Explanation of i to the i power?

Question

Could somebody give me a good explanation for how $i^i$ works? I'm a junior and just now getting to this. I'm also too hard pressed for time to dive into exploring it myself.

Answer 1

Choosing a definite argument for $\;i\;$ , say $\;\frac\pi2\;$ , we get

$$i^i=e^{i\log i}=e^{i\left(\log|i|+i\frac\pi2\right)}=e^{-\pi/2}$$