Imaginary Unit to Power of Itself/Complete

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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:

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


Sources