1.4: The Principal Argument

The Argument

---

In this text the notation $arg(z)$ is used to designate an arbitrary argument of $z$, which means that $arg(z)$ is a set rather than a number. In particular, the relation

$arg(z_{1})=arg(z_{2})$

is not an equation, but expresses equality of two sets.

As a consequence, two non-zero complex numbers  $r_{1}(cos\varphi _{1}+isin\varphi _{1})$ and  $r_{2}(cos\varphi _{2}+isin\varphi _{2})$ are equal if and only if

$r_{1}=r_{2}$, and $\varphi _{1}=\varphi _{2}+2k\pi$,

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 $Arg(z)$.

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 $Arg(z)$ of a complex number $z=x+iy$ is normally given by

$\Theta =arctan(\frac{y}{x})$,

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

$Arg(z)=\left\{\begin{matrix} arctan\frac{y}{x} & if x> 0,y\in \mathbb{R}\\ arctan\frac{y}{x}+\pi &ifx< 0,y\geq 0 \\ arctan\frac{y}{x}-\pi & ifx< 0,y< 0 \\ \frac{\pi }{2}& ifx=0,y>0\\ -\frac{\pi }{2} & ifx= 0,y< 0\\ undefined& ifx=0,y= 0 \end{matrix}\right.$

Thus, if  $z=r(cos\Theta +isin\Theta )$, with  $r>0$ and  $-\pi <\Theta <\pi$, then

$arg(z)=Arg(z)+2n\pi$,    $n\in \mathbb{Z}$.

We can visualize the multiple-valued nature of $$ by using Riemann surfaces. The following interactive shows some of the infinite values of $$. Each branch is identified with a different color.