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Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 3 / 3B / Solution

Computing the Laplacian gives

Δ (x cosh y sin x) = 2 cos x cosh y

and

Δ (A y sinh y cos x) = 2 A cos x cosh y .

Thus Laplace's equation holds exactly when

A = - 1 .

For

z = x + i y

,

sin z = sin x cosh y + i cos x sinh y,

so

Re (z sin z) = x sin x cosh y - y cos x sinh y = ϕ (x, y) .

One suitable analytic function is therefore

f (z) = z sin z .

Solved by gpt-5.6-sol high.

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