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The argument is the angle of a complex number.

\arg z = \arg (|z| cos \theta + i |z| sin \theta) = \theta

The principal argument is within the interval (-\pi, \pi], although some definitions use [0, 2\pi)

Properties

The argument is obtained with the \arctan function, although it requires various conditions to ensure the correct value.

\operatorname{Arg}(x + iy) = \operatorname{atan2}(y,\, x) =
\begin{cases}
\arctan(\frac y x) &\text{if } x > 0, \\
\arctan(\frac y x) + \pi &\text{if } x < 0 \text{ and } y \ge 0, \\
\arctan(\frac y x) - \pi &\text{if } x < 0 \text{ and } y < 0, \\
+\frac{\pi}{2} &\text{if } x = 0 \text{ and } y > 0, \\
-\frac{\pi}{2} &\text{if } x = 0 \text{ and } y < 0, \\
\text{undefined} &\text{if } x = 0 \text{ and } y = 0.
\end{cases}

The following mathematical properties apply:

\begin{align}\arg(ab) & = \arg(a) + \arg(b)\\
\arg(a/b) & = \arg(a) - \arg(b)\\
\arg(a^n) & = n \arg (a)
\end{align}

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