@@ -1,540 +1,540 @@
-\documentstyle[11pt,reduce]{article}
-\title{Truncated Power Series}
-\date{}
-\author{Alan Barnes \\
-Dept. of Computer Science and Applied Mathematics \\
-Aston University, Aston Triangle, \\
-Birmingham B4 7ET \\ GREAT BRITAIN \\
-Email: barnesa@aston.ac.uk \\[0.1in]
-and \\[0.1in]
-Julian Padget \\
-School of Mathematics, University of Bath \\
-Claverton Down, Bath, BA2 7AY \\
-GREAT BRITAIN \\
-Email: jap@maths.bath.ac.uk}
-\begin{document}
-\maketitle
-\index{power series} \index{truncated power series}
-\index{Barnes, Alan} \index{Padget, Julian}
-\section{Introduction}
-\index{Power series expansions}
-This package implements formal power series expansions in one
-variable using the domain mechanism of REDUCE. This means that power
-series objects can be added, multiplied, differentiated etc. like other
-first class objects in the system. A lazy evaluation scheme is used in
-the package and thus terms of the series are not evaluated until they
-are required for printing or for use in calculating terms in other
-power series. The series are extendible giving the user the impression
-that the full infinite series is being manipulated.  The errors that
-can sometimes occur using series that are truncated at some fixed depth
-(for example when a term in the required series depends on terms of an
-intermediate series beyond the truncation depth) are thus avoided.
-
-Below we give a brief description of the operators available in the
-power series package together with some examples of their use.
-
-\subsection{PS Operator}
-
-Syntax:
-
-\noindent{\tt PS}(EXPRN:{\em algebraic},DEPVAR:{\em kernel},
-ABOUT:{\em algebraic}):{\em ps object}
-
-\index{PS operator}
-The {\tt PS} operator returns a  power series object
-(a tagged domain element)
-representing the univariate formal power series expansion of EXPRN with
-respect to the dependent variable DEPVAR about the expansion point
-ABOUT.  EXPRN may itself contain power series objects.
-
-The algebraic expression ABOUT should simplify to an expression
-which is independent of the dependent variable DEPVAR, otherwise
-an error will result.  If ABOUT is the identifier {\tt INFINITY}
-then the power series expansion about DEPVAR = $\infty$ is
-obtained in ascending powers of 1/DEPVAR.
-
-\index{PSEXPLIM operator}
-If the command is terminated by a semi-colon, a power series object
-representing EXPRN is compiled and then a number of terms of the
-power series expansion are evaluated and printed.  The expansion is
-carried out as far as the value specified by {\tt PSEXPLIM}.  If,
-subsequently, the value of {\tt PSEXPLIM} is increased, sufficient
-information is stored in the power series object to enable the
-additional terms to be calculated without recalculating the terms
-already obtained.
-
-If the command is terminated by a dollar symbol, a power series object
-is compiled, but at most one term is calculated at this stage.
-
-If the function has a pole at the expansion point then the correct
-Laurent series expansion will be produced.
-
-\noindent The following examples are valid uses of {\tt PS}:
-\begin{verbatim}
-    psexplim 6;
-    ps(log x,x,1);
-    ps(e**(sin x),x,0);
-    ps(x/(1+x),x,infinity);
-    ps(sin x/(1-cos x),x,0);
-\end{verbatim}
-
-\index{power series ! of user defined function}
-New user-defined functions may be expanded provided the user provides
-LET rules giving
-
-\begin{enumerate}
-\item the value of the function at the expansion point
-\item a differentiation rule for the new function.
-\end{enumerate}
-
-\noindent For example
-\begin{verbatim}
-    operator sech;
-    forall x let df(sech x,x)= - sech x * tanh x;
-    let sech 0 = 1;
-    ps(sech(x**2),x,0);
-\end{verbatim}
-
-\index{power series ! of integral}
-The power series expansion of an integral may also be obtained (even if
-REDUCE cannot evaluate the integral in closed form).  An example of
-this is
-
-\begin{verbatim}
-    ps(int(e**x/x,x),x,1);
-\end{verbatim}
-
-Note that if the integration variable is the same as the expansion
-variable then REDUCE's integration package is not called; if on the
-other hand the two variables are different then the integrator is
-called to integrate each of the coefficients in the power series
-expansion of the integrand.  The constant of integration is zero by
-default.
-
-\subsection{PSEXPLIM Operator}
-
-\index{PSEXPLIM Operator}
-Syntax:
-
-\hspace*{2em} {\tt PSEXPLIM}(UPTO:{\em integer}):{\em integer}
-
-\hspace*{4em} or
-
-\hspace*{2em} {\tt PSEXPLIM}():{\em integer}
-
-Calling this operator sets an internal variable of the
-TPS package to the value of UPTO (which should evaluate to an integer).
