Paper 1, Section I, $1 \mathrm{C}$

(a) Let $R$ be the set of all $z \in \mathbb{C}$ with real part 1 . Draw a picture of $R$ and the image of $R$ under the map $z \mapsto e^{z}$ in the complex plane.

(b) For each of the following equations, find all complex numbers $z$ which satisfy it:

(i) $e^{z}=e$,

(ii) $(\log z)^{2}=-\frac{\pi^{2}}{4}$.

(c) Prove that there is no complex number $z$ satisfying $|z|-z=i$.

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