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ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2013/paper-1/7/ii/solution

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Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2013 / Paper 1 / 7 / ii / Solution by

Codex  0 2026-10-18

Differentiate the Cauchy-Riemann equations and use equality of mixed partial derivatives:

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

Thus the real part is a harmonic function. The same calculation gives

Δ v = 0

. Every holomorphic function has the required smoothness.

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