"Best" Definition of Complex Argument

Question

Let $z = a + bi \in \mathbb{C} \cong \mathbb{R}^2$. Some books define the argument of $z$ as the angle between the vector $z$ and the abscissa axis, in the counter-clockwise. So, $\arg \left( z \right) \in [0,2\pi)$.

But in other books, the argument of $z$ is the lower angle between the vector $z$ and the abscissa axis. So $\arg \left( z \right) \in (- \pi , \pi]$. When the counter-clockwise, $\arg(z) \geq 0$. When clockwise, $\arg(z) < 0$.

My first question: which is the "best" definition? Which of them is the "less problematic" definition of the argument of a complex number?

My second question: is there a formula for the argument of a complex number? In each case?

Answer 1

I don't claim this is the "best" definition of argument but here is one approach.

Recall that any $w\in\mathbb{C}\setminus\{0\}$ can be expressed as $$w=|w|e^{i\theta}$$ for some $\theta\in\mathbb{R}$, where $|w|=\sqrt{\operatorname{Re}^2(w) +\operatorname{Im}^2(w)}$.

1. The set $$\operatorname{Arg}_w:=\{\theta\in\mathbb{R}: w=|w|e^{i\theta}\}$$ is the called the set of arguments of $w$. It is easy to check that if $\theta_1,\theta_2\in\operatorname{Arg}_w$, then $\theta_1-\theta_2\in 2\pi i\mathbb{Z}$. One may think of the argument as a relation $\operatorname{Arg}=\{(w,\theta)\in(\mathbb{C}\setminus\{0\})\times\mathbb{R}:w=|w|e^{i\theta}\}$.
2. Give $\theta_0$, for any $w\in\mathbb{C}\setminus\{0\}$, there is a unique $\theta_w\in[\theta_0,\theta_0+2\pi)\cap\operatorname{Arg}_w$. The map $\operatorname{arg}_{\theta_0}:\mathbb{C}\setminus\{0\}\rightarrow[\theta_0,\theta_0+2\pi)$ given by $w\mapsto\theta_w$ is called a branch of the argument.

Once argument is defined, it is possible to extend the notion of Logarithm to the complex plane. This is one way to do it:

3. The relation $\operatorname{Log}$ from $\mathbb{C}\setminus\{0\}$ to $\mathbb{C}$ such that $(w,z)\in \operatorname{Log}$ iff $e^z=w$ is called logarithm. For $w\in\mathbb{C}\setminus\{0\}$, the set $\operatorname{Log}_w=\{z\in\mathbb{C}: (w,z)\in\operatorname{Log}\}$ is called the set of logarithms of $w$. It is easy to check that $$\operatorname{Log}_w =\{ \ln|w|+i\theta:\theta\in\operatorname{Arg}_w\}$$ where $\ln:(0,\infty)\rightarrow\mathbb{R}$ is the natural logarithm function from Calculus or Real analysis.
4. Selecting a brach $\theta_0$ of $\operatorname{Arg}$ yields a function $\log_{\theta_0}:\mathbb{C}\setminus\{0\}\rightarrow\mathbb{C}$, given as $w\mapsto \ln|w|+i\operatorname{arg}_{\theta_0}(w)$, where $\theta\in[\theta_0,\theta_0+2\pi)$. The map $\log_{\theta_0}$ is called a branch of logarithm.