power::series(): handle someg (trivial) singularities of the exponent...
[ginac.git] / ginac / pseries.cpp
index 2c8aa8f602218854a17ecc632adec9418d1efda9..c290fe0a1f7b09b0c73fa2cebdf4e9cb1f609c38 100644 (file)
@@ -4,7 +4,7 @@
  *  methods for series expansion. */
 
 /*
- *  GiNaC Copyright (C) 1999-2001 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" // for Order function
 #include "mul.h"
 #include "power.h"
 #include "relational.h"
+#include "operators.h"
 #include "symbol.h"
-#include "print.h"
+#include "integral.h"
 #include "archive.h"
 #include "utils.h"
-#include "debugmsg.h"
+
+#include <limits>
+#include <numeric>
+#include <stdexcept>
 
 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 ctor, dtor, copy ctor, assignment operator and helpers
+ *  Default constructor
  */
 
-pseries::pseries() : basic(TINFO_pseries)
-{
-       debugmsg("pseries default ctor", LOGLEVEL_CONSTRUCT);
-}
-
-void pseries::copy(const pseries &other)
-{
-       inherited::copy(other);
-       seq = other.seq;
-       var = other.var;
-       point = other.point;
-}
-
-DEFAULT_DESTROY(pseries)
+pseries::pseries() { }
 
 
 /*
@@ -68,19 +62,18 @@ DEFAULT_DESTROY(pseries)
 /** 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 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 &rel_, const epvector &ops_) : basic(TINFO_pseries), seq(ops_)
+pseries::pseries(const ex &rel_, const epvector &ops_) : seq(ops_)
 {
-       debugmsg("pseries ctor from ex,epvector", LOGLEVEL_CONSTRUCT);
-       GINAC_ASSERT(is_ex_exactly_of_type(rel_, relational));
-       GINAC_ASSERT(is_ex_exactly_of_type(rel_.lhs(),symbol));
+       GINAC_ASSERT(is_a<relational>(rel_));
+       GINAC_ASSERT(is_a<symbol>(rel_.lhs()));
        point = rel_.rhs();
-       var = *static_cast<symbol *>(rel_.lhs().bp);
+       var = rel_.lhs();
 }
 
 
@@ -88,17 +81,22 @@ pseries::pseries(const ex &rel_, const epvector &ops_) : basic(TINFO_pseries), s
  *  Archiving
  */
 
-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 ctor from archive_node", LOGLEVEL_CONSTRUCT);
-       for (unsigned int i=0; true; ++i) {
+       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;
-               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_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);
 }
@@ -116,99 +114,123 @@ void pseries::archive(archive_node &n) const
        n.add_ex("point", point);
 }
 
-DEFAULT_UNARCHIVE(pseries)
 
 //////////
-// functions overriding virtual functions from bases classes
+// functions overriding virtual functions from base classes
 //////////
 
-void pseries::print(const print_context & c, unsigned level) const
+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
 {
-       debugmsg("pseries print", LOGLEVEL_PRINT);
+       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';
 
-       if (is_a<print_tree>(c)) {
+       epvector::const_iterator i = seq.begin(), end = seq.end();
+       while (i != end) {
 
-               c.s << std::string(level, ' ') << class_name()
-                   << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
-                   << std::endl;
-               unsigned delta_indent = static_cast<const print_tree &>(c).delta_indent;
-               unsigned num = seq.size();
-               for (unsigned i=0; i<num; ++i) {
-                       seq[i].rest.print(c, level + delta_indent);
-                       seq[i].coeff.print(c, level + delta_indent);
-                       c.s << std::string(level + delta_indent, ' ') << "-----" << std::endl;
-               }
-               var.print(c, level + delta_indent);
-               point.print(c, level + delta_indent);
+               // print a sign, if needed
+               if (i != seq.begin())
+                       c.s << '+';
 
