Contour Integral and Residue Theory for Inverse $z$-Transform

1. What is Z-transform and inverse Z-transform?
2. What is residue theorem formula?

What is Z-transform and inverse Z-transform?

Z-Transform is basically a discrete time counterpart of Laplace Transform. Z-transform of a general discrete time signal is expressed in the equation-1 above. The range of values of 'Z' for which above equation is defined gives ROC (Reason of Convergence) of Z-transform.

What is residue theorem formula?

From the residue theorem (which produces a factor 2πi that cancels the similar factor in the definition of J), we getf(t)=J=(residue atz=a+ik)+(residue atz=a-ik).