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### 16.34 LAPLACE: Laplace transforms

This package can calculate ordinary and inverse Laplace transforms of expressions. Documentation is in plain text.

Authors: C. Kazasov, M. Spiridonova, V. Tomov.

 Reference: Christomir Kazasov, Laplace Transformations in REDUCE 3, Proc. Eurocal ’87, Lecture Notes in Comp. Sci., Springer-Verlag (1987) 132-133.

Some hints on how to use to use this package:

Syntax:

LAPLACE(< exp >,< var - s >,< var - t > )

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:

 lmon: If on, sin, cos, sinh and cosh are converted by LAPLACE into exponentials, lhyp: If on, expressions e˜x are converted by INVLAP into hyperbolic functions sinh and cosh, ltrig: If on, expressions e˜x are converted by INVLAP into trigonometric functions sin and cos.

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 LAPLACE and lp!& for INVLAP and the third parameter should be omitted.  Also rules for transforming derivatives are entered in such a form.

Examples:

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!&)

};