 Solution

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/v/solution

\partial_{z} = (\partial_{x} - i \partial_{y}) /2

and

\partial_{\overset{z}{ˉ}} = (\partial_{x} + i \partial_{y}) /2

, for which

Δ = 4 \partial_{z} \partial_{\overset{z}{ˉ}}

. Since

H

and

G

\partial_{\overset{z}{ˉ}} F = H (z), Δ F = 4 H^{'} (z), Δ^{2} F = 16 \partial_{z} \partial_{\overset{z}{ˉ}} H^{'} (z) = 0.

The Laplacian has real coefficients, so taking real and imaginary parts proves

Δ^{2} U = 0, Δ^{2} V = 0.

F

itself is generally not holomorphic because of its

\overset{z}{ˉ}

factor.

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