Chapter 9 - Riemann Theta Functions as Ordinary Fourier Series

Abstract

One of the most important properties of Riemann theta functions for applications ( to data analysis and to numerical modeling) is that the theta function can be written as ordinary Fourier series with time-varying coefficients. This is a useful result from many points of view: (1) the “esoteric” Riemann theta function used normally by pure and applied mathematicians and theoretical physicists to solve difficult theoretical problems is now reduced to ordinary Fourier series, that is relatively well known and easy to use for most investigators; (2) the software for computing operations with Fourier series (the fart Fourier transform (FFT) and other operations) is thus now available for applications with theta functions; and (3) analogical and digital computations available for decades for computations using Fourier transforms can also be used for the computation of theta functions. A numerical example for the Korteweg–-de Vries (KdV) equation is presented.