Definition:Argument of Complex Number

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

From the definition of the polar form of $z$, we have:


 * 1) $x = r \cos \theta$
 * 2) $y = r \sin \theta$

Any value of $\theta$ which satisfies both of these equations is called an argument 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 defined as a continuous multifunction satisfying


 * $\tan \left({\arg \left({z}\right) }\right) = \dfrac{\operatorname{Im}\left({z}\right)}{\operatorname{Re}\left({z}\right)}$

where $\operatorname{Im}$ is the imaginary part, and $\operatorname{Re}$ is the real part, of $z$.

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 continous on the real axis.