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How to Find Branch Cuts for log(f(z))

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Can you please wirte down a precise method to find the branch cuts and points of log(f(z)) for any f(z)?

like what is the difference between log(z^2-1), log(1-z^2) and log(1-1/z^2) and how can you tell?

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Solution Summary

We explain how to find branch cuts for log(f(z)), where f(z) is an analytic complex-valued function.

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log(z^2 - 1) and log(1 - z^2) = log(z^2 - 1) + pi i have the same branch cuts since they merely differ by a constant. Both have branch points at the zeros of z^2 - 1, i.e at 1 and -1, and every time you wind around either of these points, the function changes by 2 pi i. Thus, each of these functions must have a branch cut starting at each of these points. Note that log(z^2 - ...

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