@@ -1,189 +1,189 @@
-\documentstyle[11pt,reduce]{article}
-\title{{\bf RESIDUE Package for {\tt REDUCE}}}
-\author{Wolfram Koepf\\ email: {\tt Koepf@zib-berlin.de}}
-\date{April 1995 : ZIB Berlin}
-\begin{document}
-\maketitle
-\def\Res{\mathop{\rm Res}\limits}
-\newcommand{\C}{{\rm {\mbox{C{\llap{{\vrule height1.52ex}\kern.4em}}}}}}
-
-This package supports the calculation of residues. The residue
-$\Res_{z=a} f(z)$ of a function $f(z)$ at the point $a\in\C$ is defined
-as
-\[
-\Res_{z=a} f(z)=
-\frac{1}{2 \pi i}\oint f(z)\,dz
-\;,
-\]
-with integration along a closed curve around $z=a$ with winding number 1.
-
-If $f(z)$ is given by a Laurent series development at $z=a$
-\[
-f(z)=\sum_{k=-\infty}^\infty a_k\,(z-a)^k
-\;,
-\]
-then
-\begin{equation}
-\Res\limits_{z=a} f(z)=a_{-1}
-\;.
-\label{eq:Laurent}
-\end{equation}
-If $a=\infty$, one defines on the other hand
-\begin{equation}
-\Res\limits_{z=\infty} f(z)=-a_{-1}
-\label{eq:Laurent2}
-\end{equation}
-for given Laurent representation
-\[
-f(z)=\sum_{k=-\infty}^\infty a_k\,\frac{1}{z^k}
-\;.
-\]
-
-The package is loaded by the statement
-
-\begin{verbatim}
-1: load residue;
-\end{verbatim}
-
-It contains two REDUCE operators:
-
-\begin{itemize}
-\item
-{\tt residue(f,z,a)} determines the residue of $f$ at the point $z=a$
-if $f$ is meromorphic at $z=a$. The calculation of residues at essential
-singularities of $f$ is not supported.
-\item
-{\tt poleorder(f,z,a)} determines the pole order of $f$ at the point $z=a$
-if $f$ is meromorphic at $z=a$.
-\end{itemize}
-Note that both functions use the {\tt taylor} package in
-connection with representations (\ref{eq:Laurent})--(\ref{eq:Laurent2}).
-
-Here are some examples:
-
-\begin{verbatim}
-2: residue(x/(x^2-2),x,sqrt(2));
-
- 1
----
- 2
-
-3: poleorder(x/(x^2-2),x,sqrt(2));
-
-1
-
-4: residue(sin(x)/(x^2-2),x,sqrt(2));
-
- sqrt(2)*sin(sqrt(2))
-----------------------
-          4
-
-5: poleorder(sin(x)/(x^2-2),x,sqrt(2));
-
-1
-
-6: residue(1/(x-1)^m/(x-2)^2,x,2);
-
- - m
-
-7: poleorder(1/(x-1)/(x-2)^2,x,2);
-
-2
-
-8: residue(sin(x)/x^2,x,0);
-
-1
-
-9: poleorder(sin(x)/x^2,x,0);
-
-1
-
-10: residue((1+x^2)/(1-x^2),x,1);
-
--1
-
-11: poleorder((1+x^2)/(1-x^2),x,1);
-
-1
-
-12: residue((1+x^2)/(1-x^2),x,-1);
-
-1
-
-13: poleorder((1+x^2)/(1-x^2),x,-1);
-
-1
-
-14: residue(tan(x),x,pi/2);
-
--1
-
-15: poleorder(tan(x),x,pi/2);
-
-1
-
-16: residue((x^n-y^n)/(x-y),x,y);
-
-0
-
-17: poleorder((x^n-y^n)/(x-y),x,y);
-
-0
-
-18: residue((x^n-y^n)/(x-y)^2,x,y);
-
-  n
- y *n
-------
-  y
-
-19: poleorder((x^n-y^n)/(x-y)^2,x,y);
-
-1
-
-20: residue(tan(x)/sec(x-pi/2)+1/cos(x),x,pi/2);
-
--2
-
-21: poleorder(tan(x)/sec(x-pi/2)+1/cos(x),x,pi/2);
-
-1
-
-22: for k:=1:2 sum residue((a+b*x+c*x^2)/(d+e*x+f*x^2),x,
-    part(part(solve(d+e*x+f*x^2,x),k),2));
-
- b*f - c*e
------------
-     2
-    f
-
-23: residue(x^3/sin(1/x)^2,x,infinity);
-
-  - 1
-------
-  15
-
-24: residue(x^3*sin(1/x)^2,x,infinity);
-
--1
-\end{verbatim}
-\iffalse
-7: for k:=1:3 sum
-7: residue((a+b*x+c*x^2+d*x^3)/(e+f*x+g*x^2+h*x^3),x,
-7: part(part(solve(e+f*x+g*x^2+h*x^3,x),k),2));
-
-
-***** CATASTROPHIC ERROR *****
-("gcdf failed" (root_of (plus e (times f x_) (times g (expt x_ 2)) (times h (
-expt x_ 3))) x_ tag_2) (times (root_of (plus e (times f x_) (times g (expt
-x_ 2)) (times h (expt x_ 3))) x_ tag_2) h))
-
-
-
-***** Please send output and input listing to A. C. Hearn
-\fi
-Note that the residues of factorial and $\Gamma$ function terms are
-not yet supported.
