@@ -1,426 +1,426 @@
-\documentstyle[11pt,reduce]{article}
-\newcommand{\MACSYMA}{{\sf MACSYMA}}
-\newcommand{\MAPLE}{{\sf MAPLE}}
-\newcommand{\Mathematica}{{\sf Mathematica}}
-\newcommand{\PSL}{{\sf PSL}}
-\title{A \REDUCE{} package for manipulation of Taylor series}
-\date{}
-\author{Rainer Sch\"opf\\
-Zentrum f\"ur Datenverarbeitung der Universit\"at Mainz\\
-Anselm-Franz-von-Bentzel-Weg~12\\
-D-55055 Mainz\\
-Germany\\
-E--mail: {\tt Schoepf@Uni-Mainz.DE}}
-\begin{document}
-\maketitle
-\index{Taylor Series} \index{TAYLOR package}
-\index{Laurent series}
-
-This short note describes a package of \REDUCE{} procedures that allow
-Taylor expansion in one or several variables, and efficient
-manipulation of the resulting Taylor series. Capabilities include
-basic operations (addition, subtraction, multiplication and division),
-and also application of certain algebraic and transcendental
-functions. To a certain extent, Laurent and Puiseux expansions can be
-performed as well. In many cases, separable singularities are detected
-and factored out.
-
-\section{Introduction}
-
-The Taylor package was written to provide \REDUCE{} with some of
-the facilities
-that \MACSYMA's \verb+TAYLOR+ function offers,
-but most of all I needed it to be faster and
-more space-efficient.
-Especially I wanted procedures that would return the logarithm or
-arc tangent of a Taylor series, again as a Taylor series.
-This turned out be more work than expected. The features absolutely
-required were (as usual) those that were hardest to implement,
-e.g., arc tangent applied to a Taylor expansion in more than
-one variable.
-
-This package is still undergoing development.
-I'll be happy if it is of any use for you.
-Tell me if you think that there is something missing.
-I invite everybody to criticize and comment and will eagerly try to
-correct any errors found.
-
-\section{How to use it}
-
-The most important operator is `\verb+TAYLOR+'. \index{TAYLOR operator}
-It is used as follows:
-
-\noindent {\tt TAYLOR}(EXP:{\em exprn}[,VAR:{\em kernel},
-            VAR$_0$:{\em exprn},ORDER:{\em integer}]\ldots):{\em exprn}
-
-where EXP is the expression to be expanded. It can be any \REDUCE{}
-object, even an expression containing other Taylor kernels.  VAR is
-the kernel with respect to which EXP is to be expanded. VAR$_0$
-denotes the point about which and ORDER the order up to which
-expansion is to take place. If more than one (VAR, VAR0, ORDER) triple
-is specified {\tt TAYLOR} will expand its first argument independently
-with respect to each variable in turn. For example,
-
-\begin{verbatim}
-  taylor(e^(x^2+y^2),x,0,2,y,0,2);
-\end{verbatim}
-
-will calculate the Taylor expansion up to order $X^{2}*Y^{2}$.
-Note that once the expansion has been done it is not possible to
-calculate higher orders.
-Instead of a kernel, VAR may also
-be a list of kernels. In this case expansion will take place in a way
-so that the {\em sum\/} of the degrees of the kernels does not exceed
-ORDER.
-If VAR$_0$ evaluates to the special identifier \verb|INFINITY|
-{\tt TAYLOR} tries to expand EXP in a series in 1/VAR.
-
-The expansion is performed variable per variable, i.e.\ in the example
-above by first expanding $\exp(x^{2}+y^{2})$ with respect to $x$ and
-then expanding every coefficient with respect to $y$.
-
-\index{IMPLICIT\_TAYLOR operator}\index{INVERSE\_TAYLOR} There are two
-extra operators to compute the Taylor expansions of implicit and
-inverse functions:
-
-\noindent {\tt IMPLICIT\_TAYLOR}(F:{\em exprn},VAR1,VAR2:{\em kernel},\\
-\hphantom{{\tt IMPLICIT\_TAYLOR}(}VAR1$_0$,VAR2$_0$:{\em exprn},
-                                 ORDER:{\em integer}):{\em exprn}
-
-takes a function F depending on two variables VAR1 and VAR2 and
-computes the Taylor series of the implicit function VAR2(VAR1)
-given by the equation F(VAR1,VAR2) = 0. For example,
-
-\begin{verbatim}
-  implicit_taylor(x^2 + y^2 - 1,x,y,0,1,5);
-\end{verbatim}
-
-\noindent {\tt INVERSE\_TAYLOR}(F:{\em exprn},VAR1,VAR2:{\em kernel},\\
-\hphantom{{\tt INVERSE\_TAYLOR}(}VAR1$_0$:{\em exprn},
-                                 ORDER:{\em integer}):{\em exprn}
-
-takes a function F depending on VAR1 and computes the Taylor series of
-the inverse of F with respect to VAR2. For example,
-
-\begin{verbatim}
-  inverse_taylor(exp(x)-1,x,y,0,8);
-\end{verbatim}
-
-
-\index{TAYLORPRINTTERMS variable}
-When a Taylor kernel is printed, only a certain number of (non-zero)
-coefficients are shown. If there are more, an expression of the form
-\verb|(|$n$\verb| terms)| is printed to indicate how many non-zero
-terms have been suppressed. The number of terms printed is given by
-the value of the shared algebraic variable \verb|TAYLORPRINTTERMS|.
