LAPLACE: Laplace transforms
Date: 2020-10-07 18:10:00

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### 16.35 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!&)
};

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

intl(< expr >,< var >,0,< obj.var >)

which means "Integral of < expr > wrt. < var > taken from 0 to < obj.var >", e.g. intl(2∗y2,y,0,x) which is formally a function in x.
We recommend reading the ﬁle LAPLACE.TST for a further introduction.

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