 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2010/paper-1/7/i/solution

Differentiate the Cauchy-Riemann equations and use equality of mixed partial derivatives, which holds because holomorphic functions are smooth:

Δ u = (v_{y})_{x} - (v_{x})_{y} = 0, Δ v = - (u_{y})_{x} + (u_{x})_{y} = 0.

Thus both parts are harmonic functions.

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