);
+				} while (real_ldegree == orderloop);
+			} else {
+				// Here it is possible that buf does not have a ldegree, therefore
+				// check only if ldegree is negative, otherwise reconsider the case
+				// in the second round.
+				real_ldegree = buf.series(r, 0, options).ldegree(sym);
+				if (real_ldegree == 0)
+					flag_redo = true;
+			}
+		}
+
+		ldegrees.push_back(factor * real_ldegree);
+		ldegree_redo.push_back(flag_redo);
+	}
+
+	int degbound = order-std::accumulate(ldegrees.begin(), ldegrees.end(), 0);
+	// Second round: determine the remaining positive ldegrees by the series
+	// method.
+	// here we can ignore ldegrees larger than degbound
+	size_t j = 0;
+	for (epvector::const_iterator it=itbeg; it!=itend; ++it) {
+		if ( ldegree_redo[j] ) {
+			ex expon = it->coeff;
+			int factor = 1;
+			ex buf;
+			if (expon.info(info_flags::integer)) {
+				buf = it->rest;
+				factor = ex_to<numeric>(expon).to_int();
+			} else {
+				buf = recombine_pair_to_ex(*it);
+			}
+			int real_ldegree = 0;
+			int orderloop = 0;
+			do {
+				orderloop++;
+				real_ldegree = buf.series(r, orderloop, options).ldegree(sym);
+			} while ((real_ldegree == orderloop)
+					&& ( factor*real_ldegree < degbound));
+			ldegrees[j] = factor * real_ldegree;
+			degbound -= factor * real_ldegree;
+		}
+		j++;
+	}
+
+	int degsum = std::accumulate(ldegrees.begin(), ldegrees.end(), 0);
+
+	if (degsum >= order) {
+		epvector epv;
+		epv.push_back(expair(Order(_ex1), order));
+		return (new pseries(r, epv))->setflag(status_flags::dynallocated);
+	}
+
+	// Multiply with remaining terms
+	std::vector<int>::const_iterator itd = ldegrees.begin();
+	for (epvector::const_iterator it=itbeg; it!=itend; ++it, ++itd) {
+
+		// do series expansion with adjusted order
+		ex op = recombine_pair_to_ex(*it).series(r, order-degsum+(*itd), options);
+
+		// Series multiplication
+		if (it == itbeg)
+			acc = ex_to<pseries>(op);
+		else
+			acc = ex_to<pseries>(acc.mul_series(ex_to<pseries>(op)));
+	}
+
+	return acc.mul_const(ex_to<numeric>(overall_coeff));
 }
 
 
@@ -620,95 +1001,312 @@ ex mul::series(const symbol & s, const ex & point, int order) const
  *  @param deg  truncation order of series calculation */
 ex pseries::power_const(const numeric &p, int deg) const
 {
-    int i;
-    const symbol *s = static_cast<symbol *>(var.bp);
-    int ldeg = ldegree(*s);
-    
-    // Calculate coefficients of powered series
-    exvector co;
-    co.reserve(deg);
-    ex co0;
-    co.push_back(co0 = power(coeff(*s, ldeg), p));
-    bool all_sums_zero = true;
-    for (i=1; i<deg; i++) {
-        ex sum = _ex0();
-        for (int j=1; j<=i; j++) {
-            ex c = coeff(*s, j + ldeg);
-            if (is_order_function(c)) {
-                co.push_back(Order(_ex1()));
-                break;
-            } else
-                sum += (p * j - (i - j)) * co[i - j] * c;
-        }
-        if (!sum.is_zero())
-            all_sums_zero = false;
-        co.push_back(co0 * sum / numeric(i));
-    }
-    
-    // Construct new series (of non-zero coefficients)
-    epvector new_seq;
-    bool higher_order = false;
-    for (i=0; i<deg; i++) {
-        if (!co[i].is_zero())
-            new_seq.push_back(expair(co[i], numeric(i) + p * ldeg));
-        if (is_order_function(co[i])) {
-            higher_order = true;
-            break;
-        }
-    }
-    if (!higher_order && !all_sums_zero)
-        new_seq.push_back(expair(Order(_ex1()), numeric(deg) + p * ldeg));
-    return pseries(var, point, new_seq);
+	// method:
+	// (due to Leonhard Euler)
+	// let A(x) be this series and for the time being let it start with a
+	// constant (later we'll generalize):
+	//     A(x) = a_0 + a_1*x + a_2*x^2 + ...