-Allowed values are integers and the special identifier \verb|ALL|. The
-latter setting specifies that all terms are to be printed. The default
-setting is $5$.
-
-
-\index{TAYLORKEEPORIGINAL switch}
-If the switch \verb|TAYLORKEEPORIGINAL| is set to \verb|ON| the
-original expression EXP is kept for later reference.
-It can be recovered by means of the operator
-
-\hspace*{2em} {\tt TAYLORORIGINAL}(EXP:{\em exprn}):{\em exprn}
-
-An error is signalled if EXP is not a Taylor kernel or if the original
-expression was not kept, i.e.\ if \verb|TAYLORKEEPORIGINAL| was
-\verb|OFF| during expansion.  The template of a Taylor kernel, i.e.\
-the list of all variables with respect to which expansion took place
-together with expansion point and order can be extracted using
-\ttindex{TAYLORTEMPLATE}
-
-\hspace*{2em} {\tt TAYLORTEMPLATE}(EXP:{\em exprn}):{\em list}
-
-This returns a list of lists with the three elements (VAR,VAR0,ORDER).
-As with \verb|TAYLORORIGINAL|,
-an error is signalled if EXP is not a Taylor kernel.
-
-\hspace*{2em} {\tt TAYLORTOSTANDARD}(EXP:{\em exprn}):{\em exprn}
-
-converts all Taylor kernels in EXP into standard form and
-\ttindex{TAYLORTOSTANDARD} resimplifies the result.
-
-\hspace*{2em} {\tt TAYLORSERIESP}(EXP:{\em exprn}):{\em boolean}
-
-may be used to determine if EXP is a Taylor kernel.
-\ttindex{TAYLORSERIESP} Note that this operator is subject to the same
-restrictions as, e.g., ORDP or NUMBERP, i.e.\ it may only be used in
-boolean expressions in \verb|IF| or \verb|LET| statements.  Finally
-there is
-
-\hspace*{2em} {\tt TAYLORCOMBINE}(EXP:{\em exprn}):{\em exprn}
-
-which tries to combine all Taylor kernels found in EXP into one.
-\ttindex{TAYLORCOMBINE}
-Operations currently possible are:
-\index{Taylor series ! arithmetic}
-\begin{itemize}
-  \item Addition, subtraction, multiplication, and division.
-  \item Roots, exponentials, and logarithms.
-  \item Trigonometric and hyperbolic functions and their inverses.
-\end{itemize}
-Application of unary operators like \verb|LOG| and \verb|ATAN| will
-nearly always succeed. For binary operations their arguments have to be
-Taylor kernels with the same template. This means that the expansion
-variable and the expansion point must match. Expansion order is not so
-important, different order usually means that one of them is truncated
-before doing the operation.
-
-\ttindex{TAYLORKEEPORIGINAL} \ttindex{TAYLORCOMBINE}
-If \verb|TAYLORKEEPORIGINAL| is set to \verb|ON| and if all Taylor
-kernels in \verb|exp| have their original expressions kept
-\verb|TAYLORCOMBINE| will also combine these and store the result
-as the original expression of the resulting Taylor kernel.
-\index{TAYLORAUTOEXPAND switch}
-There is also the switch \verb|TAYLORAUTOEXPAND| (see below).
-
-There are a few restrictions to avoid mathematically undefined
-expressions: it is not possible to take the logarithm of a Taylor
-kernel which has no terms (i.e. is zero), or to divide by such a
-beast.  There are some provisions made to detect singularities during
-expansion: poles that arise because the denominator has zeros at the
-expansion point are detected and properly treated, i.e.\ the Taylor
-kernel will start with a negative power.  (This is accomplished by
-expanding numerator and denominator separately and combining the
-results.)  Essential singularities of the known functions (see above)
-are handled correctly.
-
-\index{Taylor series ! differentiation}
-Differentiation of a Taylor expression is possible.  If you
-differentiate with respect to one of the Taylor variables the order
-will decrease by one.
-
-\index{Taylor series ! substitution}
-Substitution is a bit restricted: Taylor variables can only be replaced
-by other kernels.  There is one exception to this rule: you can always
-substitute a Taylor variable by an expression that evaluates to a
-constant.  Note that \REDUCE{} will not always be able to determine
-that an expression is constant.
