# Imaginary Unit to Power of Itself/Complete

## Theorem

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

where $i$ is the imaginary unit.

## Proof

 $\ds i^i$ $=$ $\ds \exp \paren {i \ln \paren i}$ Definition of Complex Power $\ds$ $=$ $\ds \exp \paren {i \paren {\ln 1 + i \paren {\dfrac \pi 2 + 2 k \pi} } }$ Definition of Complex Natural Logarithm: for all $k \in \Z$ $\ds$ $=$ $\ds \exp \paren {i \paren {i \paren {\dfrac \pi 2 + 2 k \pi} } }$ Logarithm of 1 is 0 $\ds$ $=$ $\ds \exp \paren {i^2 \dfrac \pi 2 + 2 k \pi i^2}$ $\ds$ $=$ $\ds \exp \paren {-\dfrac \pi 2 - 2 k \pi}$ $\ds$ $=$ $\ds \exp \paren {\dfrac {3 \pi} 2 + 2 k \pi}$ as $k$ ranges over all integers $\ds$ $=$ $\ds \exp \paren {\dfrac {4 k + 3} 2 \pi}$

$\blacksquare$

## Also presented as

This result can also be presented as:

$i^i = \set {\exp \paren {- \dfrac \pi 2 \paren {4 k + 1} }: k \in \Z}$