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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:
laplace
(\(\langle \)exp\(\rangle \),\(\langle \)var-s\(\rangle \),\(\langle \)var-t\(\rangle \))invlap
(\(\langle \)exp\(\rangle \),\(\langle \)var-s\(\rangle \),\(\langle \)var-t\(\rangle \))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
intl
(\(\langle \)expr\(\rangle \),\(\langle \)var\(\rangle \),0,\(\langle \)obj.var\(\rangle \))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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