We often want to invert the z-transform ie given an F(z) find the corresponding f(n) .
eg the z-transform method of solving DE's involves taking the z-transform of the DE, doing some
algebraic manipulation and then returning to the time-domain solution by inverting the z-transform.
Contour integral method
f(n) is the inverse transform of F(z) if the closed contour C includes all the poles of F(z).
This is mathematically general but there are easier ways for our cases of interest.
Look-up table method
Every 
we know also gives an inverse transform. Also use general properties
such as linearity.
Example : we know 
Partial fraction expansion - we use this a lot
In many cases (eg H(z)) the z-transform to be inverted is in the form of a ratio of polynomials.
Worked example
We must have a `proper fraction' to work on (order of denominator polynomial greater than that of numerator polynomial).
The poles of 
are d=0 d=1 and d=2.
Then we can write :
To find 
first multiply by z
Evaluate this at z=0 to give 
or 
To find 
first multiply by z-1
Evaluate this at z=1 to give 
or 
.
Similarly 
Now we can invert the z-transform by look-up table and linearity. There are in fact different forms of inverse transform (causal or anticausal sequences) for different ROC in the z-plane. We don't want to dwell on this is this course. The causal sequence is :
for an ROC of |z|>2 , 2 being the magnitude of the greatest pole.
Using MATLAB Symbolic Math Toolbox with :
Y=('(6*z^2-10*z+2)/(z^2-3*z+2)')
y=inverstranz(Y)
Gives y=Delta(n)+2+3*2^n
Matlab functions residue and residuez do inverse transforms by PFE (partial fraction expansion).