Past exam of the mathematics course of the University of Cambridge / 2012 / ia / Paper 1 / 1C / b / i / Solution

For $z=x+iy$, the complex exponential has complex modulus $e^x$. Equality to the positive real number $e$ therefore forces $x=1$ and $\cos y+i\sin y=1$. The latter holds precisely when $y$ is an integer multiple of $2\pi$. Hence all solutions areSubstitution verifies every listed value; the modulus and angle arguments exclude any others.