Question

`i ^i =` ?

Answer 1

`i^i=0.20788`

Explanation:

Any complex number can be written as

`x +i y = (sqrt(x^2+y^2))e^{i phi}` with `{x,y}in RR^2`

where `phi = arctan(y/x)`

then

`(x +i y)^{x+iy} equiv ((sqrt(x^2+y^2))e^{i phi})^{(sqrt(x^2+y^2))e^{i phi}}`

making `x = 0, y = 1`

`i^i = (e^{i pi/2})^{e^{i pi/2}} = (e^{i pi/2})^i = e^{-pi/2} = 0.20788`