transform::laplace,
transform::invlaplace -- Laplace and inverse Laplace
transform
Introductiontransform::laplace(f, t, s) computes the Laplace
transform int(f*exp(-s*t), t=0..infinity) of the expression
f = f(t) with respect to the variable t at the
point s.
transform::invlaplace(F, S, T) computes the inverse
Laplace transform of the expression F = F(S) with respect to
the variable S at the point T.
Call(s)transform::laplace(f, t, s)
transform::invlaplace(F, S, T)
Parametersf, F |
- | arithmetical expressions |
t, S |
- | identifiers (the transformation variables) |
s, T |
- | arithmetical expressions (the evaluation points) |
Returnsan arithmetical expression or an unevaluated function call of domain
type transform::laplace or
transform::invlaplace, respectively.
f, F
Details
Example
1The following call produces the Laplace transform as an
expression in the variable s:
>> transform::laplace(exp(-a*t), t, s)
1
-----
a + s
>> transform::invlaplace(%, s, t)
exp(-a t)
Note that the Laplace transform can be evaluated directly at a specific point such as s = 2*a or s = 5:
>> transform::laplace(t^10*exp(-a*t), t, s), transform::laplace(t^10*exp(-a*t), t, 2*a), transform::laplace(t^10*exp(-a*t), t, 1 + PI)
3628800 44800 3628800
---------, --------, --------------
11 11 11
(a + s) 2187 a (a + PI + 1)
Some further examples:
>> transform::laplace(1 + exp(-a*t)*sin(b*t), t, s)
1 b
- + -------------
s 2 2
b + (a + s)
>> transform::invlaplace(1/(s^3 + s^5), s, t)
2
t
cos(t) + -- - 1
2
>> transform::invlaplace(exp(-2*s)/(s^2 + 1) + s/(s^3 + 1), s, t)
exp(-t)
sin(t - 2) heaviside(t - 2) - ------- +
3
/ / 1/2 \ / 1/2 \ \
/ t \ | | t 3 | 1/2 | t 3 | |
exp| - | | cos| ------ | + 3 sin| ------ | |
\ 2 / \ \ 2 / \ 2 / /
-----------------------------------------------
3
Example
2An unevaluated call is returned, if no explicit representation of the transform is found:
>> transform::laplace(exp(-t^3), t, s)
3
transform::laplace(exp(- t ), t, s)
Note that this is not an ordinary expression, but a
domain element of domain type transform::laplace:
>> domtype(%)
transform::laplace
The inverse of the formal transform yields the original expression:
>> transform::invlaplace(%2, s, t)
3
exp(- t )
Example
3The distribution dirac and the Heaviside function
heaviside are
handled:
>> transform::laplace(dirac(t - 3), t, s)
exp(-3 s)
>> transform::invlaplace(1, s, t)
dirac(t)
>> transform::laplace(heaviside(t - PI), t, s)
exp(-s PI)
----------
s
Example
4The Laplace transform of a function is related to the Laplace transform of its derivative:
>> transform::laplace(diff(f(t), t), t, s)
s transform::laplace(f(t), t, s) - f(0)
transform::invlaplace used to be called
transform::ilaplace.