Our goal is efficient and robust numerical evaluation of Fourier integrals of the form \[f(\vec{r}) = N_n(a,b) \int\, d^n k\, e^{+b i (\vec{k}\cdot\vec{r})}\, \tilde{f}(\vec{k})
\quad , \quad
\tilde{f}(\vec{k}) = \tilde{N}_n(a,b) \int\, d^n r\, e^{-b i (\vec{k}\cdot\vec{r})}\,f(\vec{r})
\; ,
\label{eqn:fourier}\] with normalization factors \[N_n(a,b) = |b|^{n/2} (2\pi)^{-n(1+a)/2} \quad , \quad
\tilde{N}_n(a,b) = |b|^{n/2} (2\pi)^{-n(1-a)/2} \; ,\] where the constants \(a\) and \(b\) establish our choice of Fourier convention^{1}. We focus on two- and three-dimensional (\(n=2,3\)) transforms of funct

## Share on Social Media