-
-\end{document}
-
+\documentstyle[11pt,reduce]{article}
+\title{{\bf RESIDUE Package for {\tt REDUCE}}}
+\author{Wolfram Koepf\\ email: {\tt Koepf@zib-berlin.de}}
+\date{April 1995 : ZIB Berlin}
+\begin{document}
+\maketitle
+\def\Res{\mathop{\rm Res}\limits}
+\newcommand{\C}{{\rm {\mbox{C{\llap{{\vrule height1.52ex}\kern.4em}}}}}}
+
+This package supports the calculation of residues. The residue
+$\Res_{z=a} f(z)$ of a function $f(z)$ at the point $a\in\C$ is defined
+as
+\[
+\Res_{z=a} f(z)=
+\frac{1}{2 \pi i}\oint f(z)\,dz
+\;,
+\]
+with integration along a closed curve around $z=a$ with winding number 1.
+
+If $f(z)$ is given by a Laurent series development at $z=a$
+\[
+f(z)=\sum_{k=-\infty}^\infty a_k\,(z-a)^k
+\;,
+\]
+then
+\begin{equation}
+\Res\limits_{z=a} f(z)=a_{-1}
+\;.
+\label{eq:Laurent}
+\end{equation}
+If $a=\infty$, one defines on the other hand
+\begin{equation}
+\Res\limits_{z=\infty} f(z)=-a_{-1}
+\label{eq:Laurent2}
+\end{equation}
+for given Laurent representation
+\[
+f(z)=\sum_{k=-\infty}^\infty a_k\,\frac{1}{z^k}
+\;.
+\]
+
+The package is loaded by the statement
+
+\begin{verbatim}
+1: load residue;
+\end{verbatim}
+
+It contains two REDUCE operators:
+
+\begin{itemize}
+\item
+{\tt residue(f,z,a)} determines the residue of $f$ at the point $z=a$
+if $f$ is meromorphic at $z=a$. The calculation of residues at essential
+singularities of $f$ is not supported.
+\item
+{\tt poleorder(f,z,a)} determines the pole order of $f$ at the point $z=a$
+if $f$ is meromorphic at $z=a$.
+\end{itemize}
+Note that both functions use the {\tt taylor} package in
+connection with representations (\ref{eq:Laurent})--(\ref{eq:Laurent2}).
+
+Here are some examples:
+
+\begin{verbatim}
+2: residue(x/(x^2-2),x,sqrt(2));
+
+ 1
+---
+ 2
+
+3: poleorder(x/(x^2-2),x,sqrt(2));
+
+1
+
+4: residue(sin(x)/(x^2-2),x,sqrt(2));
+
+ sqrt(2)*sin(sqrt(2))
+----------------------
+          4
+
+5: poleorder(sin(x)/(x^2-2),x,sqrt(2));
+
+1
+
+6: residue(1/(x-1)^m/(x-2)^2,x,2);
+
+ - m
+
+7: poleorder(1/(x-1)/(x-2)^2,x,2);
+
+2
+
+8: residue(sin(x)/x^2,x,0);
+
+1
+
+9: poleorder(sin(x)/x^2,x,0);
+
+1
+
+10: residue((1+x^2)/(1-x^2),x,1);
+
+-1
+
+11: poleorder((1+x^2)/(1-x^2),x,1);
+
+1
+
+12: residue((1+x^2)/(1-x^2),x,-1);
+
+1
+
+13: poleorder((1+x^2)/(1-x^2),x,-1);
+
+1
+
+14: residue(tan(x),x,pi/2);
+
+-1
+
+15: poleorder(tan(x),x,pi/2);
+
+1
+
+16: residue((x^n-y^n)/(x-y),x,y);
+
+0
+
+17: poleorder((x^n-y^n)/(x-y),x,y);
+
+0
+
+18: residue((x^n-y^n)/(x-y)^2,x,y);
+
+  n
+ y *n
+------
+  y
+
+19: poleorder((x^n-y^n)/(x-y)^2,x,y);
+
+1
+
+20: residue(tan(x)/sec(x-pi/2)+1/cos(x),x,pi/2);
+
+-2
+
+21: poleorder(tan(x)/sec(x-pi/2)+1/cos(x),x,pi/2);
+
+1
+
+22: for k:=1:2 sum residue((a+b*x+c*x^2)/(d+e*x+f*x^2),x,
+    part(part(solve(d+e*x+f*x^2,x),k),2));
+
+ b*f - c*e
+-----------
+     2
+    f
+
+23: residue(x^3/sin(1/x)^2,x,infinity);
+
+  - 1
+------
+  15
+
+24: residue(x^3*sin(1/x)^2,x,infinity);
+
+-1
+\end{verbatim}
+\iffalse
+7: for k:=1:3 sum
+7: residue((a+b*x+c*x^2+d*x^3)/(e+f*x+g*x^2+h*x^3),x,
+7: part(part(solve(e+f*x+g*x^2+h*x^3,x),k),2));
+
+
+***** CATASTROPHIC ERROR *****
+("gcdf failed" (root_of (plus e (times f x_) (times g (expt x_ 2)) (times h (
+expt x_ 3))) x_ tag_2) (times (root_of (plus e (times f x_) (times g (expt
+x_ 2)) (times h (expt x_ 3))) x_ tag_2) h))
+
+
+
+***** Please send output and input listing to A. C. Hearn
+\fi
+Note that the residues of factorial and $\Gamma$ function terms are
+not yet supported.
+
+\end{document}
+