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Year 2 of the natural sciences course of the University of Cambridge

Mathematics course of the year 2 of the natural sciences course of the University of Cambridge

Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge

2021

Paper 2

By the definition of <convolution> and <Fourier inversion>,
$$
\begin{aligned}
\mathcal F^{-1}(\widetilde f*\widetilde g)(t)
&=\frac1{2\pi}\int_{\mathbb R}\int_{\mathbb R}\widetilde f(\eta)\widetilde g(\omega-\eta)e^{i\omega t}\,d\eta\,d\omega\\
&=\frac1{2\pi}\left[\int_{\mathbb R}\widetilde f(\eta)e^{i\eta t}d\eta\right]\left[\int_{\mathbb R}\widetilde g(\nu)e^{i\nu t}d\nu\right]\\
&=\boxed{2\pi f(t)g(t)}.
\end{aligned}
$$
The second line uses $\nu=\omega-\eta$. The factor $2\pi$ comes from the inverse-transform normalization. This is the product form of the <convolution theorem>, initially justified for sufficiently decaying functions and extended when the indicated products and convolutions exist.

Solution

Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2021 / Paper 2 / 5 / a / ii / Solution

Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2021 / Paper 2 / 5 / a / ii / Solution

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Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2021 / Paper 2 / 5 / a / ii / Solution

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