 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2015/paper-1/7/b/solution

For the real part specified in the PDF,

u_{x} = (cos x - x sin x) cosh y + y cos x sinh y .

Integrating

v_{y} = u_{x}

gives

v = y cos x cosh y - x sin x sinh y + h (x) .

Substitution into

v_{x} = - u_{y}

shows

h^{'} (x) = 0

. Thus

f (z) = z cos z + i C, C \in R .

Indeed

cos (x + i y) = cos x cosh y - i sin x sinh y

, so the real part of

z cos z

is exactly the prescribed expression. The arbitrary imaginary constant is the only freedom on a connected domain.

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