How does sage deal with choosing branches? Examples?

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There are lots of ways, but basically it usually makes a choice based on whatever software is involved in the computation. For instance, see the latest Pynac update (#9901) for some information about this where conjugation doesn't commute with a lot of functions on the branch cut line segment.

I don't think there is some coherent underlying philosophy of branch cuts, other than agreeing with other systems when there is such agreement.