+	// We want to compute
+	//     C(x) = A(x)^p
+	//     C(x) = c_0 + c_1*x + c_2*x^2 + ...
+	// Taking the derivative on both sides and multiplying with A(x) one
+	// immediately arrives at
+	//     C'(x)*A(x) = p*C(x)*A'(x)
+	// Multiplying this out and comparing coefficients we get the recurrence
+	// formula
+	//     c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
+	//                    ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
+	// which can easily be solved given the starting value c_0 = (a_0)^p.
+	// For the more general case where the leading coefficient of A(x) is not
+	// a constant, just consider A2(x) = A(x)*x^m, with some integer m and
+	// repeat the above derivation.  The leading power of C2(x) = A2(x)^2 is
+	// then of course x^(p*m) but the recurrence formula still holds.
+	
+	if (seq.empty()) {
+		// as a special case, handle the empty (zero) series honoring the
+		// usual power laws such as implemented in power::eval()
+		if (p.real().is_zero())
+			throw std::domain_error("pseries::power_const(): pow(0,I) is undefined");
+		else if (p.real().is_negative())
+			throw pole_error("pseries::power_const(): division by zero",1);
+		else
+			return *this;
+	}
+	
+	const int ldeg = ldegree(var);
+	if (!(p*ldeg).is_integer())
+		throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
+
+	// adjust number of coefficients
+	int numcoeff = deg - (p*ldeg).to_int();
+	if (numcoeff <= 0) {
+		epvector epv;
+		epv.reserve(1);
+		epv.push_back(expair(Order(_ex1), deg));
+		return (new pseries(relational(var,point), epv))
+		       ->setflag(status_flags::dynallocated);
+	}
+	
+	// O(x^n)^(-m) is undefined
+	if (seq.size() == 1 && is_order_function(seq[0].rest) && p.real().is_negative())
+		throw pole_error("pseries::power_const(): division by zero",1);
+	
+	// Compute coefficients of the powered series
+	exvector co;
+	co.reserve(numcoeff);
+	co.push_back(power(coeff(var, ldeg), p));
+	for (int i=1; i<numcoeff; ++i) {
+		ex sum = _ex0;
+		for (int j=1; j<=i; ++j) {
+			ex c = coeff(var, j + ldeg);
+			if (is_order_function(c)) {
+				co.push_back(Order(_ex1));
+				break;
+			} else
+				sum += (p * j - (i - j)) * co[i - j] * c;
+		}
+		co.push_back(sum / coeff(var, ldeg) / i);
+	}
+	
+	// Construct new series (of non-zero coefficients)
+	epvector new_seq;
+	bool higher_order = false;
+	for (int i=0; i<numcoeff; ++i) {
+		if (!co[i].is_zero())
+			new_seq.push_back(expair(co[i], p * ldeg + i));
+		if (is_order_function(co[i])) {
+			higher_order = true;
+			break;
+		}
+	}
+	if (!higher_order)
+		new_seq.push_back(expair(Order(_ex1), p * ldeg + numcoeff));
+
+	return pseries(relational(var,point), new_seq);
+}
+
+
+/** Return a new pseries object with the powers shifted by deg. */
+pseries pseries::shift_exponents(int deg) const
+{
+	epvector newseq = seq;
+	epvector::iterator i = newseq.begin(), end  = newseq.end();
+	while (i != end) {
+		i->coeff += deg;
+		++i;
+	}
+	return pseries(relational(var, point), newseq);
 }
 
 
 /** Implementation of ex::series() for powers. This performs Laurent expansion
  *  of reciprocals of series at singularities.
  *  @see ex::series */
-ex power::series(const symbol & s, const ex & point, int order) const
-{
-    ex e;
-    if (!is_ex_exactly_of_type(basis, pseries)) {
-        // Basis is not a series, may there be a singulary?
