 Solution

ID: past-exam-of-the-metric-spaces-and-complex-analysis-course-of-the-university-of-oxford/2017/5/b/i/solution

For

z = x + i y

e^{i z} = e^{- y} e^{i x}

, so

∣ e^{i z} ∣ = e^{- y}

. Therefore

∣ e^{i z} ∣ \leq 1 ⟺ Im z \geq 0,

the closed upper half-plane. Equality holds on the real axis.

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