 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2017/paper-2/2/a/solution

e^{i z} = e^{- y} (cos x + i sin x)

. Multiplying by

x + i y

gives

f (z) = e^{- y} [(x cos x - y sin x) + i (x sin x + y cos x)] .

Thus its real part is

Re f = e^{- y} (x cos x - y sin x) .

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