Principal value of i raised to i

Taking the principal branch of the logarithm, the value of $i^i$ can be written as $e^{-c\pi}$. Find $c$.

$1/2$. On the principal branch $\log i=i\pi/2$, so $i^i=e^{i\log i}=e^{i\cdot i\pi/2}=e^{-\pi/2}$, a real number, and $c=\tfrac12$.