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

AI Thread Summary
Non-integer exponents can lead to multi-valued expressions, particularly in complex analysis. The discussion highlights the need for specific conditions under which these exponents can be treated as single-valued. It emphasizes that the initial equation presented is incorrect and suggests that there are no universal conditions that allow non-integer exponents to be single-valued. Further exploration in the field of complex functions may provide clarity on this issue. Understanding these principles is crucial for accurate mathematical modeling involving non-integer exponents.
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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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