Question 6

Let $f(z) = z^4- z^3+12z-16$.
Determine, with proof, the integral:

$$\mathcal{J} = \frac{1}{2\pi i} \int_{\partial \mathcal{R}} \frac{f'(z)}{f(z)} dz.$$

Here $\mathcal{R}$ is the region $\{ z \in \mathbb{C}: 1 \le \vert z\vert \le 3\}$.
(Hint: use the theorem of Rouche').
What can you say about the value of the integral:

$$\mathcal{K} = \frac{1}{2\pi i} \int_{\partial \mathcal{R}} \frac{zf'(z)}{f(z)} dz.$$