Artifact 6c176c18f5263ac3230855b2e03739a67e2916749a54c02a2e4333d5eebf9d71:
- Executable file
r38/packages/laplace/laplace.tex
— part of check-in
[f2fda60abd]
at
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: 2882) [annotate] [blame] [check-ins using] [more...]
\documentclass{article} \usepackage[dvipdfm]{graphicx} \usepackage[dvipdfm]{color} \usepackage[dvipdfm]{hyperref} \setlength{\parindent}{0cm} \title{SOFIA LAPLACE AND INVERSE LAPLACE TRANSFORM PACKAGE} \author{C. Kazasov\and M. Spiridonova \and V. Tomov} \date{} \begin{document} \maketitle \begin{center} \begin{tabular}{lp{10cm}} Reference: & {\bf Christomir Kazasov}, Laplace Transformations in REDUCE 3, Proc. Eurocal '87, Lecture Notes in Comp. Sci., Springer-Verlag (1987) 132-133. \end{tabular} \end{center} \ \\ \ \\ Some hints on how to use to use this package: \\ \ \\ Syntax: \\ \ \\ {\tt LAPLACE($<exp>,<var-s>,<var-t>$ }) \\ \ \\ {\tt INVLAP($<exp>,<var-s>,<var-t>$)} \\ \ \\ where $<exp>$ is the expression to be transformed, $<var-s>$ is the source variable (in most cases $<exp>$ depends explicitly of this variable) and $<var-t>$ is the target variable. If $<var-t>$ is omitted, the package uses an internal variable lp!\& or il!\&, respectively. \\ \ \\ The following switches can be used to control the transformations: \\ \begin{center} \begin{tabular}{lp{10cm}} {\tt lmon}: & If on, sin, cos, sinh and cosh are converted by {\tt LAPLACE} into exponentials, \\ {\tt lhyp}: & If on, expressions $e^{\tilde{}x}$ are converted by {\tt INVLAP} into hyperbolic functions sinh and cosh, \\ {\tt ltrig}: & If on, expressions $e^{\tilde{}x}$ are converted by {\tt INVLAP} into trigonometric functions sin and cos. \\ \end{tabular} \end{center} \ \\ The system can be extended by adding Laplace transformation rules for single functions by rules or rule sets.~ In such a rule the source variable MUST be free, the target variable MUST be il!\& for {\tt LAPLACE} and lp!\& for {\tt INVLAP} and the third parameter should be omitted.~ Also rules for transforming derivatives are entered in such a form. \\ \pagebreak {\bf Examples:} \begin{verbatim} let {laplace(log(~x),x) => -log(gam * il!&)/il!&, invlap(log(gam * ~x)/x,x) => -log(lp!&)}; operator f; let{ laplace(df(f(~x),x),x) => il!&*laplace(f(x),x) - sub(x=0,f(x)), laplace(df(f(~x),x,~n),x) => il!&**n*laplace(f(x),x) - for i:=n-1 step -1 until 0 sum sub(x=0, df(f(x),x,n-1-i)) * il!&**i when fixp n, laplace(f(~x),x) = f(il!&) }; \end{verbatim} Remarks about some functions: \\ \ \\ The DELTA and GAMMA functions are known. \\ ONE is the name of the unit step function. \\ INTL is a parametrized integral function \begin{center} {\tt intl($<expr>,<var>,0,<obj.var>$)} \end{center} which means \char`\"{}Integral of $<expr>$ wrt.~ $<var>$ taken from 0 to $<obj.var>$\char`\"{}, e.g. {\tt intl($2{*}y^2,y,0,x$)} which is formally a function in $x$. \ \\ \ \\ We recommend reading the file LAPLACE.TST for a further introduction. \end{document}