-       } else {
+               if (!is_order_function(i->rest)) {
 
-               if (precedence() <= level)
-                       c.s << "(";
-               
-               std::string par_open = is_a<print_latex>(c) ? "{(" : "(";
-               std::string par_close = is_a<print_latex>(c) ? ")}" : ")";
-               
-               // 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 << par_open;
-                                       i->rest.print(c);
-                                       c.s << par_close;
-                               }
-                               // print 'coeff', something like (x-1)^42
-                               if (!i->coeff.is_zero()) {
-                                       if (is_a<print_latex>(c))
-                                               c.s << ' ';
-                                       else
-                                               c.s << '*';
-                                       if (!point.is_zero()) {
-                                               c.s << par_open;
-                                               (var-point).print(c);
-                                               c.s << par_close;
+                       // 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
-                                               var.print(c);
-                                       if (i->coeff.compare(_ex1())) {
-                                               c.s << '^';
-                                               if (i->coeff.info(info_flags::negative)) {
-                                                       c.s << par_open;
-                                                       i->coeff.print(c);
-                                                       c.s << par_close;
-                                               } else {
-                                                       if (is_a<print_latex>(c)) {
-                                                               c.s << '{';
-                                                               i->coeff.print(c);
-                                                               c.s << '}';
-                                                       } else
-                                                               i->coeff.print(c);
-                                               }
-                                       }
+                                               i->coeff.print(c);
+                                       c.s << closebrace;
                                }
-                       } else
-                               Order(power(var-point,i->coeff)).print(c);
-                       ++i;
-               }
+                       }
+               } else
+                       Order(power(var-point,i->coeff)).print(c);
+               ++i;
+       }
 
-               if (precedence() <= level)
-                       c.s << ")";
+       if (precedence() <= level)
+               c.s << ')';
+}
+
+void pseries::do_print(const print_context & c, unsigned level) const
+{
+       print_series(c, "", "", "*", "^", level);
+}
+
+void pseries::do_print_latex(const print_latex & c, unsigned level) const
+{
+       print_series(c, "{", "}", " ", "^", level);
+}
+
+void pseries::do_print_python(const print_python & c, unsigned level) const
+{
+       print_series(c, "", "", "*", "**", level);
+}
+
+void pseries::do_print_tree(const print_tree & c, unsigned level) const
+{
+       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 << "])";
 }
 
 int pseries::compare_same_type(const basic & other) const
 {
-       GINAC_ASSERT(is_of_type(other, pseries));
+       GINAC_ASSERT(is_a<pseries>(other));
        const pseries &o = static_cast<const pseries &>(other);
        
        // first compare the lengths of the series...
@@ -240,22 +262,20 @@ int pseries::compare_same_type(const basic & other) const
 }
 
 /** Return the number of operands including a possible order term. */
-unsigned pseries::nops(void) const
+size_t pseries::nops() const
 {
        return seq.size();
 }
 
 /** Return the ith term in the series when represented as a sum. */
-ex pseries::op(int i) const
+ex pseries::op(size_t i) const
 {
-       if (i < 0 || unsigned(i) >= seq.size())
+       if (i >= seq.size())
                throw (std::out_of_range("op() out of range"));
-       return seq[i].rest * power(var - point, seq[i].coeff);
-}
 
-ex &pseries::let_op(int i)
-{
-       throw (std::logic_error("let_op not defined for pseries"));
+       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);
 }
 
