Can we have a complex number in the exponent?

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  • #1
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TL;DR Summary
Is it possible to exponentiate a number that is not e to a complex number?
Does it make sense to write ##r^c##
where ## r\in R/e ## and ##c\in C## ?
 

Answers and Replies

  • #2
FactChecker
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Yes. It makes sense. Remember that using only traditional real functions, ##r = e^{\ln{r}}##. So you can get the complex power of any real number by way of complex powers of ##e##. In fact, it makes sense to raise a complex number to a complex power.
 
  • #3
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##r^c = e^{c \log(r)}##
Both r and c can be complex here. As a notable example, ##i^i = e^{i \log{i}} = e^{i (\frac{\pi}{2} i)} = e^{- \frac{\pi}{2}}## which is real.
 

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