Imaginary Unit to Power of Itself

Theorem

 * $i^i = e^{-\pi / 2}$

where $i$ is the imaginary unit.

Its decimal expansion is:
 * $0 \cdotp 20787 \, 95763 \, 50761 \, 90854 \, 6955 \ldots$

Complete Result
The full result is actually more complicated than that: