I have a complex number $z = -\alpha_1 \alpha_2 \sin^{2}(\theta/2)+ i\alpha_3\sin(\theta/2)|\sin(\theta/2)|$, where $\alpha_i \in \Re, i \in 1, 2, 3.$ I wish to find the absolute value and argument of this complex number. Whether the absolute value is $\sqrt{\alpha_{1}^2\alpha_{2}^2+\alpha_{3}^2}$ and whether the argument is well-defined? For finding the argument, Should I resort to case by case analysis?

  • $\begingroup$ Whether the absolute value is ... $\;z=0$ for $\theta=0\,$, so that formula is missing something. Should I resort to case by case analysis Yes, there are just two cases.. $\endgroup$ – dxiv Jun 4 '17 at 6:33
  • $\begingroup$ So $|z| = \sqrt{\alpha_{1}^2\alpha_{2}^2+\alpha_{3}^2}$ if $z \neq 0$ else if $0$ for $z = 0$. The $\arg(z) = \pm \frac{\alpha_3}{\alpha_1\alpha_2}$ depending upon the sign of $|\sin(\theta/2)|$ or $\arg(z) $ is undefined if $|\sin(\theta/2)|=0$. Is this right? $\endgroup$ – Jerry Jun 4 '17 at 7:11
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    $\begingroup$ No, I think it's not the idea in the comment. The idea is that the formula is wrong and this fact was exposed by means of an counterexample. You have to redo it from the start. Maybe your formula for the absolute value can be "tuned" by some factor $\sin^2(\theta/2)$. $\endgroup$ – Rafa Budría Jun 4 '17 at 8:13
  • $\begingroup$ I forgot the factor $\sin^{2} \theta/2$ in the absolute value. $\endgroup$ – Jerry Jun 4 '17 at 9:10

$z = -\alpha_1 \alpha_2 \sin^{2}(\theta/2)+ i\alpha_3\sin(\theta/2)|\sin(\theta/2)|$

For the absolute value:

$\vert z\vert^2=z\bar z=\alpha_1^2\alpha_2^2\sin^4(\theta/2)+\alpha_3^2\sin^2(\theta/2)|\sin(\theta/2)|^2$

$\vert z\vert=\sin^2(\theta/2)\sqrt{\alpha_1^2\alpha_2^2+\alpha_3^2}$

For the argument, $\arg z=\text{atan2}\left(\Im(z),\Re(z)\right)$,

we have to consider $\sin(\theta/2)/\vert\sin(\theta/2)\vert$, that it is $\pm1$ depending on the value of $\theta/2$

$\arg z= \begin{cases} \text{atan2}\left(\alpha_3,-\alpha_1\alpha_2\right)&\text{for}&4k\pi\leq\theta\lt(4k+2)\pi,\;k\in\mathbb Z\\\ \text{atan2}\left(\alpha_3,\alpha_1\alpha_2\right)&\text{for}&(4k+2)\pi\leq\theta\lt(4k+4)\pi,\;k\in\mathbb Z \end{cases}$


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