Use the <Fourier transform> convention
$$
\boxed{\widehat v(\omega)=\int_{-\infty}^{\infty}v(x)e^{-i\omega x}\,dx.}
$$
The <absolute integrability> of $v$ makes this integral exist for every real $\omega$, with $|\widehat v(\omega)|\leq\|v\|_1$. With this convention the <inverse Fourier transform> has factor $1/(2\pi)$; maintaining that normalization is essential in part (e).

Solution

Past exam of the algebra 1 and differential equations 1 course of the University of Oxford / 2014 / 6 / a / Solution

 Solution (source code)