-        if (!exponent.info(info_flags::negint))
-            return basic::series(s, point, order);
-        
-        // Expression is of type something^(-int), check for singularity
-        if (!basis.subs(s == point).is_zero())
-            return basic::series(s, point, order);
-        
-        // Singularity encountered, expand basis into series
-        e = basis.series(s, point, order);
-    } else {
-        // Basis is a series
-        e = basis;
-    }
-    
-    // Power e
-    return ex_to_pseries(e).power_const(ex_to_numeric(exponent), order);
+ex power::series(const relational & r, int order, unsigned options) const
+{
+	// If basis is already a series, just power it
+	if (is_exactly_a<pseries>(basis))
+		return ex_to<pseries>(basis).power_const(ex_to<numeric>(exponent), order);
+
+	// Basis is not a series, may there be a singularity?
+	bool must_expand_basis = false;
+	try {
+		basis.subs(r, subs_options::no_pattern);
+	} catch (pole_error) {
+		must_expand_basis = true;
+	}
+
+	bool exponent_is_regular = true;
+	try {
+		exponent.subs(r, subs_options::no_pattern);
+	} catch (pole_error) {
+		exponent_is_regular = false;
+	}
+
+	if (!exponent_is_regular) {
+		ex l = exponent*log(basis);
+		// this == exp(l);
+		ex le = l.series(r, order, options);
+		// Note: expanding exp(l) won't help, since that will attempt
+		// Taylor expansion, and fail (because exponent is "singular")
+		// Still l itself might be expanded in Taylor series.
+		// Examples:
+		// sin(x)/x*log(cos(x))
+		// 1/x*log(1 + x)
+		return exp(le).series(r, order, options);
+		// Note: if l happens to have a Laurent expansion (with
+		// negative powers of (var - point)), expanding exp(le)
+		// will barf (which is The Right Thing).
+	}
+
+	// Is the expression of type something^(-int)?
+	if (!must_expand_basis && !exponent.info(info_flags::negint)
+	 && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
+		return basic::series(r, order, options);
+
+	// Is the expression of type 0^something?
+	if (!must_expand_basis && !basis.subs(r, subs_options::no_pattern).is_zero()
+	 && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
+		return basic::series(r, order, options);
+
+	// Singularity encountered, is the basis equal to (var - point)?
+	if (basis.is_equal(r.lhs() - r.rhs())) {
+		epvector new_seq;
+		if (ex_to<numeric>(exponent).to_int() < order)
+			new_seq.push_back(expair(_ex1, exponent));
+		else
+			new_seq.push_back(expair(Order(_ex1), exponent));
+		return pseries(r, new_seq);
+	}
+
+	// No, expand basis into series
+
+	numeric numexp;
+	if (is_a<numeric>(exponent)) {
+		numexp = ex_to<numeric>(exponent);
+	} else {
+		numexp = 0;
+	}
+	const ex& sym = r.lhs();
+	// find existing minimal degree
+	ex eb = basis.expand();
+	int real_ldegree = 0;
+	if (eb.info(info_flags::rational_function))
+		real_ldegree = eb.ldegree(sym-r.rhs());
+	if (real_ldegree == 0) {
+		int orderloop = 0;
+		do {
+			orderloop++;
+			real_ldegree = basis.series(r, orderloop, options).ldegree(sym);
+		} while (real_ldegree == orderloop);
+	}
+
+	if (!(real_ldegree*numexp).is_integer())
+		throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
+	ex e = basis.series(r, (order + real_ldegree*(1-numexp)).to_int(), options);
+	
+	ex result;
+	try {
+		result = ex_to<pseries>(e).power_const(numexp, order);
+	} catch (pole_error) {
+		epvector ser;
+		ser.push_back(expair(Order(_ex1), order));
+		result = pseries(r, ser);
+	}
+
+	return result;
+}
+
+
+/** Re-expansion of a pseries object. */
+ex pseries::series(const relational & r, int order, unsigned options) const
+{
+	const ex p = r.rhs();
+	GINAC_ASSERT(is_a<symbol>(r.lhs()));
+	const symbol &s = ex_to<symbol>(r.lhs());
+	
+	if (var.is_equal(s) && point.is_equal(p)) {
+		if (order > degree(s))
+			return *this;
+		else {
+			epvector new_seq;
+			epvector::const_iterator it = seq.begin(), itend = seq.end();
+			while (it != itend) {
+				int o = ex_to<numeric>(it->coeff).to_int();
+				if (o >= order) {
+					new_seq.push_back(expair(Order(_ex1), o));
+					break;
+				}
+				new_seq.push_back(*it);
+				++it;
+			}
+			return pseries(r, new_seq);
+		}
+	} else
+		return convert_to_poly().series(r, order, options);
+}
+
+ex integral::series(const relational & r, int order, unsigned options) const
+{
+	if (x.subs(r) != x)
+		throw std::logic_error("Cannot series expand wrt dummy variable");
+	
+	// Expanding integrant with r substituted taken in boundaries.
