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\title{SOFIA LAPLACE AND INVERSE LAPLACE TRANSFORM PACKAGE}
\author{C. Kazasov\and M. Spiridonova \and V. Tomov}
\date{}
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\maketitle
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Reference: & {\bf Christomir Kazasov}, Laplace Transformations in REDUCE 3, Proc.
Eurocal '87, Lecture Notes in Comp. Sci., Springer-Verlag
(1987) 132-133.
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\ \\
\ \\
Some hints on how to use to use this package: \\
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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: \\
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{\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. \\
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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: \\
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The DELTA and GAMMA functions are known. \\
ONE is the name of the unit step function. \\
INTL is a parametrized integral function
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{\tt intl($<expr>,<var>,0,<obj.var>$)}
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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 $x$.
\ \\
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We recommend reading the file LAPLACE.TST for a further introduction.
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