-The value returned is the previous value of this variable.
-The default value is six.
-
-If {\tt PSEXPLIM} is called with no argument, the current value for
-the expansion limit is returned.
-
-
-\subsection{PSORDLIM Operator}
-
-\index{PSORDLIM operator}
-Syntax:
-
-\hspace*{2em} {\tt PSORDLIM}(UPTO:{\em integer}):{\em integer}
-
-\hspace*{4em} or
-
-\hspace*{2em} {\tt PSORDLIM}():{\em integer}
-
-An internal variable is set to the value of {\tt UPTO} (which should
-evaluate to an integer). The value returned is the previous value of
-the variable.  The default value is 15.
-
-If {\tt PSORDLIM} is called with no argument, the current value is
-returned.
-
-The significance of this control is that the system attempts to find
-the order of the power series required, that is the order is the
-degree of the first non-zero term in the power series.  If the order
-is greater than the value of this variable an error message is given
-and the computation aborts. This prevents infinite loops in examples
-such as
-
-\begin{verbatim}
-        ps(1 - (sin x)**2 - (cos x)**2,x,0);
-\end{verbatim}
-
-where the expression being expanded is identically zero, but is not
-recognized as such by REDUCE.
-
-
-\subsection{PSTERM Operator}
-
-\index{PSTERM operator}
-Syntax:
-
-\hspace*{2em} {\tt PSTERM}(TPS:{\em power series object},
-NTH:{\em integer}):{\em algebraic}
-
-The operator {\tt PSTERM} returns the NTH term of the existing
-power series object TPS. If NTH does not evaluate to
-an integer or TPS to a power series object an error results.  It
-should be noted that an integer is treated as a power series.
-
-
-\subsection{PSORDER Operator}
-
-\index{PSORDER operator}
-Syntax:
-
-\hspace*{2em} {\tt PSORDER}(TPS:{\em power series object}):{\em integer}
-
-The operator {\tt PSORDER} returns the order, that is the degree of
-the first non-zero term, of the power series object TPS.
-TPS should evaluate to a power series object or an error results. If
-TPS is zero, the identifier {\tt UNDEFINED} is returned.
-
-\subsection{PSSETORDER Operator}
-
-\index{PSSETORDER operator}
-Syntax:
-
-\hspace*{2em} {\tt PSSETORDER}(TPS:{\em power series object},
-ORD:{\em integer}):{\em integer}
-
-The operator {\tt PSSETORDER} sets the order of the power series
-TPS to the value ORD, which should evaluate to an integer. If
-TPS does not evaluate to a power series object, then an error
-occurs. The value returned by this operator is the previous order of
-TPS, or 0 if the order of TPS was undefined.  This
-operator is useful for setting the order of the power series of a
-function defined by a differential equation in cases where the power
-series package is inadequate to determine the order automatically.
-
-
-\subsection{PSDEPVAR Operator}
-
-\index{PSDEPVAR operator}
-Syntax:
-
-\hspace*{2em} {\tt PSDEPVAR}(TPS:{\em power series object})
-:{\em identifier}
-
-The operator {\tt PSDEPVAR} returns the expansion variable of the
-power series object TPS. TPS should evaluate to a power
-series object or an integer, otherwise an error results. If TPS
-is an integer, the identifier {\tt UNDEFINED} is returned.
-
-\subsection{PSEXPANSIONPT operator}
-
-\index{PSEXPANSIONPT operator}
-Syntax:
-
-\hspace*{2em}
-{\tt PSEXPANSIONPT}(TPS:{\em power series object}):{\em algebraic}
-
-The operator {\tt PSEXPANSIONPT} returns the expansion point of the
-power series object TPS. TPS should evaluate to a power
-series object or an integer, otherwise an error results. If TPS
-is integer, the identifier {\tt UNDEFINED} is returned. If the
-expansion is about infinity, the identifier {\tt INFINITY} is
-returned.
-
-\subsection{PSFUNCTION Operator}
-
-\index{PSFUNCTION operator}
-Syntax:
-
-\hspace*{2em}
-{\tt PSFUNCTION}(TPS:{\em power series object}):{\em algebraic}
-
-The operator {\tt PSFUNCTION} returns the function whose expansion
-gave rise to the power series object TPS. TPS should
-evaluate to a power series object or an integer, otherwise an error
-results.
-
-\subsection{PSCHANGEVAR Operator}
-
-\index{PSCHANGEVAR operator}
-Syntax:
-
-\hspace*{2em} {\tt PSCHANGEVAR}(TPS:{\em power series object},
-X:{\em kernel}):{\em power series object}
-
-The operator {\tt PSCHANGEVAR} changes the dependent variable of the
-power series object TPS to the variable X. TPS
-should evaluate to a power series object and X to a kernel,
-otherwise an error results.  Also X should not appear as a
-parameter in TPS. The power series with the new dependent
-variable is returned.
