Numerical evaluation of Heun functions
Edgardo Cheb-Terrab, Maplesoft
Abstract: The five multiparameter Heun equations have been popping up with surprising frequency in applications during the last 10 years. Heun equations include as particular cases the Lame, Mathieu, spheroidal wave, hypergeometric, and with them most of the known equations of mathematical physics. Five Heun functions are defined as the solutions to each of these five Heun equations. In this talk, the difficulties for numerically evaluating these functions are summarized and an a hybrid approach resolving the problem, exploring exact Heun function identites and numerical evaluation techniques, is presented. The mathematical tools involved will be introduced together, so that the presentation is understandable for a 3rd-4th year undergraduate student. For those more familiar with linear ODE topics, in more technical words, this presentation is about a numerical approach for tackling the "two point connection problem" (TPCP), for a function with four singularities and depending on 7 complex parameters, in a case where the exact solution to the TPCP is not known.