Paper 1, Section I, 1C1 \mathrm{C}

Vectors and Matrices | Part IA, 2012

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

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

(i) ez=ee^{z}=e,

(ii) (logz)2=π24(\log z)^{2}=-\frac{\pi^{2}}{4}.

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

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