-
-\subsection{PSREVERSE Operator}
-
-\index{PSREVERSE operator}
-Syntax:
-
-\hspace*{2em}
-{\tt PSREVERSE}(TPS:{\em power series object}):{\em power series}
-
-Power series reversion.  The power series TPS is functionally
-inverted.  Four cases arise:
-
-\begin{enumerate}
-\item If the order of the series is 1, then the expansion point of the
-inverted series is 0.
-
-\item If the order is 0 {\em and} if the first order term in TPS
-is non-zero, then the expansion point of the inverted series is taken
-to be the coefficient of the zeroth order term in TPS.
-
-\item If the order is -1 the expansion point of the inverted series
-is the point at infinity.  In all other cases a REDUCE error is
-reported because the series cannot be inverted as a power series.
-Puiseux \index{Puiseux expansion} expansion would be required to
-handle these cases.
-
-\item If the expansion point of TPS is finite it becomes the
-zeroth order term in the inverted series. For expansion about 0 or the
-point at infinity the order of the inverted series is one.
-\end{enumerate}
-
-If TPS is not a power series object after evaluation an error results.
-
-\noindent Here are some examples:
-\begin{verbatim}
-        ps(sin x,x,0);
-        psreverse(ws); % produces series for asin x about x=0.
-        ps(exp x,x,0);
-        psreverse ws; % produces series for log x about x=1.
-        ps(sin(1/x),x,infinity);
-        psreverse(ws); % produces series for 1/asin(x) about x=0.
-\end{verbatim}
-
-\subsection{PSCOMPOSE Operator}
-
-\index{PSCOMPOSE operator}
-Syntax:
-
-\hspace*{2em} {\tt PSCOMPOSE}(TPS1:{\em power series},
-TPS2:{\em power series}):{\em power series}
-
-\index{power series ! composition}
-{\tt PSCOMPOSE} performs power series composition.
-The power series TPS1 and TPS2 are functionally composed.
-That is to say that TPS2 is substituted for the expansion
-variable in TPS1 and the result expressed as a power series. The
-dependent variable and expansion point of the result coincide with
-those of TPS2.  The following conditions apply to power series
-composition:
-
-\begin{enumerate}
-\item If the expansion point of TPS1 is 0 then the order of the
-TPS2 must be at least 1.
-
-\item If the expansion point of TPS1 is finite, it should
-coincide with the coefficient of the zeroth order term in TPS2.
-The order of TPS2 should also be non-negative in this case.
-
-\item If the expansion point of TPS1 is the point at infinity
-then the order of TPS2 must be less than or equal to -1.
-
-\end{enumerate}
-
-If these conditions do not hold the series cannot be composed (with
-the current algorithm terms of the inverted series would involve
-infinite sums) and a REDUCE error occurs.
-
-\noindent Examples of power series composition include the following.
-
-\begin{verbatim}
-    a:=ps(exp y,y,0);  b:=ps(sin x,x,0);
-    pscompose(a,b);
-    % Produces the power series expansion of exp(sin x)
-    % about x=0.
-
-    a:=ps(exp z,z,1); b:=ps(cos x,x,0);
-    pscompose(a,b);
-    % Produces the power series expansion of exp(cos x)
-    % about x=0.
-
-    a:=ps(cos(1/x),x,infinity);  b:=ps(1/sin x,x,0);
-    pscompose(a,b);
-    % Produces the power series expansion of cos(sin x)
-    % about x=0.
-\end{verbatim}
-
-\subsection{PSSUM Operator}
-
-\index{PSSUM operator}
-Syntax:
-
-\begin{tabbing}
-\hspace*{2em} {\tt PSSUM}(\=J:{\em kernel} = LOWLIM:{\em integer},
-COEFF:{\em algebraic}, X:{\em kernel}, \\
-\> ABOUT:{\em algebraic}, POWER:{\em algebraic}):{\em power series}
-\end{tabbing}
-
-The formal power series sum for J from LOWLIM to {\tt INFINITY} of
-
-\begin{verbatim}
-      COEFF*(X-ABOUT)**POWER
-\end{verbatim}
-
-or if ABOUT is given as {\tt INFINITY}
-
-\begin{verbatim}
-      COEFF*(1/X)**POWER
-\end{verbatim}
-
-is constructed and returned. This enables power series whose general
-term is known to be constructed and manipulated using the other
-procedures of the power series package.
-
-J and X should be distinct simple kernels. The algebraics
-ABOUT,  COEFF and POWER should not depend on the
-expansion variable X, similarly the algebraic ABOUT should
-not depend on the summation variable J.  The algebraic POWER should be
-a strictly increasing integer valued function of J for J in the range
-LOWLIM to {\tt INFINITY}.