 /** Return degree of highest power of the series.  This is usually the exponent
@@ -266,14 +286,14 @@ int pseries::degree(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();
+                       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;
+               int max_pow = std::numeric_limits<int>::min();
                while (it != itend) {
                        int pow = it->rest.degree(s);
                        if (pow > max_pow)
@@ -294,14 +314,14 @@ int pseries::ldegree(const ex &s) const
        if (var.is_equal(s)) {
                // Return first exponent
                if (seq.size())
-                       return ex_to<numeric>((*(seq.begin())).coeff).to_int();
+                       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;
+               int min_pow = std::numeric_limits<int>::max();
                while (it != itend) {
                        int pow = it->rest.ldegree(s);
                        if (pow < min_pow)
@@ -323,14 +343,14 @@ ex pseries::coeff(const ex &s, int n) const
 {
        if (var.is_equal(s)) {
                if (seq.empty())
-                       return _ex0();
+                       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_ex_exactly_of_type(seq[mid].coeff, numeric));
+                       GINAC_ASSERT(is_exactly_a<numeric>(seq[mid].coeff));
                        int cmp = ex_to<numeric>(seq[mid].coeff).compare(looking_for);
                        switch (cmp) {
                                case -1:
@@ -345,7 +365,7 @@ ex pseries::coeff(const ex &s, int n) const
                                        throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
                        }
                }
-               return _ex0();
+               return _ex0;
        } else
                return convert_to_poly().coeff(s, n);
 }
@@ -356,7 +376,7 @@ ex pseries::collect(const ex &s, bool distributed) const
        return *this;
 }
 
-/** Evaluate coefficients. */
+/** Perform coefficient-wise automatic term rewriting rules in this class. */
 ex pseries::eval(int level) const
 {
        if (level == 1)
@@ -396,13 +416,115 @@ ex pseries::evalf(int level) const
        return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
 }
 
-ex pseries::subs(const lst & ls, const lst & lr, bool no_pattern) const
+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 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, no_pattern);
+       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
@@ -410,10 +532,10 @@ ex pseries::subs(const lst & ls, const lst & lr, bool no_pattern) const
        newseq.reserve(seq.size());
        epvector::const_iterator it = seq.begin(), itend = seq.end();
        while (it != itend) {
-               newseq.push_back(expair(it->rest.subs(ls, lr, no_pattern), it->coeff));
+               newseq.push_back(expair(it->rest.subs(m, options), it->coeff));
                ++it;
        }
-       return (new pseries(relational(var,point.subs(ls, lr, no_pattern)), newseq))->setflag(status_flags::dynallocated);
+       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
@@ -429,17 +551,17 @@ ex pseries::expand(unsigned options) const
                ++i;
        }
        return (new pseries(relational(var,point), newseq))
-               ->setflag(status_flags::dynallocated | status_flags::expanded);
+               ->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
 {
+       epvector new_seq;
+       epvector::const_iterator it = seq.begin(), itend = seq.end();
+
        if (s == var) {
-               epvector new_seq;
-               epvector::const_iterator it = seq.begin(), itend = seq.end();
                
                // FIXME: coeff might depend on var
                while (it != itend) {
@@ -452,10 +574,22 @@ ex pseries::derivative(const symbol & s) const
                        }
                        ++it;
                }
-               return pseries(relational(var,point), new_seq);
+
        } else {
-               return *this;
+
+               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;
+               }
        }
+
+       return pseries(relational(var,point), new_seq);
 }
 
 ex pseries::convert_to_poly(bool no_order) const
@@ -474,11 +608,25 @@ ex pseries::convert_to_poly(bool no_order) const
        return e;
 }
 
-bool pseries::is_terminating(void) const
+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;
+}
+
 
 /*
  *  Implementations of series expansion
@@ -489,31 +637,42 @@ bool pseries::is_terminating(void) const
 ex basic::series(const relational & r, int order, unsigned options) const
 {
        epvector seq;
-       numeric fac(1);
+       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);
-       const symbol &s = static_cast<symbol &>(*r.lhs().bp);
-       
-       if (!coeff.is_zero())
-               seq.push_back(expair(coeff, _ex0()));
-       
+       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(numeric(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
+               if (deriv.is_zero())  // Series terminates
                        return pseries(r, seq);
-               }
-               coeff = deriv.subs(r);
+
+               coeff = deriv.subs(r, subs_options::no_pattern);
                if (!coeff.is_zero())
-                       seq.push_back(expair(fac.inverse() * coeff, numeric(n)));
+                       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()), numeric(n)));
+               seq.push_back(expair(Order(_ex1), n));
        return pseries(r, seq);
 }
 
