Imaginary raised to imaginary?

Question

How do you show that

$i^i = -i^{-i} = 0.207879576350761908546955619834978770033877841631769608075...$

likewise for

$-i^i = i^{-i} = 4.810477380965351655473035666703833126390170874664534940020...$

?

Answer 1

$$x^y = \exp(y \log(x))$$

so you want $$\exp(i \log(i)) = \exp\left(i \times \frac{i \pi}{2}\right) = \exp(-\pi/2)$$

The others are similar.