$i^i$ having infinitely many values. Significance? History? Applications?

So if $i^i = {(e^{{\pi \over 2}i })}^{i} = e^{-\pi/2}$ it also equals $e^{-5\pi/2}, e^{-9\pi/2}, ... $ and $e^{3\pi/2},e^{7\pi/2},....$ and has infinitely many values.

What is the significance of that? Is this a common element of more advanced mathematics or is it unique to $i$? In terms of real world applications of complex numbers in physics and other fields, is there an example where this feature of $i$ manifests itself? Historically did mathematicians find this as offputting and confusing as I do?

Solution 1:

@Yalikesifulei is right. The significance is that $z^w=\exp(w\ln z)$ is multivalued whenever $w$ isn't an integer, because $\ln z=\ln(z\cdot1)=\ln(z+e^{2\pi i})$ causes the complex logarithm to be multivalued. So there's nothing special about the choice $z=i$ beyond $z\ne0$ ($0^w$ is a hole other can of worms I'll skip), nor about $w=i$ beyond $i\notin\Bbb Z$.

The identity $\exp(z+2\pi i)=\exp z$ is a more useful perspective on this than considering e.g. $i^i$. It's the basis of complex exponential representations of oscillations, which is useful whenever quantities oscillate or are otherwise periodic. In particular, $\exp(z+ik\cdot x)=\exp z$ whenever $k\cdot x\in2\pi\Bbb Z$, which makes this important in crystallography. It's also integral${}^\dagger$ to e.g. Fourier analysis.

${}^\dagger$ Since Fourier analysis uses integrals, this is a pun, albeit an unintended one.