= Argument (complex analysis)
{title2=$\arg z$}
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= Complex argument
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= Argument
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An <argument> of a nonzero <complex number> $z$ is any <angle> $\theta$ for which $z=|z|e^{i\theta}$. All <arguments> are $\theta+2\pi k$, with $k$ an <integer>. Zero has no <argument>: it has no distinguished direction. A commonly chosen representative, the principal <argument> $\operatorname{Arg}z$, lies in $(-\pi,\pi]$. Choosing representatives is part of specifying a <branch of a multivalued function>. These conventions are described in https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/6eae09885b0fab3f068e31752198abea_MIT18_04S18_topic1.pdf[Jeremy Orloff, MIT 18.04, topic 1, sections 1.5 and 1.14].
For $3+4i$, the principal <argument> is $\arctan(4/3)\approx0.927295$ <radians>, computed with the <inverse tangent>. But the coordinate ratio alone loses the quadrant: $-3-4i$ has the same ratio $4/3$, yet its principal <argument> is $\arctan(4/3)-\pi\approx-2.214297$ <radians>.
For nonzero $z,w$, the correct rule is
$$
\operatorname{Arg}(zw)\equiv\operatorname{Arg}z+\operatorname{Arg}w\pmod{2\pi}.
$$
Ordinary equality can fail. With $z=w=e^{3\pi i/4}$, their principal <arguments> sum to $3\pi/2$, but $zw=-i$ has principal <argument> $-\pi/2$. They differ by one full turn. As a point crosses the negative horizontal axis, the principal <argument> jumps by $2\pi$. A continuous choice on a suitable region therefore requires a <branch cut>; the jump belongs to the choice of representative, although the point moves smoothly.
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