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Construct a branch $f(z)$ of $\log z$ such that $f(z)$ is analytic at $z=-1$ and takes on the value $5\pi i$ there.

I'm not really sure what I need to do to find this branch. I would greatly appreciate any help.

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You're trying to find an argument $\arg{z}$ such that $$\log(-1) = \log|-1|+i\arg{(-1)} = 5\pi i.$$ That means you want $$\arg(-1) = 5\pi.$$ For exemple you can put the cut's branch on $[0,+\infty[$ and define your argument to be in $[4\pi,6\pi[.$

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