 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2021/paper-1/7/b/solution

Insert the specified real part into the polar Cauchy-Riemann equations. The first gives

v_{r} = n r^{n - 1} sin (n θ)

, and integration with respect to

r

gives

v (r, θ) = r^{n} sin (n θ) + C (θ) .

The second equation then requires

v_{θ} = n r^{n} cos (n θ)

, so

C^{'} (θ) = 0

. The remaining imaginary additive constant is fixed by

f (0) = 0

. Thus

v (r, θ) = r^{n} sin (n θ), f (z) = r^{n} e^{in θ} = z^{n} .

The resulting polynomial is holomorphic at the origin too, consistently extending the calculation made in polar coordinates with

r > 0

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