What are the conditions for non-integer exponents to be single valued?

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SUMMARY

The discussion centers on the conditions under which non-integer exponents yield single-valued results in complex functions. It is established that for complex variables ψ and ϕ, there are inherent limitations that prevent non-integer exponents from being single valued. The participants agree that the initial equation presented is incorrect, and they emphasize that there are no scenarios where non-integer exponents can be single valued.

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Nothing much, I have this:
gif.latex?\dpi{150}%20e^\theta=(e^{i\theta})^{-i}=(e^{i(\theta+2\pi)})^{-i}=e^{\theta+2\pi}.gif


I have studied (myself) about this for many days.
And I believe that, for some conditions
gif.latex?\dpi{150}%20(e^\psi)^\phi\neq%20e^{\psi\phi}.gif
for a complex ψ,ϕ

What are those conditions I mentioned about? And which field of study I should go to see?
 
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Your first equation is obviously wrong, as you can see. For the second question, there are no conditions where it doesn't hold.

The problem lies in the fact that for non-integer exponents, the expression is not single valued.
 

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