-
-\begin{verbatim}
-   pssum(n=0,1,x,0,n*n);
-   % Produces the power series summation for n=0 to
-   % infinity of x**(n*n).
-
-   pssum(m=1,(-1)**(m-1)/(2m-1),y,1,2m-1);
-   % Produces the power series expansion of atan(y-1)
-   % about y=1.
-
-   pssum(j=1,-1/j,x,infinity,j);
-   % Produces the power series expansion of log(1-1/x)
-   % about the point at infinity.
-
-   pssum(n=0,1,x,0,2n**2+3n) + pssum(n=1,1,x,0,2n**2-3n);
-   % Produces the power series summation for n=-infinity
-   % to +infinity of x**(2n**2+3n).
-\end{verbatim}
-
-\subsection{PSCOPY Operator}
-
-\index{PSCOPY operator}
-Syntax:
-
-\hspace*{2em} {\tt PSCOPY}(TPS:{\em power series}):{\em power series}
-
-This procedure returns a copy of the power series {\tt TPS}. The copy
-has no shared sub-structures in common with the original series.  This
-enables substitutions to be performed on the series without
-side-effects on previously computed objects. For example:
-
-\begin{verbatim}
-    clear a;
-    b := ps(sin(a*x)), x, 0);
-    b where a => 1;
-\end{verbatim}
-
-will result in {\tt a} being set to 1 in each of the terms of the
-power series and the resulting expressions being simplified. Owing to
-the way power series objects are implemented using Lisp vectors, this
-has the side-effect that the value of {\tt b} is changed.  This may be
-avoided by copying the series with {\tt PSCOPY} before applying the
-substitution, thus:
-
-\begin{verbatim}
-    b := ps(sin(a*x)), x, 0);
-    pscopy b where a => 1;
-\end{verbatim}
-
-
-\subsection{Arithmetic Operations}
-
-\index{power series ! arithmetic}
-As power series objects are domain elements they may be combined
-together in algebraic expressions in algebraic mode of REDUCE in the
-normal way.
-
-For example if A and B are power series objects then the commands
-such as:
-
-\index{+ ! power series} \index{- ! power series}
-\index{/ ! power series} \index{* ! power series}
-\index{** ! power series}
-\begin{verbatim}
-    a*b;
-    a/b;
-    a**2+b**2;
-\end{verbatim}
-
-will produce power series objects representing the product,quotient
-and the sum of the squares of the power series objects A and B
-respectively.
-
-\subsection{Differentiation}
-
-\index{power series ! differentiation}
-If A is a power series object depending on X then the input
-{\tt df(a, x);} will produce the power series expansion of the
-derivative of A with respect to X.
-
-{\em Note} however that currently the input  {\tt int(a, x);} will
-not work as intended; instead one must input
-{\tt ps(int(a, x),x,0);}
-in order to obtain the power series expansion of the integral of
-{\tt a}.
-
-\section{Restrictions and Known Bugs}
-
-If A is a power series object and X is a variable
-which evaluates to itself then currently expressions such as {\tt a*x}
-do not evaluate to a single power series object (although the
-result is formally valid).  Instead use {\tt ps(a*x,x,0)} {\em etc.}.
-
-Similarly expressions such as {\tt sin(A)} where {\tt A} is a PS object
-currently will not be expanded. For example:
-
-\begin{verbatim}
-    a:=ps(1/(1+x),x,0);
-    b:=sin a;
-\end{verbatim}
-
-will not expand {\tt sin(1/(1+x))} as a power series. In fact
-
-\begin{verbatim}
-          SIN(1 - X + X**2 - X**3 + .....)
-\end{verbatim}
-
-will be returned. However,
-
-\begin{verbatim}
-    b:=ps(sin(a),x,0);
-\end{verbatim}
-
-or
-
-\begin{verbatim}
-    b:=ps(sin(1/(1+x)),x,0);
-\end{verbatim}
-
-should work as intended.
-
-The handling of functions with essential singularities is currently
-erratic: usually an error message
-
-\hspace*{2em} {\tt ***** Essential Singularity}
-
-or
-
-\hspace*{2em} {\tt ***** Logarithmic Singularity}
-
-occurs but occasionally a division by
-zero error or some drastic error like (for PSL) binding stack
-overflow may occur.
-
-Mixed mode arithmetic of power series objects with other domain
-elements is quite restricted: only integers and floats can currently
-be converted to power series objects.
-
-The printing of power series currently leaves something to be
-desired: often line-breaks appear in the middle of terms.
-
-There is no simple way to write the results of power series
-calculation to a file and read them back into REDUCE at a later
-stage.
-\end{document}
-
+\documentstyle[11pt,reduce]{article}
+\title{Truncated Power Series}
+\date{}
+\author{Alan Barnes \\
+Dept. of Computer Science and Applied Mathematics \\
+Aston University, Aston Triangle, \\
+Birmingham B4 7ET \\ GREAT BRITAIN \\
+Email: barnesa@aston.ac.uk \\[0.1in]
+and \\[0.1in]
+Julian Padget \\
+School of Mathematics, University of Bath \\
+Claverton Down, Bath, BA2 7AY \\
+GREAT BRITAIN \\
+Email: jap@maths.bath.ac.uk}
+\begin{document}
+\maketitle
+\index{power series} \index{truncated power series}
+\index{Barnes, Alan} \index{Padget, Julian}
+\section{Introduction}
+\index{Power series expansions}
+This package implements formal power series expansions in one
+variable using the domain mechanism of REDUCE. This means that power
+series objects can be added, multiplied, differentiated etc. like other
+first class objects in the system. A lazy evaluation scheme is used in
+the package and thus terms of the series are not evaluated until they
+are required for printing or for use in calculating terms in other
+power series. The series are extendible giving the user the impression
+that the full infinite series is being manipulated.  The errors that
+can sometimes occur using series that are truncated at some fixed depth
+(for example when a term in the required series depends on terms of an
+intermediate series beyond the truncation depth) are thus avoided.
+
+Below we give a brief description of the operators available in the
+power series package together with some examples of their use.
+
+\subsection{PS Operator}
+
+Syntax:
+
+\noindent{\tt PS}(EXPRN:{\em algebraic},DEPVAR:{\em kernel},
+ABOUT:{\em algebraic}):{\em ps object}
+
+\index{PS operator}
+The {\tt PS} operator returns a  power series object
+(a tagged domain element)
+representing the univariate formal power series expansion of EXPRN with
+respect to the dependent variable DEPVAR about the expansion point
+ABOUT.  EXPRN may itself contain power series objects.
+
+The algebraic expression ABOUT should simplify to an expression
+which is independent of the dependent variable DEPVAR, otherwise
+an error will result.  If ABOUT is the identifier {\tt INFINITY}
+then the power series expansion about DEPVAR = $\infty$ is
+obtained in ascending powers of 1/DEPVAR.
+
+\index{PSEXPLIM operator}
+If the command is terminated by a semi-colon, a power series object
+representing EXPRN is compiled and then a number of terms of the
+power series expansion are evaluated and printed.  The expansion is
+carried out as far as the value specified by {\tt PSEXPLIM}.  If,
+subsequently, the value of {\tt PSEXPLIM} is increased, sufficient
+information is stored in the power series object to enable the
+additional terms to be calculated without recalculating the terms
+already obtained.
+
+If the command is terminated by a dollar symbol, a power series object
+is compiled, but at most one term is calculated at this stage.
+
+If the function has a pole at the expansion point then the correct
+Laurent series expansion will be produced.
+
+\noindent The following examples are valid uses of {\tt PS}:
+\begin{verbatim}
+    psexplim 6;
+    ps(log x,x,1);
+    ps(e**(sin x),x,0);
+    ps(x/(1+x),x,infinity);
+    ps(sin x/(1-cos x),x,0);
+\end{verbatim}
+
+\index{power series ! of user defined function}
+New user-defined functions may be expanded provided the user provides
+LET rules giving
+
+\begin{enumerate}
+\item the value of the function at the expansion point
+\item a differentiation rule for the new function.
+\end{enumerate}
+
+\noindent For example
+\begin{verbatim}
+    operator sech;
+    forall x let df(sech x,x)= - sech x * tanh x;
+    let sech 0 = 1;
+    ps(sech(x**2),x,0);
+\end{verbatim}
+
+\index{power series ! of integral}
+The power series expansion of an integral may also be obtained (even if
+REDUCE cannot evaluate the integral in closed form).  An example of
+this is
+
+\begin{verbatim}
+    ps(int(e**x/x,x),x,1);
+\end{verbatim}
+
+Note that if the integration variable is the same as the expansion
+variable then REDUCE's integration package is not called; if on the
+other hand the two variables are different then the integrator is
+called to integrate each of the coefficients in the power series
+expansion of the integrand.  The constant of integration is zero by
+default.
+
+\subsection{PSEXPLIM Operator}
+
+\index{PSEXPLIM Operator}
+Syntax:
+
+\hspace*{2em} {\tt PSEXPLIM}(UPTO:{\em integer}):{\em integer}
+
+\hspace*{4em} or
+
+\hspace*{2em} {\tt PSEXPLIM}():{\em integer}
+
+Calling this operator sets an internal variable of the
+TPS package to the value of UPTO (which should evaluate to an integer).
