Q: How do I algebraically find the appropriate branch cut of a function? For example, lets consider

.

If we choose to use the principle branch of the log function then

which is undefined at

. I know that much, but the text I am using says

is analytic on the set

and

or

. I am not seeing why it is this set.

In general, is there a way to assume nothing about your function other than maybe the log is defined on the principle branch and solve for the appropriate branch cuts? I can find the branch points, but not sure how to construct the proper cuts.

Thanks