@@ -1,188 +1,188 @@
-\documentstyle[11pt,reduce]{article}
-\title{{\tt FPS}\\
-A Package for the\\
-Automatic Calculation \\
-of Formal Power Series}
-\date{}
-\author{Wolfram Koepf\\
-        ZIB Berlin \\
-        Email: {\tt  Koepf@ZIB-Berlin.de}
-\\
-\\
-        Present \REDUCE{} form by \\
-        Winfried Neun \\
-        ZIB Berlin \\
-        Email: {\tt Neun@ZIB-Berlin.de}}
-\begin{document}
-\maketitle
-
-\section{Introduction}
-This package can expand functions of certain type into
-their corresponding Laurent-Puiseux series as a sum of terms of the form
-\begin{displaymath}
-\sum_{k=0}^{\infty} a_{k} (x-x_{0})^{m k/n + s}
-\end{displaymath}
-where $m$ is the `symmetry number', $s$ is the `shift number',
-$n$ is the `Puiseux number',
-and $x_0$ is the `point of development'. The following types are
-supported:
-\begin{itemize}
-\item
-{\bf functions of `rational type'}, which are either rational or have a
-rational derivative of some order;
-\item
-{\bf functions of `hypergeometric type'} where $a(k+m)/a(k)$ is a rational
-function for some integer $m$;
-\item
-{\bf functions of `explike type'} which satisfy a linear homogeneous
-differential equation with constant coefficients.
-\end{itemize}
-
-The FPS package is an implementation of the method
-presented in \cite{Koepf:92}. The implementations of this package
-for {\sc Maple} (by D.\ Gruntz) and {\sc Mathematica} (by W.\ Koepf)
-served as guidelines for this one.
-
-Numerous examples can be found in \cite{Koepf:93a}--\cite{Koepf:93b},
-most of which are contained in the test file {\tt fps.tst}. Many
-more examples can be found in the extensive bibliography of Hansen \cite{Han}.
-
-
-\section{\REDUCE{} operator {\tt FPS}}
-The FPS Package must be loaded first by:
-\begin{verbatim}
-load FPS;
-\end{verbatim}
-{\tt FPS(f,x,x0)} tries to find a formal power
-series expansion for {\tt f} with respect to the variable {\tt x}
-at the point of development {\tt x0}.
-It also works for formal Laurent (negative exponents) and Puiseux series
-(fractional exponents). If the third
-argument is omitted, then {\tt x0:=0} is assumed.
-
-Examples: {\tt FPS(asin(x)\verb+^+2,x)} results in
-\begin{verbatim}
-
-         2*k  2*k             2  2
-        x   *2   *factorial(k) *x
-infsum(----------------------------,k,0,infinity)
-        factorial(2*k + 1)*(k + 1)
-\end{verbatim}
-{\tt FPS(sin x,x,pi)} gives
-\begin{verbatim}
-                   2*k       k
-        ( - pi + x)   *( - 1) *( - pi + x)
-infsum(------------------------------------,k,0,infinity)
-                factorial(2*k + 1)
-\end{verbatim}
-and {\tt FPS(sqrt(2-x\verb+^+2),x)} yields
-\begin{verbatim}
-            2*k
-         - x   *sqrt(2)*factorial(2*k)
-infsum(--------------------------------,k,0,infinity)
-           k             2
-          8 *factorial(k) *(2*k - 1)
-\end{verbatim}
-Note: The result contains one or more {\tt infsum} terms such that it does
-not interfere with the {\REDUCE} operator {\tt sum}. In graphical oriented
-REDUCE interfaces this operator results in the usual $\sum$ notation.
-
-If possible, the output is given using factorials. In some cases, the
-use of the Pochhammer symbol {\tt pochhammer(a,k)}$:=a(a+1)\cdots(a+k-1)$
-is necessary.
-
-The operator {\tt FPS} uses the operator {\tt SimpleDE} of the next section.
