Moving the location of the branch cut in Mathematica

Question

According to the documentation, Mathematica chooses the branch cut for $\log(z)$ to lie along the negative real axis. It it possible to change this so that it lies along the positive axis or elsewhere in the complex plane?

Answer 1

Let me join the fun and see if I can write my answer without any omissions... both of the previous ones had little errors you can easily check by inverting the newly defined log function myLog, i.e., doing Exp[myLog[...]].

So here is my definition that I just verified:

myLog[z_, θ_: 0] := Log[Abs[z]] + I (Arg[z Exp[I θ]] - θ)

Note that the sign in front of the branch angle θ has to be different in the two places where it appears, so that when you do the inverse the two instances of θ cancel.

Answer 2

I think the correct way to achieve this is

    myLog[z_, θ_: 0] := Log[Abs[z]] + I (Arg[z Exp[I θ]] + θ)

Positive or negative θ corresponds to moving up or down the Riemann surface.