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**1. Homework Statement**

Find a branch of log(z − 1) that is analytic inside the unit circle. What is the value of this branch at z = 0?

**2. The attempt at a solution**

Alright so clearly the log(z) function is analytic at all points accept for the negative real axis.

So log(z-1) will be analytic at all points x ≤ 1. My problem is choosing a branch. I'm really bad at doing this. I understand if the function was something like log(z+1) it would already be analytic inside the unit circle using the principle branch. I just don't see how I can choose a branch that is analytic inside the unit circle. I think if I understand this problem I can be prepared for my test tomorrow. Thanks for any help in advance.