Let $f:\mathbb{D}\to \mathbb{C}$ be defined by:

We find that we can simplify this to have factors of the form $\omega$ a $5$th root of unity, and $e^{\pi i/5}\omega$ a fifth root of $-1$.

By observing our conditions on the integral, we find that all the fifth roots of $1$ lie on some outer circle, and the other roots on some inner circle. We can also calculate the ${\alpha }_k$. from the above formula. For example the angle at the point $a_k=\omega$ is $\frac{1}{5}\pi$. The interior angle at each point $a_k=e^{\pi i/5}\omega$ is $\frac{7}{5}\pi$. We get a picture that looks like the following (blue points are fifth roots of 1, green roots of -1):