+The value returned is the previous value of this variable.
+The default value is six.
+
+If {\tt PSEXPLIM} is called with no argument, the current value for
+the expansion limit is returned.
+
+
+\subsection{PSORDLIM Operator}
+
+\index{PSORDLIM operator}
+Syntax:
+
+\hspace*{2em} {\tt PSORDLIM}(UPTO:{\em integer}):{\em integer}
+
+\hspace*{4em} or
+
+\hspace*{2em} {\tt PSORDLIM}():{\em integer}
+
+An internal variable is set to the value of {\tt UPTO} (which should
+evaluate to an integer). The value returned is the previous value of
+the variable.  The default value is 15.
+
+If {\tt PSORDLIM} is called with no argument, the current value is
+returned.
+
+The significance of this control is that the system attempts to find
+the order of the power series required, that is the order is the
+degree of the first non-zero term in the power series.  If the order
+is greater than the value of this variable an error message is given
+and the computation aborts. This prevents infinite loops in examples
+such as
+
+\begin{verbatim}
+        ps(1 - (sin x)**2 - (cos x)**2,x,0);
+\end{verbatim}
+
+where the expression being expanded is identically zero, but is not
+recognized as such by REDUCE.
+
+
+\subsection{PSTERM Operator}
+
+\index{PSTERM operator}
+Syntax:
+
+\hspace*{2em} {\tt PSTERM}(TPS:{\em power series object},
+NTH:{\em integer}):{\em algebraic}
+
+The operator {\tt PSTERM} returns the NTH term of the existing
+power series object TPS. If NTH does not evaluate to
+an integer or TPS to a power series object an error results.  It
+should be noted that an integer is treated as a power series.
+
+
+\subsection{PSORDER Operator}
+
+\index{PSORDER operator}
+Syntax:
+
+\hspace*{2em} {\tt PSORDER}(TPS:{\em power series object}):{\em integer}
+
+The operator {\tt PSORDER} returns the order, that is the degree of
+the first non-zero term, of the power series object TPS.
+TPS should evaluate to a power series object or an error results. If
+TPS is zero, the identifier {\tt UNDEFINED} is returned.
+
+\subsection{PSSETORDER Operator}
+
+\index{PSSETORDER operator}
+Syntax:
+
+\hspace*{2em} {\tt PSSETORDER}(TPS:{\em power series object},
+ORD:{\em integer}):{\em integer}
+
+The operator {\tt PSSETORDER} sets the order of the power series
+TPS to the value ORD, which should evaluate to an integer. If
+TPS does not evaluate to a power series object, then an error
+occurs. The value returned by this operator is the previous order of
+TPS, or 0 if the order of TPS was undefined.  This
+operator is useful for setting the order of the power series of a
+function defined by a differential equation in cases where the power
+series package is inadequate to determine the order automatically.
+
+
+\subsection{PSDEPVAR Operator}
+
+\index{PSDEPVAR operator}
+Syntax:
+
+\hspace*{2em} {\tt PSDEPVAR}(TPS:{\em power series object})
+:{\em identifier}
+
+The operator {\tt PSDEPVAR} returns the expansion variable of the
+power series object TPS. TPS should evaluate to a power
+series object or an integer, otherwise an error results. If TPS
+is an integer, the identifier {\tt UNDEFINED} is returned.
+
+\subsection{PSEXPANSIONPT operator}
+
+\index{PSEXPANSIONPT operator}
+Syntax:
+
+\hspace*{2em}
+{\tt PSEXPANSIONPT}(TPS:{\em power series object}):{\em algebraic}
+
+The operator {\tt PSEXPANSIONPT} returns the expansion point of the
+power series object TPS. TPS should evaluate to a power
+series object or an integer, otherwise an error results. If TPS
+is integer, the identifier {\tt UNDEFINED} is returned. If the
+expansion is about infinity, the identifier {\tt INFINITY} is
+returned.
+
+\subsection{PSFUNCTION Operator}
+
+\index{PSFUNCTION operator}
+Syntax:
+
+\hspace*{2em}
+{\tt PSFUNCTION}(TPS:{\em power series object}):{\em algebraic}
+
+The operator {\tt PSFUNCTION} returns the function whose expansion
+gave rise to the power series object TPS. TPS should
+evaluate to a power series object or an integer, otherwise an error
+results.
+
+\subsection{PSCHANGEVAR Operator}
+
+\index{PSCHANGEVAR operator}
+Syntax:
+
+\hspace*{2em} {\tt PSCHANGEVAR}(TPS:{\em power series object},
+X:{\em kernel}):{\em power series object}
+
+The operator {\tt PSCHANGEVAR} changes the dependent variable of the
+power series object TPS to the variable X. TPS
+should evaluate to a power series object and X to a kernel,
+otherwise an error results.  Also X should not appear as a
+parameter in TPS. The power series with the new dependent
+variable is returned.