-
-\index{Taylor series ! integration}
-Only simple taylor kernels can be integrated. More complicated
-expressions that contain Taylor kernels as parts of themselves are
-automatically converted into a standard representation by means of the
-TAYLORTOSTANDARD operator. In this case a suitable warning is printed.
-
-\index{Taylor series ! reversion} It is possible to revert a Taylor
-series of a function $f$, i.e., to compute the first terms of the
-expansion of the inverse of $f$ from the expansion of $f$. This is
-done by the operator
-
-\hspace*{2em} {\tt TAYLORREVERT}(EXP:{\em exprn},OLDVAR:{\em kernel},
-                                 NEWVAR:{\em kernel}):{\em exprn}
-
-EXP must evaluate to a Taylor kernel with OLDVAR being one of its
-expansion variables. Example:
-
-\begin{verbatim}
-  taylor (u - u**2, u, 0, 5);
-  taylorrevert (ws, u, x);
-\end{verbatim}
-
-This package introduces a number of new switches:
-\begin{itemize}
-
-\index{TAYLORAUTOCOMBINE switch}
-\item If you set \verb|TAYLORAUTOCOMBINE| to \verb|ON| \REDUCE{}
-    automatically combines Taylor expressions during the simplification
-    process.  This is equivalent to applying \verb|TAYLORCOMBINE| to
-    every expression that contains Taylor kernels.
-    Default is \verb|ON|.
-
-\index{TAYLORAUTOEXPAND switch}
-\item \verb|TAYLORAUTOEXPAND| makes Taylor expressions ``contagious''
-    in the sense that \verb|TAYLORCOMBINE| tries to Taylor expand
-    all non-Taylor subexpressions and to combine the result with the
-    rest. Default is \verb|OFF|.
-
-\index{TAYLORKEEPORIGINAL switch}
-\item \verb|TAYLORKEEPORIGINAL|, if set to \verb|ON|, forces the
-    package to keep the original expression, i.e.\ the expression
-    that was Taylor expanded.  All operations performed on the
-    Taylor kernels are also applied to this expression  which can
-    be recovered using the operator \verb|TAYLORORIGINAL|.
-    Default is \verb|OFF|.
-
-\index{TAYLORPRINTORDER switch}
-\item \verb|TAYLORPRINTORDER|, if set to \verb|ON|, causes the
-    remainder to be printed in big-$O$ notation.  Otherwise, three
-    dots are printed. Default is \verb|ON|.
-
-\index{VERBOSELOAD switch}
-\item There is also the switch \verb|VERBOSELOAD|.  If it is set to
-    \verb|ON|
-    \REDUCE{} will print some information when the Taylor package is
-    loaded.  This switch is already present in \PSL{} systems.
-    Default is \verb|OFF|.
-
-\end{itemize}
-\index{defaults ! TAYLOR package}
-
-\section{Caveats}
-\index{caveats ! TAYLOR package}
-
-\verb|TAYLOR| should now always detect non-analytical expressions in
-its first argument. As an example, consider the function $xy/(x+y)$
-that is not analytical in the neighborhood of $(x,y) = (0,0)$: Trying
-to calculate
-\begin{verbatim}
-  taylor(x*y/(x+y),x,0,2,y,0,2);
-\end{verbatim}
-causes an error
-\begin{verbatim}
-***** Not a unit in argument to QUOTTAYLOR
-\end{verbatim}
-Note that it is not generally possible to apply the standard \REDUCE{}
-operators to a Taylor kernel. For example, \verb|PART|, \verb|COEFF|,
-or \verb|COEFFN| cannot be used. Instead, the expression at hand has
-to be converted to standard form first using the \verb|TAYLORTOSTANDARD|
-operator.
-
-\section{Warnings and error messages}
-\index{errors ! TAYLOR package}
-\begin{itemize}
-
-\item \verb|Branch point detected in ...|\\
-    This occurs if you take a rational power of a Taylor kernel
-    and raising the lowest order term of the kernel to this
-    power yields a non analytical term (i.e.\ a fractional power).
-
-\item \verb|Cannot expand further... truncation done|\\
-    You will get this warning if you try to expand a Taylor kernel to
-    a higher order.
-
-\item \verb|Computation loops (recursive definition?): ...|\\
-    Most probably the expression to be expanded contains an operator
-    whose derivative involves the operator itself.
-
-\item \verb|Converting Taylor kernels to standard representation|\\
-    This warning appears if you try to integrate an expression that
-    contains Taylor kernels.
-
-\item \verb|Error during expansion (possible singularity)|\\
-    The expression you are trying to expand caused an error.
-    As far as I know this can only happen if it contains a function
-    with a pole or an essential singularity at the expansion point.
-    (But one can never be sure.)
-
-\item \verb|Essential singularity in ...|\\
-    An essential singularity was detected while applying a
-    special function to a Taylor kernel.
-
-\item \verb|Expansion point lies on branch cut in ...|\\
-    The only functions with branch cuts this package knows of are
-    (natural) logarithm, inverse circular and hyperbolic tangent and
-    cotangent.  The branch cut of the logarithm is assumed to lie on
-    the negative real axis.  Those of the arc tangent and arc
-    cotangent functions are chosen to be compatible with this: both
-    have essential singularities at the points $\pm i$.  The branch
-    cut of arc tangent is the straight line along the imaginary axis
-    connecting $+1$ to $-1$ going through $\infty$ whereas that of arc
-    cotangent goes through the origin.  Consequently, the branch cut
-    of the inverse hyperbolic tangent resp.\ cotangent lies on the
-    real axis and goes from $-1$ to $+1$, that of the latter across
-    $0$, the other across $\infty$.
-
-    The error message can currently only appear when you try to
-    calculate the inverse tangent or cotangent of a Taylor
-    kernel that starts with a negative degree.
-    The case of a logarithm of a Taylor kernel whose constant term
-    is a negative real number is not caught since it is
-    difficult to detect this in general.
-
-\item \verb|Invalid substitution in Taylor kernel: ...|\\
-    You tried to substitute a variable that is already present in the
-    Taylor kernel or on which one of the Taylor variables depend.
-
-\item \verb|Not a unity in ...|\\
-    This will happen if you try to divide by or take the logarithm of
-    a Taylor series whose constant term vanishes.
-
-\item \verb|Not implemented yet (...)|\\
-    Sorry, but I haven't had the time to implement this feature.
-    Tell me if you really need it, maybe I have already an improved
-    version of the package.
-
-\item \verb|Reversion of Taylor series not possible: ...|\\
-\ttindex{TAYLORREVERT}
-    You tried to call the \verb|TAYLORREVERT| operator with
-    inappropriate arguments. The second half of this error message
-    tells you why this operation is not possible.
-
-\item \verb|Taylor kernel doesn't have an original part|\\
-\ttindex{TAYLORORIGINAL} \ttindex{TAYLORKEEPORIGINAL}
-    The Taylor kernel upon which you try to use \verb|TAYLORORIGINAL|
-    was created with the switch \verb|TAYLORKEEPORIGINAL|
-    set to \verb|OFF|
-    and does therefore not keep the original expression.
-
-\item \verb|Wrong number of arguments to TAYLOR|\\
-    You try to use the operator \verb|TAYLOR| with a wrong number of
-    arguments.
-
-\item \verb|Zero divisor in TAYLOREXPAND|\\
-    A zero divisor was found while an expression was being expanded.
-    This should not normally occur.
-
-\item \verb|Zero divisor in Taylor substitution|\\
-    That's exactly what the message says.  As an example consider the
-    case of a Taylor kernel containing the term \verb|1/x| and you try
-    to substitute \verb|x| by \verb|0|.
-
-\item \verb|... invalid as kernel|\\
-    You tried to expand with respect to an expression that is not a
-    kernel.
-
-\item \verb|... invalid as order of Taylor expansion|\\
-    The order parameter you gave to \verb|TAYLOR| is not an integer.
-
-\item \verb|... invalid as Taylor kernel|\\
-\ttindex{TAYLORORIGINAL} \ttindex{TAYLORTEMPLATE}
-    You tried to apply \verb|TAYLORORIGINAL| or \verb|TAYLORTEMPLATE|
-    to an expression that is not a Taylor kernel.
-
-\item \verb|... invalid as Taylor variable|\\
-    You tried to substitute a Taylor variable by an expression that is
-    not a kernel.
-
-\item \verb|... invalid as value of TaylorPrintTerms|\\
-\ttindex{TAYLORPRINTTERMS}
-    You have assigned an invalid value to \verb|TAYLORPRINTTERMS|.
-    Allowed values are: an integer or the special identifier
-    \verb|ALL|.
-
-\item \verb|TAYLOR PACKAGE (...): this can't happen ...|\\
-    This message shows that an internal inconsistency was detected.
-    This is not your fault, at least as long as you did not try to
-    work with the internal data structures of \REDUCE. Send input
-    and output to me, together with the version information that is
-    printed out.
-
-\end{itemize}
-
-\section{Comparison to other packages}
-
-At the moment there is only one \REDUCE{} package that I know of:
-the truncated power series package by Alan Barnes and Julian Padget.
-In my opinion there are two major differences:
-\begin{itemize}
-  \item The interface. They use the domain mechanism for their power
-        series, I decided to invent a special kind of kernel. Both
-        approaches have advantages and disadvantages: with domain
-        modes, it is easier
-        to do certain things automatically, e.g., conversions.
-  \item The concept of a truncated series. Their idea is to remember
-        the original expression and to compute more coefficients when
-        more of them are needed. My approach is to truncate at a
-        certain order and forget how the unexpanded expression
-        looked like.  I think that their method is more widely
-        usable, whereas mine is more efficient when you know in
-        advance exactly how many terms you need.
-\end{itemize}
-
-\end{document}
+\documentstyle[11pt,reduce]{article}
+\newcommand{\MACSYMA}{{\sf MACSYMA}}
+\newcommand{\MAPLE}{{\sf MAPLE}}
+\newcommand{\Mathematica}{{\sf Mathematica}}
+\newcommand{\PSL}{{\sf PSL}}
+\title{A \REDUCE{} package for manipulation of Taylor series}
+\date{}
+\author{Rainer Sch\"opf\\
+Zentrum f\"ur Datenverarbeitung der Universit\"at Mainz\\
+Anselm-Franz-von-Bentzel-Weg~12\\
+D-55055 Mainz\\
+Germany\\
+E--mail: {\tt Schoepf@Uni-Mainz.DE}}
+\begin{document}
+\maketitle
+\index{Taylor Series} \index{TAYLOR package}
+\index{Laurent series}
+
+This short note describes a package of \REDUCE{} procedures that allow
+Taylor expansion in one or several variables, and efficient
+manipulation of the resulting Taylor series. Capabilities include
+basic operations (addition, subtraction, multiplication and division),
+and also application of certain algebraic and transcendental
+functions. To a certain extent, Laurent and Puiseux expansions can be
+performed as well. In many cases, separable singularities are detected
+and factored out.
+
+\section{Introduction}
+
+The Taylor package was written to provide \REDUCE{} with some of
+the facilities
+that \MACSYMA's \verb+TAYLOR+ function offers,
+but most of all I needed it to be faster and
+more space-efficient.
+Especially I wanted procedures that would return the logarithm or
+arc tangent of a Taylor series, again as a Taylor series.
+This turned out be more work than expected. The features absolutely
+required were (as usual) those that were hardest to implement,
+e.g., arc tangent applied to a Taylor expansion in more than
+one variable.
+
+This package is still undergoing development.
+I'll be happy if it is of any use for you.
+Tell me if you think that there is something missing.
+I invite everybody to criticize and comment and will eagerly try to
+correct any errors found.
+
+\section{How to use it}
+
+The most important operator is `\verb+TAYLOR+'. \index{TAYLOR operator}
+It is used as follows:
+
+\noindent {\tt TAYLOR}(EXP:{\em exprn}[,VAR:{\em kernel},
+            VAR$_0$:{\em exprn},ORDER:{\em integer}]\ldots):{\em exprn}
+
+where EXP is the expression to be expanded. It can be any \REDUCE{}
+object, even an expression containing other Taylor kernels.  VAR is
+the kernel with respect to which EXP is to be expanded. VAR$_0$
+denotes the point about which and ORDER the order up to which
+expansion is to take place. If more than one (VAR, VAR0, ORDER) triple
+is specified {\tt TAYLOR} will expand its first argument independently
+with respect to each variable in turn. For example,
+
+\begin{verbatim}
+  taylor(e^(x^2+y^2),x,0,2,y,0,2);
+\end{verbatim}
+
+will calculate the Taylor expansion up to order $X^{2}*Y^{2}$.
+Note that once the expansion has been done it is not possible to
+calculate higher orders.
+Instead of a kernel, VAR may also
+be a list of kernels. In this case expansion will take place in a way
+so that the {\em sum\/} of the degrees of the kernels does not exceed
+ORDER.
+If VAR$_0$ evaluates to the special identifier \verb|INFINITY|
+{\tt TAYLOR} tries to expand EXP in a series in 1/VAR.
+
+The expansion is performed variable per variable, i.e.\ in the example
+above by first expanding $\exp(x^{2}+y^{2})$ with respect to $x$ and
+then expanding every coefficient with respect to $y$.
+
+\index{IMPLICIT\_TAYLOR operator}\index{INVERSE\_TAYLOR} There are two
+extra operators to compute the Taylor expansions of implicit and
+inverse functions:
+
+\noindent {\tt IMPLICIT\_TAYLOR}(F:{\em exprn},VAR1,VAR2:{\em kernel},\\
+\hphantom{{\tt IMPLICIT\_TAYLOR}(}VAR1$_0$,VAR2$_0$:{\em exprn},
+                                 ORDER:{\em integer}):{\em exprn}
+
+takes a function F depending on two variables VAR1 and VAR2 and
+computes the Taylor series of the implicit function VAR2(VAR1)
+given by the equation F(VAR1,VAR2) = 0. For example,
+
+\begin{verbatim}
+  implicit_taylor(x^2 + y^2 - 1,x,y,0,1,5);
+\end{verbatim}
+
+\noindent {\tt INVERSE\_TAYLOR}(F:{\em exprn},VAR1,VAR2:{\em kernel},\\
+\hphantom{{\tt INVERSE\_TAYLOR}(}VAR1$_0$:{\em exprn},
+                                 ORDER:{\em integer}):{\em exprn}
+
+takes a function F depending on VAR1 and computes the Taylor series of
+the inverse of F with respect to VAR2. For example,
+
+\begin{verbatim}
+  inverse_taylor(exp(x)-1,x,y,0,8);
+\end{verbatim}
+
+
+\index{TAYLORPRINTTERMS variable}
+When a Taylor kernel is printed, only a certain number of (non-zero)
+coefficients are shown. If there are more, an expression of the form
+\verb|(|$n$\verb| terms)| is printed to indicate how many non-zero
+terms have been suppressed. The number of terms printed is given by
+the value of the shared algebraic variable \verb|TAYLORPRINTTERMS|.
+Allowed values are integers and the special identifier \verb|ALL|. The
+latter setting specifies that all terms are to be printed. The default
+setting is $5$.
+
+
+\index{TAYLORKEEPORIGINAL switch}
+If the switch \verb|TAYLORKEEPORIGINAL| is set to \verb|ON| the
+original expression EXP is kept for later reference.
+It can be recovered by means of the operator
+
+\hspace*{2em} {\tt TAYLORORIGINAL}(EXP:{\em exprn}):{\em exprn}
+
+An error is signalled if EXP is not a Taylor kernel or if the original
+expression was not kept, i.e.\ if \verb|TAYLORKEEPORIGINAL| was
+\verb|OFF| during expansion.  The template of a Taylor kernel, i.e.\
+the list of all variables with respect to which expansion took place
+together with expansion point and order can be extracted using
+\ttindex{TAYLORTEMPLATE}
+
+\hspace*{2em} {\tt TAYLORTEMPLATE}(EXP:{\em exprn}):{\em list}
+
+This returns a list of lists with the three elements (VAR,VAR0,ORDER).
+As with \verb|TAYLORORIGINAL|,
+an error is signalled if EXP is not a Taylor kernel.
+
+\hspace*{2em} {\tt TAYLORTOSTANDARD}(EXP:{\em exprn}):{\em exprn}
+
+converts all Taylor kernels in EXP into standard form and
+\ttindex{TAYLORTOSTANDARD} resimplifies the result.
+
+\hspace*{2em} {\tt TAYLORSERIESP}(EXP:{\em exprn}):{\em boolean}
+
+may be used to determine if EXP is a Taylor kernel.
+\ttindex{TAYLORSERIESP} Note that this operator is subject to the same
+restrictions as, e.g., ORDP or NUMBERP, i.e.\ it may only be used in
+boolean expressions in \verb|IF| or \verb|LET| statements.  Finally
+there is
+
+\hspace*{2em} {\tt TAYLORCOMBINE}(EXP:{\em exprn}):{\em exprn}
+
+which tries to combine all Taylor kernels found in EXP into one.
+\ttindex{TAYLORCOMBINE}
+Operations currently possible are:
+\index{Taylor series ! arithmetic}
+\begin{itemize}
+  \item Addition, subtraction, multiplication, and division.
+  \item Roots, exponentials, and logarithms.
+  \item Trigonometric and hyperbolic functions and their inverses.
+\end{itemize}
+Application of unary operators like \verb|LOG| and \verb|ATAN| will
+nearly always succeed. For binary operations their arguments have to be
+Taylor kernels with the same template. This means that the expansion
+variable and the expansion point must match. Expansion order is not so
+important, different order usually means that one of them is truncated
+before doing the operation.
+
+\ttindex{TAYLORKEEPORIGINAL} \ttindex{TAYLORCOMBINE}
+If \verb|TAYLORKEEPORIGINAL| is set to \verb|ON| and if all Taylor
+kernels in \verb|exp| have their original expressions kept
+\verb|TAYLORCOMBINE| will also combine these and store the result
+as the original expression of the resulting Taylor kernel.
+\index{TAYLORAUTOEXPAND switch}
+There is also the switch \verb|TAYLORAUTOEXPAND| (see below).
+
+There are a few restrictions to avoid mathematically undefined
+expressions: it is not possible to take the logarithm of a Taylor
+kernel which has no terms (i.e. is zero), or to divide by such a
+beast.  There are some provisions made to detect singularities during
+expansion: poles that arise because the denominator has zeros at the
+expansion point are detected and properly treated, i.e.\ the Taylor
+kernel will start with a negative power.  (This is accomplished by
+expanding numerator and denominator separately and combining the
+results.)  Essential singularities of the known functions (see above)
+are handled correctly.
+
+\index{Taylor series ! differentiation}
+Differentiation of a Taylor expression is possible.  If you
+differentiate with respect to one of the Taylor variables the order
+will decrease by one.
+
+\index{Taylor series ! substitution}
+Substitution is a bit restricted: Taylor variables can only be replaced
+by other kernels.  There is one exception to this rule: you can always
+substitute a Taylor variable by an expression that evaluates to a
+constant.  Note that \REDUCE{} will not always be able to determine
+that an expression is constant.
