The question

Where is the function $\displaystyle Log(z^2 - 1)$ analytic?

My attempt

The branch cut is on the negative real axis, so:

$\displaystyle z^2 - 1 = (x^2 - y^2 - 1) + i(2xy)$

$\displaystyle x^2 - y^2 - 1 \le 0$

and

$\displaystyle xy = 0$

Therefore, not analytic when $\displaystyle x = \pm 1$

Is this correct? Thank you.