**Power Series ExpansionMethod:-**

can be expressed either in positive powers of Z or negative powers of Z.

if the sequence is causal has negative powers of Z similarly the Non-causal sequence negative powers of Z.

Let .

is causal it has ROC then can be expressed as

is non-causal it has ROC then can be expressed as

**Partial fraction Method:-**

Let and

if , is a proper function .

if , is improper function so convert the improper function to proper function as

.

.

express into powers of Z as follows

then divide by Z

.

Now express into partial fractions using different cases and find out the inverse Z-transform for the function

.

**Convolution Method:-**

express as a product of two functions and as follows

then find the inverse Z- transforms of individual functions

by using convolution method find convolution of and

i.e,

now is the inverse Z-transform of .

**Cauchy Residue Theorem:-**

a function in Z if the derivative exists on and inside contour C and has no poles at then.

** .**

if the derivative of exists on and has no poles at then.

** .**

the values on the right hand side are called Residue’s of the pole .

if there are n no of poles inside C** .**

** **

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