@@ -524,18 +683,17 @@ ex symbol::series(const relational & r, int order, unsigned options) const
 {
        epvector seq;
        const ex point = r.rhs();
-       GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
-       ex s = r.lhs();
-       
-       if (this->is_equal(*s.bp)) {
+       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()));
+               seq.push_back(expair(*this, _ex0));
        return pseries(r, seq);
 }
 
@@ -551,7 +709,7 @@ ex pseries::add_series(const pseries &other) const
        // results in an empty (constant) series 
        if (!is_compatible_to(other)) {
                epvector nul;
-               nul.push_back(expair(Order(_ex1()), _ex0()));
+               nul.push_back(expair(Order(_ex1), _ex0));
                return pseries(relational(var,point), nul);
        }
        
@@ -561,7 +719,7 @@ ex pseries::add_series(const pseries &other) const
        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;
+       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) {
@@ -599,7 +757,7 @@ ex pseries::add_series(const pseries &other) const
                } 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));
+                               new_seq.push_back(expair(Order(_ex1), (*a).coeff));
                                break;  // Order term ends the sequence
                        } else {
                                ex sum = (*a).rest + (*b).rest;
@@ -629,11 +787,11 @@ ex add::series(const relational & r, int order, unsigned options) const
        epvector::const_iterator itend = seq.end();
        for (; it!=itend; ++it) {
                ex op;
-               if (is_ex_exactly_of_type(it->rest, pseries))
+               if (is_exactly_a<pseries>(it->rest))
                        op = it->rest;
                else
                        op = it->rest.series(r, order, options);
-               if (!it->coeff.is_equal(_ex1()))
+               if (!it->coeff.is_equal(_ex1))
                        op = ex_to<pseries>(op).mul_const(ex_to<numeric>(it->coeff));
                
                // Series addition
@@ -676,13 +834,17 @@ ex pseries::mul_series(const pseries &other) const
        // results in an empty (constant) series 
        if (!is_compatible_to(other)) {
                epvector nul;
-               nul.push_back(expair(Order(_ex1()), _ex0()));
+               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);
@@ -690,8 +852,8 @@ ex pseries::mul_series(const pseries &other) const
        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;
+       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)))
@@ -701,7 +863,7 @@ ex pseries::mul_series(const pseries &other) const
                cdeg_max = higher_order_c - 1;
        
        for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
-               ex co = _ex0();
+               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);
@@ -712,8 +874,8 @@ ex pseries::mul_series(const pseries &other) const
                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)));
+       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);
 }
 
@@ -723,30 +885,113 @@ ex pseries::mul_series(const pseries &other) const
  *  @see ex::series */
 ex mul::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);
-       
+       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
-       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(r, order, options);
-               if (!it->coeff.is_equal(_ex1()))
-                       op = ex_to<pseries>(op).power_const(ex_to<numeric>(it->coeff), order);
+       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
-               acc = ex_to<pseries>(acc).mul_series(ex_to<pseries>(op));
+               if (it == itbeg)
+                       acc = ex_to<pseries>(op);
+               else
+                       acc = ex_to<pseries>(acc.mul_series(ex_to<pseries>(op)));
        }
-       return acc;
+
+       return acc.mul_const(ex_to<numeric>(overall_coeff));
 }
 
 
@@ -757,6 +1002,7 @@ ex mul::series(const relational & r, int order, unsigned options) const
 ex pseries::power_const(const numeric &p, int deg) const
 {
        // 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 + ...
@@ -777,51 +1023,65 @@ ex pseries::power_const(const numeric &p, int deg) const
        // then of course x^(p*m) but the recurrence formula still holds.
        
        if (seq.empty()) {
-               // as a spacial case, handle the empty (zero) series honoring the
+               // 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"));
+                       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));
+                       throw pole_error("pseries::power_const(): division by zero",1);
                else
                        return *this;
        }
        
