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# ilaplace

Inverse Laplace transform

## Syntax

``ilaplace(F)``
``ilaplace(F,transVar)``
``ilaplace(F,var,transVar)``

## Description

example

````ilaplace(F)` returns the Inverse Laplace Transform of `F`. By default, the independent variable is `s` and the transformation variable is `t`. If `F` does not contain `s`, `ilaplace` uses the function `symvar`.```

example

````ilaplace(F,transVar)` uses the transformation variable `transVar` instead of `t`.```

example

````ilaplace(F,var,transVar)` uses the independent variable `var` and transformation variable `transVar` instead of `s` and `t`, respectively.```

## Examples

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Compute the inverse Laplace transform of `1/s^2`. By default, the inverse transform is in terms of `t`.

```syms s F = 1/s^2; ilaplace(F)```
```ans = t```

Compute the inverse Laplace transform of `1/(s-a)^2`. By default, the independent and transformation variables are `s` and `t`, respectively.

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

Specify the transformation variable as `x`. If you specify only one variable, that variable is the transformation variable. The independent variable is still `s`.

```syms x ilaplace(F,x)```
```ans = x*exp(a*x)```

Specify both the independent and transformation variables as `a` and `x` in the second and third arguments, respectively.

`ilaplace(F,a,x)`
```ans = x*exp(s*x)```

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

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

Find the inverse Laplace transform of the matrix `M`. Specify the independent and transformation variables for each matrix entry by using matrices of the same size. When the arguments are nonscalars, `ilaplace` acts on them element-wise.

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

If `ilaplace` is called with both scalar and nonscalar arguments, then it expands the scalars to match the nonscalars by using scalar expansion. Nonscalar arguments must be the same size.

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

If `ilaplace` cannot compute the inverse transform, then it returns an unevaluated call to `ilaplace`.

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

Return the original expression by using `laplace`.

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

Compute the Inverse Laplace transform of symbolic functions. When the first argument contains symbolic functions, then 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)]```

## Input Arguments

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Input, specified as a symbolic expression, function, vector, or matrix.

Independent variable, specified as a symbolic variable, expression, vector, or matrix. This variable is often called the "complex frequency variable." If you do not specify the variable, then `ilaplace` uses `s`. If `F` does not contain `s`, then `ilaplace` uses the function `symvar` to determine the independent variable.

Transformation variable, specified as a symbolic variable, expression, vector, or matrix. It is often called the "time variable" or "space variable." By default, `ilaplace` uses `t`. If `t` is the independent variable of `F`, then `ilaplace` uses `x`.

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### Inverse Laplace Transform

The inverse Laplace transform f = f(t) of F = F(s) is:

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

Here, c is a suitable complex number.

## Tips

• If any argument is an array, then `ilaplace` acts element-wise on all elements of the array.

• If the first argument contains a symbolic function, then the second argument must be a scalar.

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