20.31 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: [Kaz87].

Some hints on how to use to use this package:
Syntax:

where \(\langle \)exp\(\rangle \) is the expression to be transformed, \(\langle \)var-s\(\rangle \) is the source variable (in most cases \(\langle \)exp\(\rangle \) depends explicitly of this variable) and \(\langle \)var-t\(\rangle \) is the target variable. If \(\langle \)var-t\(\rangle \) is omitted, the above operators use 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 \(e^{\tilde \ x}\) are converted by invlap into hyperbolic functions sinh and cosh,

ltrig:

If on, expressions \(e^{\tilde \ x}\) 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(Euler_Gamma * il!&)/il!&,

         invlap(log(Euler_Gamma * ~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

which means “Integral of \(\langle \)expr\(\rangle \) w.r.t. \(\langle \)var\(\rangle \) taken from 0 to \(\langle \)obj.var\(\rangle \)”, e.g. 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.

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