# ilaplace

Inverse Laplace transform

## Syntax

`ilaplace(F,trans_var,eval_point)`

## Description

`ilaplace(F,trans_var,eval_point)` computes the inverse Laplace transform of `F` with respect to the transformation variable `trans_var` at the point `eval_point`.

## Input Arguments

 `F` Symbolic expression or function, vector or matrix of symbolic expressions or functions. `trans_var` Symbolic variable representing the transformation variable. This variable is often called the "complex frequency variable". Default: The variable `s`. If `F` does not contain `s`, then the default variable is determined by `symvar`. `eval_point` Symbolic variable or expression representing the evaluation point. This variable is often called the "time variable". Default: The variable `t`. If `t` is the transformation variable of `F`, then the default evaluation point is the variable `x`.

## Examples

Compute the inverse Laplace transform of this expression with respect to the variable `y` at the evaluation point `x`:

```syms x y F = 1/y^2; ilaplace(F, y, x)```
```ans = x```

Compute the inverse Laplace transform of this expression calling the `ilaplace` function with one argument. If you do not specify the transformation variable, `ilaplace` uses the variable `s`.

```syms a s x F = 1/(s - a)^2; ilaplace(F, x)```
```ans = x*exp(a*x)```

If you also do not specify the evaluation point, `ilaplace` uses the variable `t`:

`ilaplace(F)`
```ans = t*exp(a*t)```

Compute the following inverse Laplace transforms that involve the Dirac and Heaviside functions:

```syms s t ilaplace(1, s, t)```
```ans = dirac(t)```
`ilaplace(exp(-2*s)/(s^2 + 1) + s/(s^3 + 1), s, t)`
```ans = heaviside(t - 2)*sin(t - 2) - exp(-t)/3 +... (exp(t/2)*(cos((3^(1/2)*t)/2) + 3^(1/2)*sin((3^(1/2)*t)/2)))/3```

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

```syms F(s) t f = ilaplace(F, s, t)```
```f = ilaplace(F(s), s, t)```

`laplace` returns the original expression:

`laplace(f, t, s)`
```ans = F(s)```

Find the inverse Laplace transform of this matrix. Use matrices of the same size to specify the transformation variable and evaluation point.

```syms a b c d w x y z ilaplace([exp(x), 1; sin(y), i*z],[w, x; y, z],[a, b; c, d])```
```ans = [ exp(x)*dirac(a), dirac(b)] [ ilaplace(sin(y), y, c), dirac(1, d)*1i]```

When the input arguments are nonscalars, `ilaplace` acts on them element-wise. If `ilaplace` is called with both scalar and nonscalar arguments, then `ilaplace` expands the scalar arguments into arrays of the same size as the nonscalar arguments with all elements of the array equal to the scalar.

```syms w x y z a b c d ilaplace(x,[x, w; y, z],[a, b; c, d])```
```ans = [ dirac(1, a), x*dirac(b)] [ x*dirac(c), x*dirac(d)]```

Note that nonscalar input arguments must have the same size.

When the first argument is a symbolic function, the second argument must be a scalar.

```syms f1(x) f2(x) a b f1(x) = exp(x); f2(x) = x; ilaplace([f1, f2],x,[a, b])```
```ans = [ ilaplace(exp(x), x, a), dirac(1, b)]```

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### Inverse Laplace Transform

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

$f\left(t\right)=\frac{1}{2\pi i}\underset{c-i\infty }{\overset{c+i\infty }{\int }}F\left(s\right){e}^{st}ds.$

Here, c is a suitable complex number.

### Tips

• If you call `ilaplace` with two arguments, it assumes that the second argument is the evaluation point `eval_point`.

• If `F` is a matrix, `ilaplace` acts element-wise on all components of the matrix.

• If `eval_point` is a matrix, `ilaplace` acts element-wise on all components of the matrix.

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

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