In this case the ROC is the interior of a circle. Consequently, the signal x(n) is anti causal. To obtain a power series expansion in positive powers of z, we perform the long division in the following way:

First we eliminate the negative powers of z by multiplying both numerator and denominator by z2.

Using the power series expansion for log(1+x), with |x|<1, we have

First we express X(z) in terms of positive powers of z, in the form X(z)=\\(\\frac{z^3}{(z+1)(z-1)^2}\\)

The partial fraction expansion for the given X(z) is

Since the ROC is the exterior circle, we expect x(n) to be a causal signal. Thus we seek a power series expansion in negative powers of ‘z’. By dividing the numerator of X(z) by its denominator, we obtain the power series

All the methods mentioned above can be used to calculate the inverse z-transform of the given signal.

The partial fraction expansion for the given X(z) is

First, we note that we should reduce the numerator so that the terms z-2 and z-3 are eliminated. Thus we should carry out the long division with these two polynomials written in the reverse order. We stop the division when the order of the remainder becomes z-1. Then we obtain

To eliminate the negative powers of z, we multiply both numerator and denominator by z2. Thus,