+	ex fseries = f.series(r, order, options);
+	epvector fexpansion;
+	fexpansion.reserve(fseries.nops());
+	for (size_t i=0; i<fseries.nops(); ++i) {
+		ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
+		currcoeff = (currcoeff == Order(_ex1))
+			? currcoeff
+			: integral(x, a.subs(r), b.subs(r), currcoeff);
+		if (currcoeff != 0)
+			fexpansion.push_back(
+				expair(currcoeff, ex_to<pseries>(fseries).exponop(i)));
+	}
+
+	// Expanding lower boundary
+	ex result = (new pseries(r, fexpansion))->setflag(status_flags::dynallocated);
+	ex aseries = (a-a.subs(r)).series(r, order, options);
+	fseries = f.series(x == (a.subs(r)), order, options);
+	for (size_t i=0; i<fseries.nops(); ++i) {
+		ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
+		if (is_order_function(currcoeff))
+			break;
+		ex currexpon = ex_to<pseries>(fseries).exponop(i);
+		int orderforf = order-ex_to<numeric>(currexpon).to_int()-1;
+		currcoeff = currcoeff.series(r, orderforf);
+		ex term = ex_to<pseries>(aseries).power_const(ex_to<numeric>(currexpon+1),order);
+		term = ex_to<pseries>(term).mul_const(ex_to<numeric>(-1/(currexpon+1)));
+		term = ex_to<pseries>(term).mul_series(ex_to<pseries>(currcoeff));
+		result = ex_to<pseries>(result).add_series(ex_to<pseries>(term));
+	}
+
+	// Expanding upper boundary
+	ex bseries = (b-b.subs(r)).series(r, order, options);
+	fseries = f.series(x == (b.subs(r)), order, options);
+	for (size_t i=0; i<fseries.nops(); ++i) {
+		ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
+		if (is_order_function(currcoeff))
+			break;
+		ex currexpon = ex_to<pseries>(fseries).exponop(i);
+		int orderforf = order-ex_to<numeric>(currexpon).to_int()-1;
+		currcoeff = currcoeff.series(r, orderforf);
+		ex term = ex_to<pseries>(bseries).power_const(ex_to<numeric>(currexpon+1),order);
+		term = ex_to<pseries>(term).mul_const(ex_to<numeric>(1/(currexpon+1)));
+		term = ex_to<pseries>(term).mul_series(ex_to<pseries>(currcoeff));
+		result = ex_to<pseries>(result).add_series(ex_to<pseries>(term));
+	}
+
+	return result;
 }
 
 
 /** Compute the truncated series expansion of an expression.
- *  This function returns an expression containing an object of class pseries to
- *  represent the series. If the series does not terminate within the given
+ *  This function returns an expression containing an object of class pseries 
+ *  to represent the series. If the series does not terminate within the given
  *  truncation order, the last term of the series will be an order term.
  *
- *  @param s  expansion variable
- *  @param point  expansion point
+ *  @param r  expansion relation, lhs holds variable and rhs holds point
  *  @param order  truncation order of series calculations
+ *  @param options  of class series_options
  *  @return an expression holding a pseries object */
-ex ex::series(const symbol &s, const ex &point, int order) const
+ex ex::series(const ex & r, int order, unsigned options) const
 {
-    GINAC_ASSERT(bp!=0);
-    return bp->series(s, point, order);
+	ex e;
+	relational rel_;
+	
+	if (is_a<relational>(r))
+		rel_ = ex_to<relational>(r);
+	else if (is_a<symbol>(r))
+		rel_ = relational(r,_ex0);
+	else
+		throw (std::logic_error("ex::series(): expansion point has unknown type"));
+	
+	e = bp->series(rel_, order, options);
+	return e;
 }
 
+GINAC_BIND_UNARCHIVER(pseries);
 
-// Global constants
-const pseries some_pseries;
-const type_info & typeid_pseries = typeid(some_pseries);
-
-#ifndef NO_NAMESPACE_GINAC
 } // namespace GiNaC
-#endif // ndef NO_NAMESPACE_GINAC