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The Z-transform of the function `F(z)`

is defined
as follows:

$$F\left(z\right)={\displaystyle \sum _{k=0}^{\infty}\frac{f\left(k\right)}{{z}^{k}}}$$

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:

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

You can consider the Z-transform as a discrete equivalent of the Laplace transform.

To compute the Z-transform of an arithmetical expression, use
the `ztrans`

function.
For example, compute the Z-transform of the following expression:

S := ztrans(sinh(n), n, z)

If you know the Z-transform of an expression, you can find the
original expression or a mathematically equivalent form by computing
the inverse Z-transform. To compute the inverse Z-transform, use the `iztrans`

function. For
example, compute the inverse Z-transform of the expression `S`

:

iztrans(S, z, n)

Suppose, you compute the Z-transform of an expression, and then
compute the inverse Z-transform of the result. In this case, MuPAD^{®} can
return an expression that is mathematically equivalent to the original
one, but presented in a different form. For example, compute the Z-transform
of the following expression:

C := ztrans(exp(n), n, z)

Now, compute the inverse Z-transform of the resulting expression `C`

.
The result differs from the original expression:

invC := iztrans(C, z, n)

Simplifying the resulting expression `invC`

gives
the original expression:

simplify(invC)

Besides arithmetical expressions, the `ztrans`

and `iztrans`

functions also accept matrices
of arithmetical expressions. For example, compute the Z-transform
of the following matrix:

A := matrix(2, 2, [1, n, n + 1, 2*n + 1]): ZA := ztrans(A, n, z)

Computing the inverse Z-transform of `ZA`

gives
the original matrix `A`

:

iztrans(ZA, z, n)

The `ztrans`

and `iztrans`

functions let
you evaluate the transforms of an expression or a matrix at a particular
point. For example, evaluate the Z-transform of the following expression
for the value `z = 2`

:

ztrans(1/n!, n, 2)

Evaluate the inverse Z-transform of the following expression
for the value `n = 10`

:

iztrans(z/(z - exp(x)), z, 10)

If MuPAD cannot compute the Z-transform or the inverse Z-transform of an expression, it returns an unresolved transform:

ztrans(f(n), n, z)

iztrans(F(z), z, n)

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