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Why the principal value of complex number argument is taken within $(-\pi,\pi]?$ What is the harm in considering similar other intervals like $[0,2\pi)$ for the purpose?

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Absolutly no harm, it is just a convention. Usually one wants the argument function to be continuous around non negative reals.

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    $\begingroup$ @SugataAdhya If what you want to do is more to the left than to the right of the imaginary axis, then $[0, 2\pi)$ is better. It's all about context. $\endgroup$ – Arthur Feb 18 '13 at 11:16
  • $\begingroup$ 'Usually one wants the argument function to be continuous around non negative reals.' Can you elaborate? $\endgroup$ – Nikhil Panikkar Dec 18 '15 at 15:19

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