Definition:Argument (Complex Analysis)

Definition

Let $z = x + i y$ be a complex number.

If we represent $z$ in the complex plane, the argument of $z$, or $\arg \left({z}\right)$, is intuitively defined as the angle which $z$ yields with the real ($y = 0$) axis.

Formally, it is defined as any solution to the pair of equations:

$(1): \quad \dfrac x {\left|{z}\right|} = \cos \left({\arg \left({z}\right)}\right)$

$(2): \quad \dfrac y {\left|{z}\right|} = \sin \left({\arg \left({z}\right)}\right)$

where $\left|{z}\right|$ is the modulus of $z$.

From Sine and Cosine are Periodic on Reals, it follows that if $\theta$ is an argument of $z$, then so is $\theta + 2k\pi$ where $k \in \Z$ is any integer.

Thus, the argument of a complex number $z$ is a continuous multifunction.

Principal Argument

It is understood from the above that $\theta$ is unique only up to multiples of $2 k \pi$.

With this understanding, we can limit the choice of what $\theta$ can be for any given $z$ by requiring that $\theta$ lie in some half open interval of length $2 \pi$.

The most usual of these are:

• $\left[{0 .. 2 \pi}\right)$
• $\left({- \pi .. \pi}\right]$

but in theory any such interval may be used.

The unique value of $\theta$ in the interval $\left({-\pi.. \pi}\right]$ is known as the principal value of the argument, or just principal argument, of $z$.

This is denoted $\operatorname{Arg} \left({z}\right)$.

Note the capital $A$.

This standard practice ensures that the principal argument is continuous on the real axis.