Rational Landen Transformations, Dynamical Systems, and Hurwitz Zeta Function

Tulane University, New Orleans LA

Investigators

Abstract

Moll 0409968 Many problems in physics and engineering require the exact evaluation of integrals in terms of parameters appearing in those integrals. These problems come up in the study of particle physics and classical mechanics. While it is not always possible to find analytic expressions, the investigator and his colleagues study questions that appear in the development of an efficient and robust symbolic software package that should give such results in closed form, or decide whether such an expression is achievable. The investigator's goal is to develop algorithms that expand upon the capabilities of existing software pacakages that are widely used in industry and universities. The investigator's efforts are concentrated in two classes: rational functions, and elementary functions related to the Hurwitz zeta function. The study of this last class has produced a new approach to conjectures on the so-called multiple zeta values and the cloed-form evaluation of Tornheim-Zagier sums. The problem of definite integration of special functions started in the eighteen century with the work of J. Bernoulli. The knowledge developed from this problem has been central in many parts of analysis and applied mathematics. The development of highly sophisticated symbolic languages during the last part of the 20th century has required further study of this old problem. The investigator and his colleagues are developing the mathematical theory that is needed for the symbolic evaluation of these integrals. It is a surprising fact that this endevour has connections with many areas of mathematics. The investigator develops new algorithms that ensure the efficiency and robustness of the current symbolic languages available to the scientific community.