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\chapter[LAPLACE: Laplace transforms etc.]%
        {LAPLACE: Laplace and inverse Laplace transforms}
\label{LAPLACE}
\typeout{{LAPLACE: Laplace and inverse Laplace transforms}}

{\footnotesize
\begin{center}
C. Kazasov, M. Spiridonova, V. Tomov \\
Sofia, Bulgaria %%\\[0.05in]
%%e--mail: 
\end{center}
}
\ttindex{LAPLACE}

The LAPLACE package provides both Laplace Transforms and Inverse
Laplace Transforms, with the two operators

\noindent{\tt LAPLACE(exp, s\_var, t\_var)}\ttindex{LAPLACE} \\
{\tt INVLAP(exp, s\_var, t\_var)}\ttindex{INVLAP}

The action is to transform the expression from the {\tt s\_var} or
source variable into the {\tt t\_var} or target variable.  If {\tt
t\_var} is omitted, the package uses an internal variable {\tt lp!\&} or
{\tt il!\&} respectively. 

Three switches control the transformations.  If {\tt
lmon}\ttindex{lpon} is on then sine, cosine, hyperbolic sine and
hyperbolic cosines are converted by LAPLACE into exponentials.  If
{\tt lhyp} is on then exponential functions are converted into
hyperbolic form.   The last switch {\tt ltrig}\ttindex{ltrig} has the
same effect except it uses trigonometric functions.

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 {\bf must} be free, the target variable {\bf must} be {\tt
il!\&} for LAPLACE and {\tt lp!\&} for INVLAP, with the third parameter
omitted.  Also rules for transforming derivatives are entered in such
a form.  For example 
\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}

The LAPLACE system knows about the functions {\tt DELTA} and {\tt
GAMMA}, and used the operator {\tt ONE} for the unit step function and
{\tt INTL} stands for the parameterised integral function, for
instance  {\tt intl(2*y**2,y,0,x)} stands for $\int^x_0 2 y^2 dx$.

\begin{verbatim}
load_package laplace;

laplace(sin(17*x),x,p);

    17
----------
  2
 p  + 289

on lmon;

laplace(-1/4*e**(a*x)*(x-k)**(-1/2), x, p);

     1            a*k
  - ---*sqrt(pi)*e
     4
----------------------
  k*p
 e   *sqrt( - a + p)

invlap(c/((p-a)*(p-b)), p, t);

     a*t    b*t
 c*(e    - e   )
-----------------
      a - b

invlap(p**(-7/3), p, t);

    1/3
   t   *t
------------
        7
 gamma(---)
        3
\end{verbatim}