Transform a univariate sequence:

Transform a multivariate sequence:

Compute a typical transform:

Plot the magnitude using Plot3D, ContourPlot, or DensityPlot:

Plot the complex phase:

Generate conditions for the region of convergence:

Plot the region for :

Evaluate the transform at a point:

Plot the spectrum:

The phase:

Plot both the spectrum and the plot phase using color:

Plot the spectrum in the complex plane using ParametricPlot3D:

ZTransform will use several properties including linearity:

Shifts:

Multiplication by exponentials:

Multiplication by polynomials:

Conjugate:

ZTransform automatically threads over lists:

Equations:

Rules:

TraditionalForm typesetting:

Discrete impulses:

Discrete unit steps:

Discrete ramps:

Polynomials result in rational transforms:

Factorial polynomials:

Exponential functions:

Exponential polynomials:

Factorial exponential polynomials:

Trigonometric functions:

Trigonometric, exponential and polynomial:

Combinations of the previous input will also generate rational transforms:

Different ways of expressing piecewise defined signals:

Rational functions:

Rational exponential functions:

Hypergeometric term sequences:

The DiscreteRatio is rational for all hypergeometric term sequences:

Many functions give hypergeometric terms:

Any products are hypergeometric terms:

Transforms of hypergeometric terms:

Holonomic sequences:

A holonomic sequence is defined by a linear difference equation:

Many special function are holonomic sequences in their index:

Special sequences:

Periodic sequences:

Multivariate transforms:

Multivariate periodic sequences:

Linearity:

There are several relations to the InverseZTransform:

Shifts:

Polynomial multiplication:

Exponential multiplication:

Differences and shifts:

Sums:

Integrals:

Without assumptions, typically a general formula will be produced:

Use Assumptions to obtain the expression on a given range:

Set GenerateConditions to True to get the region of convergence:

Different methods may produce different results:

By default, convergence testing is performed:

Setting VerifyConvergence->False will avoid the verification step: