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It can be shown that $i^i = e^{-\frac{\pi}{2}}$:

$$ \begin{align} i^i &= e^{i\ln i} \\ &= e^{i\ln\left(e^\frac{i\pi}{2}\right)} \\ &= e^{i\times\frac{i\pi}{2}} \\ &= e^{-\frac{\pi}{2}} \end{align} $$

However, it could just as easily been:

$$ \begin{align} i^i &= e^{i\ln i} \\ &= e^{i\ln\left(e^\frac{5i\pi}{2}\right)} \\ &= e^{i\times\frac{5i\pi}{2}} \\ &= e^{-\frac{5\pi}{2}} \end{align} $$

since $i = e^\frac{i\pi}{2} = e^\frac{5i\pi}{2} \left(= e^\frac{\left(4n+1\right)i\pi}{2}\right)$.

This solution is obviously false, since $e^{-\frac{\pi}{2}} \neq e^{-\frac{5\pi}{2}}$.

Is there some other method of evaluating $i^i$ that eliminates this solution (and the infinite number of other similar solutions)?

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  • $\begingroup$ Exponential with complex base requires the choice of the branch of logarithm, so it is defined only up to such a choice. Moreover, none of those branches are a priori preferred. So depending on the choice, any of your answer can be made correct. $\endgroup$ – Sangchul Lee Sep 13 '17 at 8:14
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Note that the logarithm has many branches, see e.g. here: https://en.wikipedia.org/wiki/Complex_logarithm

The problem with complex numbers is that $e^{i\theta} = e^{i(\theta+2\pi k)}$ for all $k\in \mathbb Z$ (but obviously $\theta \neq \theta +2 \pi k$ for $k\neq 0$!) So you have to choose one branch of the logarithm to actually work with it. This means you have to choose one branch to assign $i^i$ some value.

The following image shows a few of the possible branches of the imaginary part of the logarithm. As you can see the imaginary part is only defined up to a multiple of $2\pi$.

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No. Complex exponentiation to any other base than $e$ is inherently multivalued. If the base is real and positive, there is one distinguished value (since the complex logarithm of a positive real number has one real value), but otherwise there is no consistent way to say that one result is objectively better / more correct / more useful than any other.

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