\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}