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Year 1 of the natural sciences course of the University of Cambridge

Mathematics course of the year 1 of the natural sciences course of the University of Cambridge

Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge

2018

Paper 2

At a fixed radius, <Parseval's identity> applied to the preceding <Fourier series> gives
$$
\int_{-\pi}^{\pi}[y_{f_0}(r,\theta)]^2\,d\theta
=\pi\sum_{\substack{n\ge1\\n\text{ odd}}}\frac{16r^{2n}}{\pi^2n^2}
=\frac{16}{\pi}\sum_{\substack{n\ge1\\n\text{ odd}}}\frac{r^{2n}}{n^2}.
$$
In <polar coordinates>, $dA=r\,dr\,d\theta$. Integrating over radius and interchanging the nonnegative sum and <integral> gives
$$
\begin{aligned}
\int_D[y_{f_0}]^2\,dA
&=\frac{16}{\pi}\sum_{\substack{n\ge1\\n\text{ odd}}}\frac1{n^2}\int_0^1r^{2n+1}\,dr\\
&=\boxed{\frac8\pi\sum_{\substack{n\ge1\\n\text{ odd}}}\frac1{n^2(n+1)},\qquad p=2.}
\end{aligned}
$$
The interchange follows from the <monotone convergence theorem>. This is the special case of the <area norm of the Poisson extension in the unit disk> for the odd step boundary data.

Solution

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

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

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