Imaginary Unit to Power of Itself

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Theorem

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

where $i$ is the imaginary unit.


Its decimal expansion is:

$0 \cdotp 20787 \, 95763 \, 50761 \, 90854 \, 6955 \ldots$

This sequence is A049006 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


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
\(\displaystyle \) \(=\) \(\displaystyle e^{\pi i^2 / 2}\)
\(\displaystyle \) \(=\) \(\displaystyle e^{-\pi / 2}\)

$\blacksquare$


Sources