# Documentation

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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.

`f`