# 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:

$i^i = \set {\exp \paren {\dfrac {4 k + 3} 2 \pi}: k \in \Z}$

where $i$ is the imaginary unit.

## Proof

 $\displaystyle i^i$ $=$ $\displaystyle \left({e^{i \pi / 2} }\right)^i$ Euler's Formula $\displaystyle$ $=$ $\displaystyle e^{\pi i^2 / 2}$ $\displaystyle$ $=$ $\displaystyle e^{-\pi / 2}$

$\blacksquare$