+
+\index{Taylor series ! integration}
+Only simple taylor kernels can be integrated. More complicated
+expressions that contain Taylor kernels as parts of themselves are
+automatically converted into a standard representation by means of the
+TAYLORTOSTANDARD operator. In this case a suitable warning is printed.
+
+\index{Taylor series ! reversion} It is possible to revert a Taylor
+series of a function $f$, i.e., to compute the first terms of the
+expansion of the inverse of $f$ from the expansion of $f$. This is
+done by the operator
+
+\hspace*{2em} {\tt TAYLORREVERT}(EXP:{\em exprn},OLDVAR:{\em kernel},
+                                 NEWVAR:{\em kernel}):{\em exprn}
+
+EXP must evaluate to a Taylor kernel with OLDVAR being one of its
+expansion variables. Example:
+
+\begin{verbatim}
+  taylor (u - u**2, u, 0, 5);
+  taylorrevert (ws, u, x);
+\end{verbatim}
+
+This package introduces a number of new switches:
+\begin{itemize}
+
+\index{TAYLORAUTOCOMBINE switch}
+\item If you set \verb|TAYLORAUTOCOMBINE| to \verb|ON| \REDUCE{}
+    automatically combines Taylor expressions during the simplification
+    process.  This is equivalent to applying \verb|TAYLORCOMBINE| to
+    every expression that contains Taylor kernels.
+    Default is \verb|ON|.
+
+\index{TAYLORAUTOEXPAND switch}
+\item \verb|TAYLORAUTOEXPAND| makes Taylor expressions ``contagious''
+    in the sense that \verb|TAYLORCOMBINE| tries to Taylor expand
+    all non-Taylor subexpressions and to combine the result with the
+    rest. Default is \verb|OFF|.
+
+\index{TAYLORKEEPORIGINAL switch}
+\item \verb|TAYLORKEEPORIGINAL|, if set to \verb|ON|, forces the
+    package to keep the original expression, i.e.\ the expression
+    that was Taylor expanded.  All operations performed on the
+    Taylor kernels are also applied to this expression  which can
+    be recovered using the operator \verb|TAYLORORIGINAL|.
+    Default is \verb|OFF|.
+
+\index{TAYLORPRINTORDER switch}
+\item \verb|TAYLORPRINTORDER|, if set to \verb|ON|, causes the
+    remainder to be printed in big-$O$ notation.  Otherwise, three
+    dots are printed. Default is \verb|ON|.
+
+\index{VERBOSELOAD switch}
+\item There is also the switch \verb|VERBOSELOAD|.  If it is set to
+    \verb|ON|
+    \REDUCE{} will print some information when the Taylor package is
+    loaded.  This switch is already present in \PSL{} systems.
+    Default is \verb|OFF|.
+
+\end{itemize}
+\index{defaults ! TAYLOR package}
+
+\section{Caveats}
+\index{caveats ! TAYLOR package}
+
+\verb|TAYLOR| should now always detect non-analytical expressions in
+its first argument. As an example, consider the function $xy/(x+y)$
+that is not analytical in the neighborhood of $(x,y) = (0,0)$: Trying
+to calculate
+\begin{verbatim}
+  taylor(x*y/(x+y),x,0,2,y,0,2);
+\end{verbatim}
+causes an error
+\begin{verbatim}
+***** Not a unit in argument to QUOTTAYLOR
+\end{verbatim}
+Note that it is not generally possible to apply the standard \REDUCE{}
+operators to a Taylor kernel. For example, \verb|PART|, \verb|COEFF|,
+or \verb|COEFFN| cannot be used. Instead, the expression at hand has
+to be converted to standard form first using the \verb|TAYLORTOSTANDARD|
+operator.
+
+\section{Warnings and error messages}
+\index{errors ! TAYLOR package}
+\begin{itemize}
+
+\item \verb|Branch point detected in ...|\\
+    This occurs if you take a rational power of a Taylor kernel
+    and raising the lowest order term of the kernel to this
+    power yields a non analytical term (i.e.\ a fractional power).
+
+\item \verb|Cannot expand further... truncation done|\\
+    You will get this warning if you try to expand a Taylor kernel to
+    a higher order.
+
+\item \verb|Computation loops (recursive definition?): ...|\\
+    Most probably the expression to be expanded contains an operator
+    whose derivative involves the operator itself.
+
+\item \verb|Converting Taylor kernels to standard representation|\\
+    This warning appears if you try to integrate an expression that
+    contains Taylor kernels.
+
+\item \verb|Error during expansion (possible singularity)|\\
+    The expression you are trying to expand caused an error.
+    As far as I know this can only happen if it contains a function
+    with a pole or an essential singularity at the expansion point.
+    (But one can never be sure.)
+
+\item \verb|Essential singularity in ...|\\
+    An essential singularity was detected while applying a
+    special function to a Taylor kernel.
