Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
Ok, I understand now that $\tau$ means that the function arg(z) jumps to $\tau$ when arg(z)$ = \tau +2\pi$. – Applied mathematician Dec 17 '12 at 19:12
up vote 1 down vote accepted

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

share|cite|improve this answer
You mean $\arg_{-\pi}$, not $\arg_{-\pi/2}$. – mrf Dec 18 '12 at 12:26
@mrf Yes I do mean that – Nameless Dec 18 '12 at 14:04

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.