# i ^i =  ?

Jun 25, 2016

${i}^{i} = 0.20788$

#### Explanation:

Any complex number can be written as

$x + i y = \left(\sqrt{{x}^{2} + {y}^{2}}\right) {e}^{i \phi}$ with $\left\{x , y\right\} \in {\mathbb{R}}^{2}$

where $\phi = \arctan \left(\frac{y}{x}\right)$

then

${\left(x + i y\right)}^{x + i y} \equiv {\left(\left(\sqrt{{x}^{2} + {y}^{2}}\right) {e}^{i \phi}\right)}^{\left(\sqrt{{x}^{2} + {y}^{2}}\right) {e}^{i \phi}}$

making $x = 0 , y = 1$

${i}^{i} = {\left({e}^{i \frac{\pi}{2}}\right)}^{{e}^{i \frac{\pi}{2}}} = {\left({e}^{i \frac{\pi}{2}}\right)}^{i} = {e}^{- \frac{\pi}{2}} = 0.20788$