Laplace transform
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laplace(f
, t
, s
)
laplace(f, t, s)
computes the Laplace transform
of the expression f = f(t)
with respect to the
variable t at
the point s.
The Laplace transform is defined as follows:
.
If laplace
cannot find an explicit representation
of the transform, it returns an unevaluated function call. See Example 3.
If f
is a matrix, laplace
applies
the Laplace transform to all components of the matrix.
To compute the inverse Laplace transform, use ilaplace
.
Compute the Laplace transforms of these expressions with respect
to the variable t
:
laplace(exp(a*t), t, s)
laplace(1 + exp(a*t)*sin(b*t), t, s)
Compute the Laplace transform of this expression with respect
to the variable t
:
F := laplace(t^10*exp(s_0*t), t, s)
Evaluate the Laplace transform of the expression at the points s =
 2 s_{0} and s =
1 + π. You can evaluate the resulting expression F
using 
(or
its functional form evalAt
):
F  s = 2*s_0
Also, you can evaluate the Laplace transform at a particular point directly:
laplace(t^10*exp(s_0*t), t, 1 + PI)
If laplace
cannot find an explicit representation
of the transform, it returns an unevaluated call:
laplace(exp(t^3), t, s)
ilaplace
returns
the original expression:
ilaplace(%, s, t)
Compute the folllowing Laplace transforms that involve the Dirac and the Heaviside functions:
laplace(dirac(t  3), t, s)
laplace(heaviside(t  PI), t, s)
The Laplace transform of a function is related to the Laplace transform of its derivative:
laplace(diff(f(t), t), t, s)

Arithmetical expression or matrix of such expressions 

Identifier or indexed identifier representing the transformation variable 

Arithmetical expression representing the evaluation point 
Arithmetical expression or unevaluated function call of type laplace
.
If the first argument is a matrix, then the result is returned as
a matrix.
f