Some Questions Coming From the Evaluation of Integrals

The problem of evaluating definite integrals is this: given a function f and an interval [a, b], provide a closed-form expression for

I(f, a, b) = ∫_a^b f(x) dx.

The talk will present a variety of interesting questions that have appeared in the search of exact formulas. As an example connected to Number Theory, consider the quartic integral

N_{0, 4} (a, m) = ∫₀ ^∞  dx/(x⁴ + 2 a x² + 1)^m+1.

It turns out that, aside from some simple factors, the integral is a polynomial in a. The coefficients of this polynomials are positive rationals and have remarkable properties. Some of these will be presented. The second part of the talk will introduce a new heuristic method, the so-called method of brackets. This method was created by Ivan Gonzalez as a procedure to evaluate Feynman diagrams. It consists of a very small number of heuristic rules, some of which have been made rigorous. The extension as a general integration method was developed by the author in joint work with Ivan Gonzalez and Karen Kohl.