Helen Barton Lecture: “Integral Tales: Some Unexpected Connections”

Victor H. Moll is a professor of mathematics at Tulane University.

During the process of learning calculus one observes that there is a well-defined list of rules to compute derivatives: product, quotient, and chain rules are among the first taught in every class. On the other hand, when one tries to compute integrals, the student is left with a feeling that now there is simply a collection of tricks. There is no clear reason why one can integrate e^x in a simple manner, but the integral of e^{x^2} is more complicated. One learns these tricks from the instructor, by talking to older classmates, or by searching for them online. At the end, there seems to be no systematic way of doing this.

It is remarkable that, in the search of producing closed-forms of definite integrals, one finds many interesting connections with apparently disjoint parts of mathematics. Examples will include (1) properties of a collection of positive integers appearing in the evaluation of a rational function, (2) a planar dynamical system connected with a variation of the arithmetic geometric mean and (3) a list of definite integrals involving the gamma function. The lecture will be suitable for undergraduates and it will include stories about how the speaker got involved in such projects.