Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2006 / Paper 2 / 3 / iii / a

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- Solution a

Solution

The Cauchy-Riemann equations give

v_{x} = 2 x

,

v_{y} = 2 - 2 y

. Integrating and matching both derivatives gives the harmonic conjugate

v = x^{2} + 2 y - y^{2} + C, f (z) = 2 z + i z^{2} + i C, C \in R .

The real part of this polynomial is exactly the required

2 x (1 - y)

.

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