How would a function mapping a complex point $z=re^{i\theta}$ to $re^{i\frac{\theta}{2}}$ be correctly written?
1 Answer
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note that $|z|=r$, so I think the answer you are looking for (and you can test it yourself) is $$f(z)=|z|\cdot \left( \frac{z}{|z|}\right)^{\frac{1}{2}}$$
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$\begingroup$ Ok, so there is no easy way to get the angle of the complex point? I have seen somewhere that $Arg~z$ might be used as a notation for that, is that correct? $\endgroup$– MattDec 12, 2018 at 20:29
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$\begingroup$ This can be more succinctly written as $(|z|z)^{1/2}$. $\endgroup$ Dec 12, 2018 at 20:36
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$\begingroup$ @Matthew Yes, it's correct. To get the angle of $a+bi$ from the real axis, just draw a diagram and notice that $\frac{b}{a}$ is its tangent. $\endgroup$– timtfjDec 12, 2018 at 20:36
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$\begingroup$ Yes I just wrote it out long way's so order of operations tells the story of what's happening. Also yes, $\theta = tan^{-1}(\frac{b}{a})$ $\endgroup$– NazimJDec 12, 2018 at 20:38
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$\begingroup$ Is it possible to define a function as $f(a+bi)=\tan^{-1}(\frac{b}{a})$, or is that incorrect notation? $\endgroup$– MattDec 12, 2018 at 20:42