The Unified Method in Polygonal Domains via the Explicit Fourier Transform of Legendre Polynomials

The recent numerical implementation by Fornberg and collaborators of the so-called unified method to linear elliptic PDEs in polygonal domains involves the computation of the finite Fourier transform of the Legendre polynomials. A variation of this approach, introduced by two of the authors, also involves the same computation. Here, instead of expressing the finite Fourier transform of the Legendre polynomials in terms of Bessel functions J n+ 1 2 , we employ an explicit formula in terms of exponentials. We illustrate the usefulness of this formula, which is considerably cheaper to use, by implementing the unified method to the modified Helmholtz equation in the interior of a square. For completeness we present an explicit formula for the finite Fourier transform of all Jacobi polynomials.