Definition:Argument of Complex Number

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Definition

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

An argument of $z$, or $\arg z$, is formally defined as a solution to the pair of equations:

$(1): \quad \dfrac x {\cmod z} = \map \cos {\arg z}$
$(2): \quad \dfrac y {\cmod z} = \map \sin {\arg z}$

where $\cmod z$ 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 $\theta + 2 k \pi$ where $k \in \Z$ is any integer.


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


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Principal Range

It is understood that the argument of a complex number $z$ 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:

$\hointr 0 {2 \pi}$
$\hointl {-\pi} \pi$

but in theory any such interval may be used.

This interval is known as the principal range.


Principal Argument

Let $R$ be the principal range of the complex numbers $\C$.

The unique value of $\theta$ in $R$ is known as the principal argument, of $z$.

This is denoted $\Arg z$.

Note the capital $A$.

The standard practice is for $R$ to be $\hointl {-\pi} \pi$.

This ensures that the principal argument is continuous on the real axis for positive numbers.

Thus, if $z$ is represented in the complex plane, the principal argument $\Arg z$ is intuitively defined as the angle which $z$ yields with the real ($y = 0$) axis.


Warning: Flawed Definition

It appears at first glance that it would be simpler to define the argument of a complex number $z = x + i y$ as:

$\theta = \arg z := \map \arctan {\dfrac y x}$

This arises from the definition of the tangent as sine divided by cosine.

This, however, does not determine $\theta$ uniquely.

The image set of $\arctan$ is usually defined as:

$\openint {-\dfrac \pi 2} {\dfrac \pi 2}$

and in any case, is a open interval of length $\pi$.

As the image set of $\arg z$ is $2 \pi$, that means that there are in general two values of $z$ which have the same $\arctan$ value.


Some more superficial sources gloss over this point, and merely suggest that $\arg z$ is one of the two values of $\map \arctan {\dfrac y x}$.


Also known as

The argument of a complex number is also seen as:

its amplitude, denoted $\am z$
its phase, denoted $\ph z$


Examples

Example: $\arg 3$

$\arg 3 = 0$


Example: $\map \arg {-3}$

$\map \arg {-3} = \pi$


Example: $\map \arg {1 + i}$

$\map \arg {1 + i} = \dfrac \pi 4$


Example: $\map \arg {-1 - i}$

$\map \arg {-1 - i} = -\dfrac {3 \pi} 4$


Example: $\map \arg {2 i}$

$\map \arg {2 i} = \dfrac \pi 2$


Example: $\map \arg {-i}$

$\map \arg {-i} = -\dfrac \pi 2$


Also see

  • Results about the argument of a complex number can be found here.


Sources

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