transform::laplace,
transform::invlaplace
-- Laplace and inverse Laplace
transformtransform::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.
transform::laplace(f, t, s)
transform::invlaplace(F, S, T)
f, F |
- | arithmetical expressions |
t, S |
- | identifiers (the transformation variables) |
s, T |
- | arithmetical expressions (the evaluation points) |
an arithmetical expression or an unevaluated function call of domain
type transform::laplace
or
transform::invlaplace
, respectively.
f
, F
The 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
An 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 )
The 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
The 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
.