 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2018/paper-2/18/d/solution

At a fixed radius, Parseval's identity applied to the preceding Fourier series gives

\int_{- π}^{π} [y_{f_{0}} (r, θ)]^{2} d θ = π \sum_{n \geq 1 n odd} \frac{16 r ^{2 n}}{π ^{2} n ^{2}} = \frac{16}{π} \sum_{n \geq 1 n odd} \frac{r ^{2 n}}{n ^{2}} .

d A = r d r d θ

. Integrating over radius and interchanging the nonnegative sum and integral gives

\int_{D} [y_{f_{0}}]^{2} d A = \frac{16}{π} n \geq 1 n odd \sum \frac{1}{n ^{2}} \int_{0}^{1} r^{2 n + 1} d r = \frac{8}{π} n \geq 1 n odd \sum \frac{1}{n ^{2} ( n + 1 )}, p = 2.

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