+
+\subsection{PSREVERSE Operator}
+
+\index{PSREVERSE operator}
+Syntax:
+
+\hspace*{2em}
+{\tt PSREVERSE}(TPS:{\em power series object}):{\em power series}
+
+Power series reversion.  The power series TPS is functionally
+inverted.  Four cases arise:
+
+\begin{enumerate}
+\item If the order of the series is 1, then the expansion point of the
+inverted series is 0.
+
+\item If the order is 0 {\em and} if the first order term in TPS
+is non-zero, then the expansion point of the inverted series is taken
+to be the coefficient of the zeroth order term in TPS.
+
+\item If the order is -1 the expansion point of the inverted series
+is the point at infinity.  In all other cases a REDUCE error is
+reported because the series cannot be inverted as a power series.
+Puiseux \index{Puiseux expansion} expansion would be required to
+handle these cases.
+
+\item If the expansion point of TPS is finite it becomes the
+zeroth order term in the inverted series. For expansion about 0 or the
+point at infinity the order of the inverted series is one.
+\end{enumerate}
+
+If TPS is not a power series object after evaluation an error results.
+
+\noindent Here are some examples:
+\begin{verbatim}
+        ps(sin x,x,0);
+        psreverse(ws); % produces series for asin x about x=0.
+        ps(exp x,x,0);
+        psreverse ws; % produces series for log x about x=1.
+        ps(sin(1/x),x,infinity);
+        psreverse(ws); % produces series for 1/asin(x) about x=0.
+\end{verbatim}
+
+\subsection{PSCOMPOSE Operator}
+
+\index{PSCOMPOSE operator}
+Syntax:
+
+\hspace*{2em} {\tt PSCOMPOSE}(TPS1:{\em power series},
+TPS2:{\em power series}):{\em power series}
+
+\index{power series ! composition}
+{\tt PSCOMPOSE} performs power series composition.
+The power series TPS1 and TPS2 are functionally composed.
+That is to say that TPS2 is substituted for the expansion
+variable in TPS1 and the result expressed as a power series. The
+dependent variable and expansion point of the result coincide with
+those of TPS2.  The following conditions apply to power series
+composition:
+
+\begin{enumerate}
+\item If the expansion point of TPS1 is 0 then the order of the
+TPS2 must be at least 1.
+
+\item If the expansion point of TPS1 is finite, it should
+coincide with the coefficient of the zeroth order term in TPS2.
+The order of TPS2 should also be non-negative in this case.
+
+\item If the expansion point of TPS1 is the point at infinity
+then the order of TPS2 must be less than or equal to -1.
+
+\end{enumerate}
+
+If these conditions do not hold the series cannot be composed (with
+the current algorithm terms of the inverted series would involve
+infinite sums) and a REDUCE error occurs.
+
+\noindent Examples of power series composition include the following.
+
+\begin{verbatim}
+    a:=ps(exp y,y,0);  b:=ps(sin x,x,0);
+    pscompose(a,b);
+    % Produces the power series expansion of exp(sin x)
+    % about x=0.
+
+    a:=ps(exp z,z,1); b:=ps(cos x,x,0);
+    pscompose(a,b);
+    % Produces the power series expansion of exp(cos x)
+    % about x=0.
+
+    a:=ps(cos(1/x),x,infinity);  b:=ps(1/sin x,x,0);
+    pscompose(a,b);
+    % Produces the power series expansion of cos(sin x)
+    % about x=0.
+\end{verbatim}
+
+\subsection{PSSUM Operator}
+
+\index{PSSUM operator}
+Syntax:
+
+\begin{tabbing}
+\hspace*{2em} {\tt PSSUM}(\=J:{\em kernel} = LOWLIM:{\em integer},
+COEFF:{\em algebraic}, X:{\em kernel}, \\
+\> ABOUT:{\em algebraic}, POWER:{\em algebraic}):{\em power series}
+\end{tabbing}
+
+The formal power series sum for J from LOWLIM to {\tt INFINITY} of
+
+\begin{verbatim}
+      COEFF*(X-ABOUT)**POWER
+\end{verbatim}
+
+or if ABOUT is given as {\tt INFINITY}
+
+\begin{verbatim}
+      COEFF*(1/X)**POWER
+\end{verbatim}
+
+is constructed and returned. This enables power series whose general
+term is known to be constructed and manipulated using the other
+procedures of the power series package.
+
+J and X should be distinct simple kernels. The algebraics
+ABOUT,  COEFF and POWER should not depend on the
+expansion variable X, similarly the algebraic ABOUT should
+not depend on the summation variable J.  The algebraic POWER should be
+a strictly increasing integer valued function of J for J in the range
+LOWLIM to {\tt INFINITY}.
