Question Number 27603 by lizan 123 last updated on 10/Jan/18
$${Find}\:\:{the}\:{value}\:{of}\:\:\:{i}^{{i}} \:\:? \\ $$$$ \\ $$$$ \\ $$
Commented by abdo imad last updated on 10/Jan/18
$${if}\:{you}\:{mean}\:{i}/{i}^{\mathrm{2}} =−\mathrm{1}\:{we}\:{have}\:{i}^{{i}} =\:{e}^{{iln}\left({i}\right)} \\ $$$${and}\:{ln}\left({i}\right)\:{mean}\:{the}\:{pricipal}\:{determination}\:{of}\:{complex} \\ $$$${logarithme}\:{but}\:\:{i}={e}^{{i}\frac{\pi}{\mathrm{2}}} \Rightarrow{ln}\left({i}\right)\:=\:\frac{{i}\pi}{\mathrm{2}}\:\Rightarrow{e}^{{iln}\left({i}\right)} =\:{e}^{−\frac{\pi}{\mathrm{2}}} \\ $$$${i}^{{i}} \:\:={e}^{−\frac{\pi}{\mathrm{2}}} \:\:\:\:\:. \\ $$