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$e^{i \frac{2\pi}{2} }=(e^{i 2\pi})^{1/2}=1^{1/2}=1$ where the last equality follows from $1*1=1$.

Meanwhile, $e^{i \frac{2\pi}{2} }=e^{i\pi }=-1$ by canceling out $2$.

How is this paradox related to the concept of branch in complex variable? Can some explain brach or branch cut in the context of this paradox? Thank you.

Daniel Li
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    Long story short: complex exponentiation obeys different rules than real exponentiation. – mrtaurho Aug 25 '19 at 21:22
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    as mentioned in the above comment, a lot of these types of "fake proofs" and "contradictions" arise because of a failure to not take into account the laws of exponentiation. So, really, there is no paradox at all. People often only remember the formulas $(a^b)^c = a^{bc}$, and $a^{x+y} = a^xa^y$ etc, without paying attention to WHEN it is valid. Is is valid for all complex numbers? (clearly it isn't) all real numbers? all positive numbers? Once you clarify this for yourself, you'll be able to see that it is the first step $e^{i\frac{2\pi}{2}} = (e^{i 2\pi})^{1/2}$ which is incorrect. – peek-a-boo Aug 25 '19 at 21:33
  • @peek-a-boo This should be an answer :) – mrtaurho Aug 25 '19 at 21:36
  • @mrtaurho But this answer has been given already in Isaac's answer at the duplicate (and the other answers). – Dietrich Burde Aug 25 '19 at 21:41
  • @mrtaurho yup the answer given in the duplicate is more specific as to when the rules were valid, which is why I only left this as a comment rather than an answer – peek-a-boo Aug 25 '19 at 21:43
  • @peek-a-boo Anyway, it's good to have an explaining comment here too! – mrtaurho Aug 25 '19 at 21:46

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