The project is an effort to investigate the mathematical theory and algorithms associated with the symbolic evaluation of definite integrals. The first class of functions that will be considered is the family of hypergeometric functions. This class is characterized by satisfying a differential equation of second order with exactly three singular points. The PI will attempt to develop an automatic procedure based on the Barnes representation for these functions. The natural next step is the family of Heun functions that have four singular points. The project also considers many topics that have risen from previous investigations by the PI. These include arithmetical properties of sequences coming from the iteration of integrals, questions of logconcavity of sequences appearing in the integration of rational functions and general questions on the p-adic properties of naturally occurring sequences. A general theory for these valuations will be developed. A third component of the project is the implementation and justification of a heuristic method developed by the PI and his coworkers to evaluate integrals coming from Feynman diagrams.
Many problems in physics and engineering require the exact evaluation of definite integrals in terms of the parameters appearing in them. 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 expression is achievable. The goal of this project is to develop algorithms that will expand the capabilities of existing software packages that are widely used in industry and universities. A second goal is to develop mathematical theories for theoretical questions that appear in the development of the previously mentioned algorithms.