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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.


Example 1

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)

Example 2

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 s0 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)

Example 3

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)

Example 4

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)

Example 5

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

Return Values

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

Overloaded By


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