 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Natural sciences course of the University of Cambridge

Year of the natural sciences course of the University of Cambridge

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

2017

Paper 1

Since $\gamma>0$, the causal damped sine has an absolutely convergent <Fourier transform> for real $\omega$. Resolving the sine into complex exponentials gives
$$
\widetilde f(\gamma,p,\omega)
=\frac1{2i}\int_0^\infty\left[e^{-[\gamma+i(\omega-p)]t}-e^{-[\gamma+i(\omega+p)]t}\right]dt
=\frac1{2i}\left[\frac1{\gamma+i(\omega-p)}-\frac1{\gamma+i(\omega+p)}\right].
$$
Combining the two fractions yields
$$
\boxed{\widetilde f(\gamma,p,\omega)=\frac{p}{(\gamma+i\omega)^2+p^2}.}
$$
The formula also includes $p=0$, for which both the original function and its transform vanish. Its sign reverses when the real parameter $p$ reverses.

Solution

Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2017 / Paper 1 / 4 / c / Solution

Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2017 / Paper 1 / 4 / c / Solution

 Ancestors (13)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2017 / Paper 1 / 4 / c / Solution

 Ancestors (13)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)