Rational Landen Transformations and Hurwitz Zeta Function: from theory to algorithms

Tulane University, New Orleans LA

Investigators

Abstract

.The problem of integration of rational functions was considered by J. Bernoulli and Leibnitz in the 18th century. The main difficulty associated with this procedure is to obtain a complete factorization of the denominator. The long term goal of this project is to develop a complete solution to the problem of definite integration in finite terms. This work will complement the work done by Lazard, Rioboo, Risch and Trager in the case of indefinite integration. The implementation of this solution will have a profound impact on the current symbolic integrators. This project is an effort to investigate the mathematical theory and algorithms associated with the symbolic evaluation of definite integrals. Two classes of functions are considered: rational functions and elementary functions of the Hurwitz zeta function. The work on rational integrands centers around the rational Landen transformations developed by the PI. The analysis and implementation of these transformations is complete for definite integrals on the whole real line. We will concentrate on the finite interval case. These transformations will be incorporated into an efficient and robust symbolic algorithm for the evaluation of rational integrals, as well as some simple extensions of this class. Many problems in physics and engineering require the exact evaluation of integrals in terms of the parameters appearing in those integrals. These integrals come up in the study of particle physics and classical mechanics. While it is not always possible to find such an expression, an efficient and robust symbolic software package should give the result in closed form, or decide whether such an expression is achievable. The goal of this project is to develop algorithms that will expand upon the capabilities of existing software packages that are widely used in industry and universities.