Computing Fourier Integrals

Fast Fourier Transforms (or FFT's) are the first-stop for the numerical computation of Fourier transforms, but I recently learned that they are not well suited to evaluating the Fourier integral $latex \tilde{f}(\omega) = \int_{-\infty}^{\infty} f(t) e^{i\omega t} \,\textrm{d} t. $ This is stated explicitly in Numerical Recipes, and to borrow the words of a 1920's…

Gaussian Quadrature

Gaussian quadrature is a technique to evaluate integrals by summing a weighted integrand at a finite number of points. It can correctly evaluate some forms of integral to machine precision, rather than the approximations produced by techniques like the trapezium rule. When I first looked into using it, I found a huge number of explanations…

Conformal Mapping Example – The Eccentric Coax

1.0 - Introduction I recently had the chance to try out conformal mapping for boundary value problems. This allows a problem with complicated boundary conditions to be solved by transforming with a mapping function into a simpler problem. In some sense this method is a competitor to the 2D finite element techniques, since both are…