-       int ldeg = ldegree(var);
+       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(deg);
+       co.reserve(numcoeff);
        co.push_back(power(coeff(var, ldeg), p));
-       bool all_sums_zero = true;
-       for (int i=1; i<deg; ++i) {
-               ex sum = _ex0();
+       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()));
+                               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(sum / coeff(var, ldeg) / numeric(i));
+               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<deg; ++i) {
+       for (int i=0; i<numcoeff; ++i) {
                if (!co[i].is_zero())
-                       new_seq.push_back(expair(co[i], numeric(i) + p * ldeg));
+                       new_seq.push_back(expair(co[i], p * ldeg + i));
                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));
+       if (!higher_order)
+               new_seq.push_back(expair(Order(_ex1), p * ldeg + numcoeff));
+
        return pseries(relational(var,point), new_seq);
 }
 
@@ -844,33 +1104,97 @@ pseries pseries::shift_exponents(int deg) const
  *  @see ex::series */
 ex power::series(const relational & r, int order, unsigned options) const
 {
-       ex e;
-       if (!is_ex_exactly_of_type(basis, pseries)) {
-               // Basis is not a series, may there be a singularity?
-               bool must_expand_basis = false;
-               try {
-                       basis.subs(r);
-               } catch (pole_error) {
-                       must_expand_basis = true;
-               }
-               
-               // Is the expression of type something^(-int)?
-               if (!must_expand_basis && !exponent.info(info_flags::negint))
-                       return basic::series(r, order, options);
-               
-               // Is the expression of type 0^something?
-               if (!must_expand_basis && !basis.subs(r).is_zero())
-                       return basic::series(r, order, options);
-               
-               // Singularity encountered, expand basis into series
-               e = basis.series(r, order, options);
+       // 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 {
-               // Basis is a series
-               e = basis;
+               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);
        
-       // Power e
-       return ex_to<pseries>(e).power_const(ex_to<numeric>(exponent), order);
+       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;
 }
 
 
@@ -878,8 +1202,8 @@ ex power::series(const relational & r, int order, unsigned options) const
 ex pseries::series(const relational & r, int order, unsigned options) const
 {
        const ex p = r.rhs();
-       GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
-       const symbol &s = static_cast<symbol &>(*r.lhs().bp);
+       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))
@@ -890,7 +1214,7 @@ ex pseries::series(const relational & r, int order, unsigned options) const
                        while (it != itend) {
                                int o = ex_to<numeric>(it->coeff).to_int();
                                if (o >= order) {
-                                       new_seq.push_back(expair(Order(_ex1()), o));
+                                       new_seq.push_back(expair(Order(_ex1), o));
                                        break;
                                }
                                new_seq.push_back(*it);
@@ -902,6 +1226,61 @@ ex pseries::series(const relational & r, int order, unsigned options) const
                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 
@@ -914,23 +1293,20 @@ ex pseries::series(const relational & r, int order, unsigned options) const
  *  @return an expression holding a pseries object */
 ex ex::series(const ex & r, int order, unsigned options) const
 {
-       GINAC_ASSERT(bp!=0);
        ex e;
        relational rel_;
        
-       if (is_ex_exactly_of_type(r,relational))
+       if (is_a<relational>(r))
                rel_ = ex_to<relational>(r);
-       else if (is_ex_exactly_of_type(r,symbol))
-               rel_ = relational(r,_ex0());
+       else if (is_a<symbol>(r))
+               rel_ = relational(r,_ex0);
        else
                throw (std::logic_error("ex::series(): expansion point has unknown type"));
        
-       try {
-               e = bp->series(rel_, order, options);
-       } catch (std::exception &x) {
-               throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")"));
-       }
+       e = bp->series(rel_, order, options);
        return e;
 }
 
+GINAC_BIND_UNARCHIVER(pseries);
+
 } // namespace GiNaC