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

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- Solution 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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