+
+\begin{verbatim}
+   pssum(n=0,1,x,0,n*n);
+   % Produces the power series summation for n=0 to
+   % infinity of x**(n*n).
+
+   pssum(m=1,(-1)**(m-1)/(2m-1),y,1,2m-1);
+   % Produces the power series expansion of atan(y-1)
+   % about y=1.
+
+   pssum(j=1,-1/j,x,infinity,j);
+   % Produces the power series expansion of log(1-1/x)
+   % about the point at infinity.
+
+   pssum(n=0,1,x,0,2n**2+3n) + pssum(n=1,1,x,0,2n**2-3n);
+   % Produces the power series summation for n=-infinity
+   % to +infinity of x**(2n**2+3n).
+\end{verbatim}
+
+\subsection{PSCOPY Operator}
+
+\index{PSCOPY operator}
+Syntax:
+
+\hspace*{2em} {\tt PSCOPY}(TPS:{\em power series}):{\em power series}
+
+This procedure returns a copy of the power series {\tt TPS}. The copy
+has no shared sub-structures in common with the original series.  This
+enables substitutions to be performed on the series without
+side-effects on previously computed objects. For example:
+
+\begin{verbatim}
+    clear a;
+    b := ps(sin(a*x)), x, 0);
+    b where a => 1;
+\end{verbatim}
+
+will result in {\tt a} being set to 1 in each of the terms of the
+power series and the resulting expressions being simplified. Owing to
+the way power series objects are implemented using Lisp vectors, this
+has the side-effect that the value of {\tt b} is changed.  This may be
+avoided by copying the series with {\tt PSCOPY} before applying the
+substitution, thus:
+
+\begin{verbatim}
+    b := ps(sin(a*x)), x, 0);
+    pscopy b where a => 1;
+\end{verbatim}
+
+
+\subsection{Arithmetic Operations}
+
+\index{power series ! arithmetic}
+As power series objects are domain elements they may be combined
+together in algebraic expressions in algebraic mode of REDUCE in the
+normal way.
+
+For example if A and B are power series objects then the commands
+such as:
+
+\index{+ ! power series} \index{- ! power series}
+\index{/ ! power series} \index{* ! power series}
+\index{** ! power series}
+\begin{verbatim}
+    a*b;
+    a/b;
+    a**2+b**2;
+\end{verbatim}
+
+will produce power series objects representing the product,quotient
+and the sum of the squares of the power series objects A and B
+respectively.
+
+\subsection{Differentiation}
+
+\index{power series ! differentiation}
+If A is a power series object depending on X then the input
+{\tt df(a, x);} will produce the power series expansion of the
+derivative of A with respect to X.
+
+{\em Note} however that currently the input  {\tt int(a, x);} will
+not work as intended; instead one must input
+{\tt ps(int(a, x),x,0);}
+in order to obtain the power series expansion of the integral of
+{\tt a}.
+
+\section{Restrictions and Known Bugs}
+
+If A is a power series object and X is a variable
+which evaluates to itself then currently expressions such as {\tt a*x}
+do not evaluate to a single power series object (although the
+result is formally valid).  Instead use {\tt ps(a*x,x,0)} {\em etc.}.
+
+Similarly expressions such as {\tt sin(A)} where {\tt A} is a PS object
+currently will not be expanded. For example:
+
+\begin{verbatim}
+    a:=ps(1/(1+x),x,0);
+    b:=sin a;
+\end{verbatim}
+
+will not expand {\tt sin(1/(1+x))} as a power series. In fact
+
+\begin{verbatim}
+          SIN(1 - X + X**2 - X**3 + .....)
+\end{verbatim}
+
+will be returned. However,
+
+\begin{verbatim}
+    b:=ps(sin(a),x,0);
+\end{verbatim}
+
+or
+
+\begin{verbatim}
+    b:=ps(sin(1/(1+x)),x,0);
+\end{verbatim}
+
+should work as intended.
+
+The handling of functions with essential singularities is currently
+erratic: usually an error message
+
+\hspace*{2em} {\tt ***** Essential Singularity}
+
+or
+
+\hspace*{2em} {\tt ***** Logarithmic Singularity}
+
+occurs but occasionally a division by
+zero error or some drastic error like (for PSL) binding stack
+overflow may occur.
+
+Mixed mode arithmetic of power series objects with other domain
+elements is quite restricted: only integers and floats can currently
+be converted to power series objects.
+
+The printing of power series currently leaves something to be
+desired: often line-breaks appear in the middle of terms.
+
+There is no simple way to write the results of power series
+calculation to a file and read them back into REDUCE at a later
+stage.
+\end{document}
+