1.4: The Principal Argument

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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.