Inverse $Z$-transform $X (z) = \log \left( \frac{z}{z-a} \right)$

Question

I need help to find the inverse $Z$-transform of the following function

$$X (z) = \log \left( \frac{z}{z-a} \right)$$

I get to the point

$$x[n] = \frac{\delta[n]}{n}+\frac{\epsilon[n]*a^n}{n}$$

But how do I find the value when $n = 0$?

Answer 1

Using the Taylor series $\log(1-x)=\sum_{n=1}^\infty-\frac{x^n}n$, we have $$\log(z/(z-a))=-\log(1-a/z)=\sum_{n=1}^\infty \frac{(a/z)^n}n$$ for $\left| \frac{a}{z} \right| < 1.$ Therefore, the inverse $z$-transform should be $$ x[n] = \begin{cases}\frac1na^{n} & n>0 \\ 0 & \text{otherwise}\end{cases} $$