Numerical evaluation of Heun functions

Edgardo Cheb-Terrab, Maplesoft

November 1st, 2004 at 3:30pm in K9509.

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.