Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2017 / Paper 1 / 11 / b / i / Solution

For $z\ne 0$, the complex logarithm is $\ln z=\ln r+i(\theta +2\pi k)$, where $k\in \mathbb{Z}$. ConsequentlyA specified branch chooses one continuous determination of the complex argument on an appropriate cut domain. For the usual principal-value convention use the principal argument $\mathrm{Arg}(x+iy)\in (-\pi ,\pi ]$. A formula using only $\arctan (y/x)$ would lose the quadrant information; it is a valid local angle only with the appropriate additive multiple of $\pi$.