B Can we have a complex number in the exponent?

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It is valid to express a real number raised to a complex exponent, as demonstrated by the equation r^c = e^{c \log(r)}. This approach allows for the computation of complex powers of real numbers using the exponential function. Additionally, raising a complex number to a complex power is also meaningful. An example provided is i^i, which results in a real number, specifically e^{-π/2}. The discussion confirms that both r and c can indeed be complex.
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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## ?
 
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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.
 
##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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