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.

$\ds i^i$

$=$

$\ds \exp \paren {i \ln \paren i}$

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 $

$=$

$\ds \exp \paren {i \paren {i \paren {\dfrac \pi 2 + 2 k \pi} } }$

$\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 $

$=$

$\ds \exp \paren {\dfrac {4 k + 3} 2 \pi}$

$\blacksquare$

This result can also be presented as:

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

1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4$. Elementary Functions of a Complex Variable: Exercise $6 \ \text{(iv)}$
1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.21$: Euler ($\text {1707}$ – $\text {1783}$): Footnote $2$