# Documentation

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# `ilaplace`

Inverse Laplace transform

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## Syntax

```ilaplace(`F`, `s`, `t`)
```

## Description

`ilaplace(F, s, t)` computes the inverse Laplace transform of the expression `F = F(s)` with respect to the variable `s` at the point `t`.

The inverse Laplace transform can be defined by a contour integral in the complex plane:

,

where c is a suitable complex number.

If `ilaplace` cannot find an explicit representation of the transform, it returns an unevaluated function call. See Example 3.

If `F` is a matrix, `ilaplace` applies the inverse Laplace transform to all components of the matrix.

To compute the direct Laplace transform, use `laplace`.

## Examples

### Example 1

Compute the inverse Laplace transforms of these expressions with respect to the variable `s`:

`ilaplace(1/(a + s), s, t)`

`ilaplace(1/(s^3 + s^5), s, t)`

`ilaplace(exp(-2*s)/(s^2 + 1) + s/(s^3 + 1), s, t)`

### Example 2

Compute the inverse Laplace transform of this expression with respect to the variable `s`:

`f := ilaplace(1/(1 + s)^2, s, t)`

Evaluate the inverse Laplace transform of the expression at the points t = - 2 t0 and t = 1. You can evaluate the resulting expression `f` using `|` (or its functional form `evalAt`):

`f | t = -2*t_0`

Also, you can evaluate the inverse Laplace transform at a particular point directly:

`ilaplace(1/(1 + s)^2, s, 1)`

### Example 3

If `laplace` cannot find an explicit representation of the transform, it returns an unevaluated call:

`ilaplace(1/(1 + sqrt(t)), t, s)`

`laplace` returns the original expression:

`laplace(%, s, t)`

### Example 4

Compute this inverse Laplace transform. The result is the Dirac function:

`ilaplace(1, s, t)`

## Parameters

 `F` Arithmetical expression or matrix of such expressions `s` Identifier or indexed identifier representing the transformation variable `t` Arithmetical expression representing the evaluation point

## Return Values

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

`F`