Past exam of the algebra and analysis I for mathematics and philosophy course of the University of Oxford / 2013 / Analysis / 5 / b / Solution

Choose the continuous argument with values in $(-\pi /2,3\pi /2)$ and defineIts branch cut is the nonpositive imaginary axis, including zero, exactly the removed ray. Locally away from that ray, the Cauchy-Riemann equations or the local inverse of the exponential show that it is a holomorphic logarithm, with derivative $1/z$ and $e^{\mathrm{Log}z}=z$. The chosen interval fixes the additive ambiguity by integer multiples of $2\pi i$.