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I understand that in complex analysis $\arg(z) = \operatorname{Arg}(z) + 2k\pi i$.
In some texts about complex analysis I read things like $\arg_{\tau}(z)$. What does $\tau$ mean? In addition, what does $\operatorname{Arg}(z) = \arg_{−\pi}(z)$ mean? What is $\arg_{0}$?
What are principal values? Is $\operatorname{Arg}(z)$ the principal value of $\arg(z)$?
How do I solve problems like: "Determine a branch of $f(z)=\log(z^3-2)$ that is analytic in $z_0$"?
Thanks in advance.
As you said the symbol $\arg_r$ denotes the branch of the argument function that has image $(r,2\pi+r]$. Now for your second question remember that the principal $\log$ (defined for $\arg_{-\pi}$) is complex differentiable everywhere but $S=\left\{z\in \mathbb{C}:z\leq 0\right\}$.
If $z_0\notin S$ choose the principal log.
If $z_0\in S$, choose the branch of $\log$ defined for $\arg_{0}$ that is complex differentiable everywhere but $P=\left\{z\in \mathbb{C}:z\ge 0\right\}$.
That way, $\log z$ will be holomorphic at $z_0$ everytime (unless of course $z_0=0$ where no $\log$ can even be defined). I think you can know work your problem