-
-If an error message of type
-\begin{verbatim}
-Could not find the limit of:
-\end{verbatim}
-occurs, you can set the corresponding limit yourself and try a
-recalculation. In the computation of {\tt FPS(atan(cot(x)),x,0)},
-REDUCE is not able to find the value for the limit
-{\tt limit(atan(cot(x)),x,0)} since the {\tt atan} function is multi-valued.
-One can choose the branch of {\tt atan} such that this limit equals
-$\pi/2$ so that we may set
-\begin{verbatim}
-let limit(atan(cot(~x)),x,0)=>pi/2;
-\end{verbatim}
-and a recalculation of {\tt FPS(atan(cot(x)),x,0)}
-yields the output {\tt pi - 2*x} which is
-the correct local series representation.
-
-\section{\REDUCE{} operator {\tt SimpleDE}}
-
-{\tt SimpleDE(f,x)} tries to find a homogeneous linear differential
-equation with polynomial coefficients for $f$ with respect to $x$.
-Make sure that $y$ is not a used variable.
-The setting {\tt factor df;} is recommended to receive a nicer output form.
-
-Examples: {\tt SimpleDE(asin(x)\verb+^+2,x)} then results in
-\begin{verbatim}
-            2
-df(y,x,3)*(x  - 1) + 3*df(y,x,2)*x + df(y,x)
-\end{verbatim}
-{\tt SimpleDE(exp(x\verb+^+(1/3)),x)} gives
-\begin{verbatim}
-              2
-27*df(y,x,3)*x  + 54*df(y,x,2)*x + 6*df(y,x) - y
-\end{verbatim}
-and {\tt SimpleDE(sqrt(2-x\verb+^+2),x)} yields
-\begin{verbatim}
-          2
-df(y,x)*(x  - 2) - x*y
-\end{verbatim}
-The depth for the search of a differential equation for {\tt f} is
-controlled by the variable {\tt fps\verb+_+search\verb+_+depth};
-higher values for {\tt fps\verb+_+search\verb+_+depth}
-will increase the chance to find the solution, but increases the
-complexity as well. The default value for {\tt fps\verb+_+search\verb+_+depth}
-is 5. For {\tt FPS(sin(x\verb+^+(1/3)),x)}, or
-{\tt SimpleDE(sin(x\verb+^+(1/3)),x)} e.\ g., a setting
-{\tt fps\verb+_+search\verb+_+depth:=6} is necessary.
-
-The output of the FPS package can be influenced by the
-switch {\tt tracefps}. Setting {\tt on tracefps} causes various
-prints of intermediate results.
-
-\section{Problems in the current version}
-The handling of logarithmic singularities is not yet implemented.
-
-The rational type implementation is not yet complete.
-
-The support of special functions \cite{Koepf:94}
-will be part of the next version.
-
-\begin{thebibliography}{9}
-
-\bibitem{Han}
-E.\ R. Hansen, {\em A table of series and products.}
-Prentice-Hall, Englewood Cliffs, NJ, 1975.
-
-\bibitem{Koepf:92} Wolfram Koepf,
-{\em Power Series in Computer Algebra},
-J.\ Symbolic Computation 13 (1992)
-
-\bibitem{Koepf:93a} Wolfram Koepf,
-{\em Examples for the Algorithmic Calculation of Formal
-Puiseux, Laurent and Power series},
-SIGSAM Bulletin 27, 1993, 20-32.
-
-\bibitem{Koepf:93b} Wolfram Koepf,
-{\em Algorithmic development of power series.} In:
-Artificial intelligence and symbolic mathematical computing,
-ed.\ by J.\ Calmet and J.\ A.\ Campbell,
-International Conference AISMC-1, Karlsruhe, Germany, August 1992, Proceedings,
-Lecture Notes in Computer Science {\bf 737}, Springer-Verlag,
-Berlin--Heidelberg, 1993, 195--213.
-
-\bibitem{Koepf:94} Wolfram Koepf,
-{\em Algorithmic work with orthogonal polynomials and special functions.}
-Konrad-Zuse-Zentrum Berlin (ZIB), Preprint SC 94-5, 1994.
