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laplace

Laplace transform

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Syntax

laplace(f, t, s)

Description

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.

Examples

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)

Parameters

f

Arithmetical expression or matrix of such expressions

t

Identifier or indexed identifier representing the transformation variable

s

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

f

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