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r38/packages/residue/residue.tex
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2011-09-02 18:13:33
on branch master
— Some historical releases purely for archival purposes
git-svn-id: https://svn.code.sf.net/p/reduce-algebra/code/trunk/historical@1375 2bfe0521-f11c-4a00-b80e-6202646ff360 (user: arthurcnorman@users.sourceforge.net, size: 3497) [annotate] [blame] [check-ins using]
\documentstyle[11pt,reduce]{article} \title{{\bf RESIDUE Package for {\tt REDUCE}}} \author{Wolfram Koepf\\ email: {\tt Koepf@zib.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}