Complex Analysis

The Argument

In this text the notation $\textbf{arg} (z)$ is used to designate an arbitrary argument of $z$, which means that $\textbf{arg} (z)$ is a set rather than a number. In particular, the relation $$\textbf{arg} (z_1) = \textbf{arg} (z_2)$$ is not an equation, but expresses equality of two sets.

As a consequence, two non-zero complex numbers $r_1 (\cos \varphi_1 + i \sin\varphi_1)$ and $r_2 (\cos \varphi_2 + i \sin\varphi_2)$ are equal if and only if $$r_1=r_2,\quad \text{and}\quad \varphi_1 = \varphi_2+ 2 k \pi,$$ where $k \in \mathbb Z$.

In order to make the argument of $z$ a well-defined number, it is sometimes restricted to the interval $(−\pi, \pi]$. This special choice is called the principal value or the main branch of the argument and is written as $\textbf{Arg}(z)$.

Note that there is no general convention about the definition of the principal value, sometimes its values are supposed to be in the interval $[0, 2\pi)$. This ambiguity is a perpetual source of misunderstandings and errors.

The Principal Argument

The principal value $\textbf{Arg}(z)$ of a complex number $z=x+iy$ is normally given by $$\Theta=\arctan\left(\frac{y}{x}\right),$$ where $y/x$ is the slope, and $\arctan$ converts slope to angle. But this is correct only when $x > 0$, so the quotient is defined and the angle lies between $-\pi/2$ and $\pi/2$. We need to extend this definition to cases where $x$ is not positive, considering the principal value of the argument separately on the four quadrants.

The function $\textbf{Arg}(z)$ $:\mathbb C \setminus \{0\} \rightarrow \left(-\pi,\pi\right]$ is defined as follows:

\begin{eqnarray*} \textbf{Arg}(z)= \left\{\def\arraystretch{1.2}% \begin{array}{@{}c@{\quad}r@{}} \arctan \frac{y}{x} & \;\text{if $x>0$, $y\in \mathbb R$}\\ \arctan \frac{y}{x}+\pi & \;\text{if $x <0$, $y\geq 0$}\\ \arctan \frac{y}{x}-\pi & \;\text{if $x < 0$, $y < 0$}\\ \frac{\pi}{2} & \;\text{if $x=0$, $y> 0$}\\ -\frac{\pi}{2} & \;\text{if $x=0$, $y < 0$}\\ \text{undefined}& \;\text{if $x=0$, $y=0$}\\ \end{array}\right. \end{eqnarray*} Thus, if $z=r(\cos \Theta +i\sin \Theta)$, with $r>0$ and $-\pi < \Theta \leq \pi$, then \begin{eqnarray*} \textbf{arg}(z)= \textbf{Arg}(z)+2n\pi, \quad n \in \mathbb Z. \end{eqnarray*}

We can visualize the multiple-valued nature of $\textbf{arg}(z)$ by using Riemann surfaces. The following interactive shows some of the infinite values of $\textbf{arg}(z)$. Each branch is identified with a different color.

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