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February 8 Math Colloquium

Victor Moll, Tulane University,

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) = ∫af(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

N0, 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.

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