 Solution

ID: 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/solution

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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