# Documentation

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# iztrans

Inverse Z transform

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## Syntax

```iztrans(`F`, `z`, `k`)
```

## Description

`iztrans(F, z, k)` computes the inverse Z transform of the expression `F = F(z)` with respect to the variable `z` at the point `k`.

If `R` is a positive number, such that the function `F(Z)` is analytic on and outside the circle ```|z| = R```, then the inverse Z-transform is defined as follows:

`$f\left(k\right)=\frac{1}{2\pi i}\underset{|z|=R}{\oint }F\left(z\right){z}^{k-1}dz,\text{ }k=0,1,2...$`

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

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

To compute the direct Z transform, use `ztrans`.

## Examples

### Example 1

Compute the inverse Z transform of these expressions:

`iztrans(exp(1/z), z, k)`
``` ```
`iztrans((z*sin(1))/(z^2 - 2*cos(1)*z + 1), z, k)`
``` ```

### Example 2

Compute the inverse Z transform of this expression with respect to the variable `z`:

`f := iztrans((3*z)/(z - 1) + (2*z)/(z - 1)^2, z, k)`
``` ```

Evaluate the inverse Z transform of the expression at the points k = 2 a + 3 and k = 1 + i. You can evaluate the resulting expression `f` using `|` (or its functional form `evalAt`):

`f | k = 2*a + 3`
``` ```

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

`iztrans((3*z)/(z - 1) + (2*z)/(z - 1)^2, z, 1 + I)`
``` ```

### Example 3

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

`iztrans(F(z), z, k)`
``` ```

`ztrans` returns the original expression:

`ztrans(%, k, z)`
``` ```

### Example 4

Compute the inverse Z transforms of these expressions. The results involve the `kroneckerDelta` function:

`iztrans(1/z, z, k)`
``` ```
`iztrans((z^3 + 3*z^2 + 6*z + 5)/z^5, z, k)`
``` ```

### Example 5

Compute the inverse Z tranform of this expression:

`iztrans(z*diff(g(z), z), z, k)`
``` ```

## Parameters

 `F` Arithmetical expression or matrix of such expressions `z` `k` Arithmetical expression representing the evaluation point

## Return Values

Arithmetical expression or unevaluated function call of type `iztrans`. An explicit result can be a `piecewise` object. If the first argument is a matrix, then the result is returned as a matrix.

`F`