power::series(): handle someg (trivial) singularities of the exponent...
[ginac.git] / ginac / pseries.cpp
index 7fbf73113a74f293c3cca3b151083ff29dfa2bb9..c290fe0a1f7b09b0c73fa2cebdf4e9cb1f609c38 100644 (file)
@@ -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
  *
  *  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