Past exam of the mathematical methods I course of the University of Oxford / 2015 / 2 / Solution

All complex logarithms of $i$ are $i(\pi /2+2\pi k)$ for integer $k$. By complex exponentiation,Every value is positive and real. If the power is defined using the principal logarithm, the single value is $\boxed{e^{-\pi /2}}$. Keeping the logarithm branch explicit distinguishes this value from the full multivalued set.