Past exam of the mathematical methods I course of the University of Oxford / 2022 / 9 / a / ii / Solution

The base is $2e^{i\pi /4}$. Its complex logarithms are $\ln 2+i(\pi /4+2\pi k)$, $k\in \mathbb{Z}$. Exponentiating $i$ times each logarithm gives all values of the multivalued power:The values form a discrete geometric sequence along the Argand ray of angle $\ln 2$. Consecutive radii differ by $e^{-2\pi }$. The origin is an accumulation point but is not itself a value. If the power is instead defined using the principal logarithm, only $k=0$ is retained. The phrase “all values” requires this branch distinction.