Odd and even square roots of $z^2-1$

Question

This is a very interesting exercise (provided that it is correct).

Find two holomorphic functions $\,f_1: \Omega_1\to\mathbb C$ and $f_2:\Omega_2\to\mathbb C$, which are both square roots of $z^2-1$, with maximal domains (i.e., they can not be extended analytically any further), and $f_1$ is even while $f_2$ is odd.

The most interesting part is that $f_2$ is an odd function, and therefore, it can not be thought of as a composition of $\sqrt{}$ and $z^2-1$, which is even!

Answer 1

It is correct.

Hint: If you follow a branch of $\sqrt{z^2-1}$ along a small circle around one of the branch points, when you complete the circle, you have the negative value of that with which you started. If you make the circle larger, until it surrounds both branch points, ...

$\Omega_1 = \mathbb{C}\setminus ((-\infty,-1]\cup [1,\infty))$, and $\Omega_2 = \mathbb{C}\setminus [-1,1]$.