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What is the most general branch cut curve for the logarithm $\log z$?

We have $$\log z = \log r + i \theta,$$ where $z = r e^{i \theta}$ is an arbitrary complex number, and $\theta \in [\theta_0, \theta_0 + 2\pi\rangle$ for some $\theta_0 \in \mathbb{R}$. Now, there's no reason why $\theta_0$ should be the same for all $r$. Therefore, we could have $\theta_0 \to \theta_0(r)$.

Is the function $\theta_0(r)$ (taken to be continuous) the most general form of the branch cut curve for the logarithm? If no, what are further restrictions/relaxations?

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1 Answer 1

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A holomorphic $\log$ can be defined on any simply connected domain, not containing $0$. So any curve $\gamma \ni 0$ such that $\Omega \setminus \gamma$ is simply connected may be used as a branch cut. (It's not even necessary that $\gamma$ is a curve.)

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