-
-\end{thebibliography}
-
-\end{document}
-
-
-
+\documentstyle[11pt,reduce]{article}
+\title{{\tt FPS}\\
+A Package for the\\
+Automatic Calculation \\
+of Formal Power Series}
+\date{}
+\author{Wolfram Koepf\\
+        ZIB Berlin \\
+        Email: {\tt  Koepf@ZIB-Berlin.de}
+\\
+\\
+        Present \REDUCE{} form by \\
+        Winfried Neun \\
+        ZIB Berlin \\
+        Email: {\tt Neun@ZIB-Berlin.de}}
+\begin{document}
+\maketitle
+
+\section{Introduction}
+This package can expand functions of certain type into
+their corresponding Laurent-Puiseux series as a sum of terms of the form
+\begin{displaymath}
+\sum_{k=0}^{\infty} a_{k} (x-x_{0})^{m k/n + s}
+\end{displaymath}
+where $m$ is the `symmetry number', $s$ is the `shift number',
+$n$ is the `Puiseux number',
+and $x_0$ is the `point of development'. The following types are
+supported:
+\begin{itemize}
+\item
+{\bf functions of `rational type'}, which are either rational or have a
+rational derivative of some order;
+\item
+{\bf functions of `hypergeometric type'} where $a(k+m)/a(k)$ is a rational
+function for some integer $m$;
+\item
+{\bf functions of `explike type'} which satisfy a linear homogeneous
+differential equation with constant coefficients.
+\end{itemize}
+
+The FPS package is an implementation of the method
+presented in \cite{Koepf:92}. The implementations of this package
+for {\sc Maple} (by D.\ Gruntz) and {\sc Mathematica} (by W.\ Koepf)
+served as guidelines for this one.
+
+Numerous examples can be found in \cite{Koepf:93a}--\cite{Koepf:93b},
+most of which are contained in the test file {\tt fps.tst}. Many
+more examples can be found in the extensive bibliography of Hansen \cite{Han}.
+
+
+\section{\REDUCE{} operator {\tt FPS}}
+The FPS Package must be loaded first by:
+\begin{verbatim}
+load FPS;
+\end{verbatim}
+{\tt FPS(f,x,x0)} tries to find a formal power
+series expansion for {\tt f} with respect to the variable {\tt x}
+at the point of development {\tt x0}.
+It also works for formal Laurent (negative exponents) and Puiseux series
+(fractional exponents). If the third
+argument is omitted, then {\tt x0:=0} is assumed.
+
+Examples: {\tt FPS(asin(x)\verb+^+2,x)} results in
+\begin{verbatim}
+
+         2*k  2*k             2  2
+        x   *2   *factorial(k) *x
+infsum(----------------------------,k,0,infinity)
+        factorial(2*k + 1)*(k + 1)
+\end{verbatim}
+{\tt FPS(sin x,x,pi)} gives
+\begin{verbatim}
+                   2*k       k
+        ( - pi + x)   *( - 1) *( - pi + x)
+infsum(------------------------------------,k,0,infinity)
+                factorial(2*k + 1)
+\end{verbatim}
+and {\tt FPS(sqrt(2-x\verb+^+2),x)} yields
+\begin{verbatim}
+            2*k
+         - x   *sqrt(2)*factorial(2*k)
+infsum(--------------------------------,k,0,infinity)
+           k             2
+          8 *factorial(k) *(2*k - 1)
+\end{verbatim}
+Note: The result contains one or more {\tt infsum} terms such that it does
+not interfere with the {\REDUCE} operator {\tt sum}. In graphical oriented
+REDUCE interfaces this operator results in the usual $\sum$ notation.
+
+If possible, the output is given using factorials. In some cases, the
+use of the Pochhammer symbol {\tt pochhammer(a,k)}$:=a(a+1)\cdots(a+k-1)$
+is necessary.
+
+The operator {\tt FPS} uses the operator {\tt SimpleDE} of the next section.
