Lecture 2: Properties of the ZTransform
In the descriptions of these properties, take

Delay property

This is very analogous to the differentiation property of Laplace transforms,
and will similarly allow us to solve differential equations.
This property is used to introduce the initial conditions when we use transforms to solve difference equations.
 Left Shift (Advance)

Similar to the last property,
 Convolution

Like the convolution property for Laplace transforms, the convolution property
for Ztransforms is very important for systems analysis and design. In words:
The transform of the convolution is the product of the transforms. This holds
for both Laplace and Ztransforms.
If and then
 Multiplication by

 Multiplication by

 Initial Value theorem

For a causal
,
 Final Value theorem

If
has no poles outside the unit circle (i.e. it is stable),