A short post on evaluating the Fourier integral numerically. https://comphysblog.wordpress.com/2023/07/06/computing-fourier-integrals/

When I tried to look up how to do this, I almost exclusively found information on the FFT. This gave inadequate results, which I now know is expected. Numerical recipes says:

"It is a sobering exercise to implement equation (13.9.6)[DFT] for an integral that can be done
analytically, and to see just how bad it is."