IMC 2004 · Problem 6

For every complex number $z \notin \{0,1\}$ define

$$f(z) := \sum (\log z)^{-4},$$

where the sum is over all branches of the complex logarithm.

a) Show that there are two polynomials $P$ and $Q$ such that $f(z) = P(z)/Q(z)$ for all $z \in \mathbb{C} \setminus \{0,1\}$.

b) Show that for all $z \in \mathbb{C} \setminus \{0,1\}$

$$f(z) = z \frac{z^2 + 4z + 1}{6(z-1)^4}.$$