# InverseZTransform

InverseZTransform[expr,z,n]

gives the inverse Z transform of expr.

InverseZTransform[expr,{z1,z2,},{n1,n2,}]

gives the multiple inverse Z transform of expr.

# Details and Options • The inverse Z transform of a function is given by the contour integral .
• The multidimensional inverse Z transform is given by .
• The following options can be given:
•  Assumptions \$Assumptions assumptions to make about parameters Method Automatic method to use
• In TraditionalForm, InverseZTransform is output using .

# Examples

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## Basic Examples(2)

Univariate inverse transforms:

Multivariate inverse transforms:

## Scope(4)

Shifted impulse sequence:

Rational transforms yield exponential and trigonometric sequences:

In some cases, additional simplification and transformations are needed:

Elementary functions:

Special functions:

## Options(1)

### Assumptions(1)

This transform will not evaluate without any constraints on the range of p:

Use Assumptions to limit the range of p:

## Applications(3)

Solve a linear difference equation:

Add an initial value equation and solve the algebraic equation for the transform:

Get the solution through inverse transformation:

Use RSolve:

Solve a linear difference-summation equation:

Use the inverse transform to get a solution to the original problem:

Use RSolve:

A discrete system transfer function:

Impulse response:

Step response:

Ramp response:

## Properties & Relations(6)

Use DiscreteAsymptotic to compute an asymptotic approximation:

ZTransform is the inverse operator:

Linearity:

Shifting:

Derivatives:

Initial value property:

Final value property:

InverseZTransform is closely related to SeriesCoefficient:

## Neat Examples(1)

Inverse transform for a hypergeometric function: