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How would a function mapping a complex point $z=re^{i\theta}$ to $re^{i\frac{\theta}{2}}$ be correctly written?

Matt
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1 Answers1

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note that $|z|=r$, so I think the answer you are looking for (and you can test it yourself) is $$f(z)=|z|\cdot \left( \frac{z}{|z|}\right)^{\frac{1}{2}}$$

NazimJ
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  • 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? – Matt Dec 12 '18 at 20:29
  • This can be more succinctly written as $(|z|z)^{1/2}$. – eyeballfrog Dec 12 '18 at 20:36
  • @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. – timtfj Dec 12 '18 at 20:36
  • Yes I just wrote it out long way's so order of operations tells the story of what's happening. Also yes, $\theta = tan^{-1}(\frac{b}{a})$ – NazimJ Dec 12 '18 at 20:38
  • Is it possible to define a function as $f(a+bi)=\tan^{-1}(\frac{b}{a})$, or is that incorrect notation? – Matt Dec 12 '18 at 20:42
  • You might actually be after this: $e^{ix}=\cos(x) + i\sin (x)$ (not as your function, but as your starting point) – timtfj Dec 12 '18 at 20:47
  • What is the domain of this function? I would be more careful using the notation $z^{1/2}$ because in order to define that, you need firstly to define a branch of $\log z $ – Beslikas Thanos Dec 12 '18 at 21:02