### DIMACS - RUTGERS EXPERIMENTAL MATHEMATICS SEMINAR

Sponsored by the Rutgers University Department of Mathematics and the
Center for Discrete Mathematics and Theoretical Computer Science (DIMACS)

Co-organizers:
Drew Sills, Rutgers University, asills {at} math [dot] rutgers [dot] edu
Doron Zeilberger, Rutgers University, zeilberg {at} math [dot] rutgers [dot] edu

Title: Definite Integrals

Speaker: Victor Moll, Tulane University

Date: March 24, 2005 4:30-5:30pm

Location: Hill Center, Room 705, Rutgers University, Busch Campus, Piscataway, NJ

Abstract:

The question of evaluating definite integrals is as old as calculus itself. In spite of that, there is no coherent theory that will tell us how to proceed when confronted with a specific integrand. In this talk I will discuss two aspects of this problem. The first one deals with a series of transformations on the parameteres of a rational integrand. This is a rational version of the classical transformation of Landen, Gauss and Legendre for an elliptic integral that produced the arithmetic-geometric mean. The second problem illustrates the initial stages of a homogeneity conjecture. We present a case study on the family $$L_n = \int_{0}^1 \ln^n \Gamma (q) dq.$$

Euler evaluated $L_1 = \ln(2 pi)^{1/2}$ and in joint work with O. Espinosa we obtained an evaluation of $L_2$. The evaluation of $L_3$ is still open.