Checking if a function is holomorphic in $\Omega$

Question

I'm stuck with this exercise of my "Cauchy-Riemann equation" problem list.

Let $\Omega=\{x\in\mathbb{C}\ | 0<Re(z)<1\}$. Find a determination of the logarithm such that $f(z)=log(z)\cdot log(1-z)$ is holomorphic over $\Omega$.

I know that I have to check for what values of $\theta=arg(z)$ it holds that $$\frac{\partial f}{\partial \bar{z}}=0.$$

I've tried it but I'm not able to simplify the monster after finding $\frac{\partial f}{\partial \bar{z}}$, and I think that I'm missing something.

Thank you.

Answer 1

Hint : multiplication of holomorphic functions is holomorphic. So you need to find a determination of $\log(z)$ which is holomorphic on $ \{z \in \Bbb C : \Re (z) > 0\}$ (for example, it's enough to find such determination on $\Bbb C \backslash \Bbb R_-$).

Do similarly with $\log(z-1)$ and multiply them together.