+
+\item \verb|Expansion point lies on branch cut in ...|\\
+    The only functions with branch cuts this package knows of are
+    (natural) logarithm, inverse circular and hyperbolic tangent and
+    cotangent.  The branch cut of the logarithm is assumed to lie on
+    the negative real axis.  Those of the arc tangent and arc
+    cotangent functions are chosen to be compatible with this: both
+    have essential singularities at the points $\pm i$.  The branch
+    cut of arc tangent is the straight line along the imaginary axis
+    connecting $+1$ to $-1$ going through $\infty$ whereas that of arc
+    cotangent goes through the origin.  Consequently, the branch cut
+    of the inverse hyperbolic tangent resp.\ cotangent lies on the
+    real axis and goes from $-1$ to $+1$, that of the latter across
+    $0$, the other across $\infty$.
+
+    The error message can currently only appear when you try to
+    calculate the inverse tangent or cotangent of a Taylor
+    kernel that starts with a negative degree.
+    The case of a logarithm of a Taylor kernel whose constant term
+    is a negative real number is not caught since it is
+    difficult to detect this in general.
+
+\item \verb|Invalid substitution in Taylor kernel: ...|\\
+    You tried to substitute a variable that is already present in the
+    Taylor kernel or on which one of the Taylor variables depend.
+
+\item \verb|Not a unity in ...|\\
+    This will happen if you try to divide by or take the logarithm of
+    a Taylor series whose constant term vanishes.
+
+\item \verb|Not implemented yet (...)|\\
+    Sorry, but I haven't had the time to implement this feature.
+    Tell me if you really need it, maybe I have already an improved
+    version of the package.
+
+\item \verb|Reversion of Taylor series not possible: ...|\\
+\ttindex{TAYLORREVERT}
+    You tried to call the \verb|TAYLORREVERT| operator with
+    inappropriate arguments. The second half of this error message
+    tells you why this operation is not possible.
+
+\item \verb|Taylor kernel doesn't have an original part|\\
+\ttindex{TAYLORORIGINAL} \ttindex{TAYLORKEEPORIGINAL}
+    The Taylor kernel upon which you try to use \verb|TAYLORORIGINAL|
+    was created with the switch \verb|TAYLORKEEPORIGINAL|
+    set to \verb|OFF|
+    and does therefore not keep the original expression.
+
+\item \verb|Wrong number of arguments to TAYLOR|\\
+    You try to use the operator \verb|TAYLOR| with a wrong number of
+    arguments.
+
+\item \verb|Zero divisor in TAYLOREXPAND|\\
+    A zero divisor was found while an expression was being expanded.
+    This should not normally occur.
+
+\item \verb|Zero divisor in Taylor substitution|\\
+    That's exactly what the message says.  As an example consider the
+    case of a Taylor kernel containing the term \verb|1/x| and you try
+    to substitute \verb|x| by \verb|0|.
+
+\item \verb|... invalid as kernel|\\
+    You tried to expand with respect to an expression that is not a
+    kernel.
+
+\item \verb|... invalid as order of Taylor expansion|\\
+    The order parameter you gave to \verb|TAYLOR| is not an integer.
+
+\item \verb|... invalid as Taylor kernel|\\
+\ttindex{TAYLORORIGINAL} \ttindex{TAYLORTEMPLATE}
+    You tried to apply \verb|TAYLORORIGINAL| or \verb|TAYLORTEMPLATE|
+    to an expression that is not a Taylor kernel.
+
+\item \verb|... invalid as Taylor variable|\\
+    You tried to substitute a Taylor variable by an expression that is
+    not a kernel.
+
+\item \verb|... invalid as value of TaylorPrintTerms|\\
+\ttindex{TAYLORPRINTTERMS}
+    You have assigned an invalid value to \verb|TAYLORPRINTTERMS|.
+    Allowed values are: an integer or the special identifier
+    \verb|ALL|.
+
+\item \verb|TAYLOR PACKAGE (...): this can't happen ...|\\
+    This message shows that an internal inconsistency was detected.
+    This is not your fault, at least as long as you did not try to
+    work with the internal data structures of \REDUCE. Send input
+    and output to me, together with the version information that is
+    printed out.
+
+\end{itemize}
+
+\section{Comparison to other packages}
+
+At the moment there is only one \REDUCE{} package that I know of:
+the truncated power series package by Alan Barnes and Julian Padget.
+In my opinion there are two major differences:
+\begin{itemize}
+  \item The interface. They use the domain mechanism for their power
+        series, I decided to invent a special kind of kernel. Both
+        approaches have advantages and disadvantages: with domain
+        modes, it is easier
+        to do certain things automatically, e.g., conversions.
+  \item The concept of a truncated series. Their idea is to remember
+        the original expression and to compute more coefficients when
+        more of them are needed. My approach is to truncate at a
+        certain order and forget how the unexpanded expression
+        looked like.  I think that their method is more widely
+        usable, whereas mine is more efficient when you know in
+        advance exactly how many terms you need.
+\end{itemize}
+
+\end{document}