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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2017

Paper 1

Use the <Fourier transform> convention $\widetilde h(k)=\int_{\mathbb R}e^{-ikx}h(x)\,dx$, with inverse $(2\pi)^{-1}\int_{\mathbb R}e^{ikx}\widetilde h(k)\,dk$. Splitting at the origin, with $a>0$ to ensure convergence, gives
$$
\widetilde h(k)=\int_{-\infty}^0e^{(a-ik)x}\,dx+\int_0^\infty e^{-(a+ik)x}\,dx=\frac1{a-ik}+\frac1{a+ik}.
$$
Thus
$$
\boxed{\widetilde h(k)=\frac{2a}{a^2+k^2}}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2017 / ib / Paper 1 / 14B / a / i / Solution

Past exam of the mathematics course of the University of Cambridge / 2017 / ib / Paper 1 / 14B / a / i / Solution

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Past exam of the mathematics course of the University of Cambridge / 2017 / ib / Paper 1 / 14B / a / i / Solution

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