i^i=e^ln(i^i)
=e^ilni
=e^iln(sqrt(-1))
=e^(i/2)ln(-1)
we also know by euler's formula
i^i=(e^((pi/2)i))^i because i=e^((pi/2)i)
i^i=e^-pi/2
so
e^-pi/2=e^(i/2)ln(-1)
-pi/2=iln(-1)/2
ln(-1)=(pi)i
i^i=e^ln(i^i)
=e^ilni
=e^iln(sqrt(-1))
=e^(i/2)ln(-1)
we also know by euler's formula
i^i=(e^((pi/2)i))^i because i=e^((pi/2)i)
i^i=e^-pi/2
so
e^-pi/2=e^(i/2)ln(-1)
-pi/2=iln(-1)/2
ln(-1)=(pi)i