branch cuts

We are asked to find a branch where the multi-valued function is analytic in the given domain.

The function: $(4+z^2)^{1/2}$ in the complex plane slit along the imaginary axis from -2i to 2i.

The principal branch is $\exp(\frac{1}{2} Log(4+z^2))$ and so we want analyticity from -2i to 2i. It seems to me that this is obeyed and the cuts appear when |z|>2i. When |z|>2i we see that the term inside Log become negative, when |z|<2i,the term inside the Log is positive. Does this mean the principal branch is a branch that is analytic in the given domain? I referred to the back of the book and it gave a different branch, namely $z \exp(\frac{1}{2} Log(1+\frac{4}{z^2})})$, what went wrong?

The function is: $(z^4-1)^{1/2}$ in |z|>1.

The principal branch is $\exp(\frac{1}{2}Log(z^4-1))$ and so we want analyticity when |z|>1. However, we see that this is already the case. Because within the unit circle, z^4 is positive, and subtracting 1 makes z^4-1 negative. We want analyticity when |z|>1 which is the case since z^4 is now positive and larger than 1, making z^4-1 positive when |z|>1. Again, I feel that the principal branch works, however, the back of the book gives a different result $z^2 \exp{\frac{1}{2} Log(1-\frac{1}{z^4})}$

The function in d is: $(z^3-1)^{1/2}$ in |z|>1.

I could make the same argument as above, and say the principal branch of $\exp(\frac{1}{2} Log(z^3-1))$ works. However, again, this disagrees with the result in the back of the text: $z\exp(\frac{1}{3} Log(1-\frac{1}{z^3}))$
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