Can we have a complex number in the exponent?

[docnet]

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 $c\in C$ ?

 

[FactChecker]

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.

 

[mfb]

$r^c = e^{c \log(r)}$
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.