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
\(\ds i^i\) | \(=\) | \(\ds \paren {e^{i \pi / 2} }^i\) | Euler's Formula | |||||||||||
\(\ds \) | \(=\) | \(\ds e^{\pi i^2 / 2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds e^{-\pi / 2}\) |
$\blacksquare$
Sources
- 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $0,20787 95763 50761 \ldots$
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $0 \cdotp 207 \, 879 \, 576 \, 350 \, 761 \, 908 \, 546 \, 955 \ldots$
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.21$: Euler ($\text {1707}$ – $\text {1783}$): Footnote $2$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $0 \cdotp 20787 \, 95763 \, 50761 \, 90854 \, 6955 \ldots$