Imaginary Unit to Power of Itself

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