+
+If an error message of type
+\begin{verbatim}
+Could not find the limit of:
+\end{verbatim}
+occurs, you can set the corresponding limit yourself and try a
+recalculation. In the computation of {\tt FPS(atan(cot(x)),x,0)},
+REDUCE is not able to find the value for the limit
+{\tt limit(atan(cot(x)),x,0)} since the {\tt atan} function is multi-valued.
+One can choose the branch of {\tt atan} such that this limit equals
+$\pi/2$ so that we may set
+\begin{verbatim}
+let limit(atan(cot(~x)),x,0)=>pi/2;
+\end{verbatim}
+and a recalculation of {\tt FPS(atan(cot(x)),x,0)}
+yields the output {\tt pi - 2*x} which is
+the correct local series representation.
+
+\section{\REDUCE{} operator {\tt SimpleDE}}
+
+{\tt SimpleDE(f,x)} tries to find a homogeneous linear differential
+equation with polynomial coefficients for $f$ with respect to $x$.
+Make sure that $y$ is not a used variable.
+The setting {\tt factor df;} is recommended to receive a nicer output form.
+
+Examples: {\tt SimpleDE(asin(x)\verb+^+2,x)} then results in
+\begin{verbatim}
+            2
+df(y,x,3)*(x  - 1) + 3*df(y,x,2)*x + df(y,x)
+\end{verbatim}
+{\tt SimpleDE(exp(x\verb+^+(1/3)),x)} gives
+\begin{verbatim}
+              2
+27*df(y,x,3)*x  + 54*df(y,x,2)*x + 6*df(y,x) - y
+\end{verbatim}
+and {\tt SimpleDE(sqrt(2-x\verb+^+2),x)} yields
+\begin{verbatim}
+          2
+df(y,x)*(x  - 2) - x*y
+\end{verbatim}
+The depth for the search of a differential equation for {\tt f} is
+controlled by the variable {\tt fps\verb+_+search\verb+_+depth};
+higher values for {\tt fps\verb+_+search\verb+_+depth}
+will increase the chance to find the solution, but increases the
+complexity as well. The default value for {\tt fps\verb+_+search\verb+_+depth}
+is 5. For {\tt FPS(sin(x\verb+^+(1/3)),x)}, or
+{\tt SimpleDE(sin(x\verb+^+(1/3)),x)} e.\ g., a setting
+{\tt fps\verb+_+search\verb+_+depth:=6} is necessary.
+
+The output of the FPS package can be influenced by the
+switch {\tt tracefps}. Setting {\tt on tracefps} causes various
+prints of intermediate results.
+
+\section{Problems in the current version}
+The handling of logarithmic singularities is not yet implemented.
+
+The rational type implementation is not yet complete.
+
+The support of special functions \cite{Koepf:94}
+will be part of the next version.
+
+\begin{thebibliography}{9}
+
+\bibitem{Han}
+E.\ R. Hansen, {\em A table of series and products.}
+Prentice-Hall, Englewood Cliffs, NJ, 1975.
+
+\bibitem{Koepf:92} Wolfram Koepf,
+{\em Power Series in Computer Algebra},
+J.\ Symbolic Computation 13 (1992)
+
+\bibitem{Koepf:93a} Wolfram Koepf,
+{\em Examples for the Algorithmic Calculation of Formal
+Puiseux, Laurent and Power series},
+SIGSAM Bulletin 27, 1993, 20-32.
+
+\bibitem{Koepf:93b} Wolfram Koepf,
+{\em Algorithmic development of power series.} In:
+Artificial intelligence and symbolic mathematical computing,
+ed.\ by J.\ Calmet and J.\ A.\ Campbell,
+International Conference AISMC-1, Karlsruhe, Germany, August 1992, Proceedings,
+Lecture Notes in Computer Science {\bf 737}, Springer-Verlag,
+Berlin--Heidelberg, 1993, 195--213.
+
+\bibitem{Koepf:94} Wolfram Koepf,
+{\em Algorithmic work with orthogonal polynomials and special functions.}
+Konrad-Zuse-Zentrum Berlin (ZIB), Preprint SC 94-5, 1994.
+
+\end{thebibliography}
+
+\end{document}
+
+
+