Definition:Argument of Complex Number

Let $$z = x + \imath y$$ 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

$$\tan \left({\arg \left({z}\right) }\right) = \tfrac{\text{Im }\left({z}\right)}{\text{Re }\left({z}\right)} \ $$

where $$\text{Im} \ $$ is the imaginary part, and $$\text{Re} \ $$ 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.