Thierry Dana-Picard

Explicit closed forms for parametric integrals

Closed forms are computed for parametric integrals, generally using induction formulas. It is shown that these integrals can be core activities, mixing hand-work, computations with a computer algebra system and experimental mathematics with an interactive website.

Parametric Integrals and Catalan Numbers.

Catalan numbers are defined by combinatorial properties, in numerous situations. Here, an integral representation for these Catalan numbers is computed.

Integral presentations of Catalan numbers

We compute in three different ways the same definite parametric integral. By-products are the derivation of a combinatorial identity and two integral presentations of Catalan numbers. One of them leads to a presentation using the γ function.

Integral presentations of Catalan numbers and Wallis formula

Looking for closed forms for an integral depending on one parameter, we obtain integral identities and new integral presentations of Catalan numbers. Wallis formula appears as an important tool.

Sequences of definite integrals

Definite integrals are computed for polynomials depending on at least one parameter. Recurrence relations are derived for the sequences of integrals and then used to derive closed forms for the general term of these sequences. In one example, a connection with an important trigonometric integral is found.

Teaching Mathematical Integration: Human Computational Skills versus Computer Algebra. Classroom Notes

Despite the ready availability of computer algebra packages, from a pedagogical point of view, the authors still feel that integration should be taught extensively the classical way, by means of carefully selected examples which combine as many fundamental techniques as possible for their evaluation. Furthermore, whenever possible, these examples should be specifically selected from those which a Computer Algebra System (CAS) either cannot solve precisely to provide an exact answer in analytical or transcendental form, or which a CAS cannot solve altogether. This approach ensures that students gain a proper understanding of the topic and, perhaps even more important, the fact that the students can evaluate precisely in closed form integrals which the computer cannot, stimulates and motivates them further to study these techniques. This note illustrates the above claims by means of both an indefinite integral and the associated improper integral which is offered to students as the ‘integral of the year’.

Parametric Improper Integrals, Wallis Formula and Catalan Numbers

We present a sequence of improper integrals, for which a closed formula can be computed using Wallis formula and a non-straightforward recurrence formula. This yields a new integral presentation for Catalan numbers.

Bisoptic curves of hyperbolas

Given a hyperbola, we study its bisoptic curves, i.e. the geometric locus of points through which passes a pair of tangents making a fixed angle θ or 180° − θ. This question has been addressed in a previous paper for parabolas and for ellipses, showing hyperbolas and spiric curves, respectively. Here the requested geometric locus can be empty. If not, it is a punctured spiric curve, and two cases occur: the curve can have either one loop or two loops. Finally, we reconstruct explicitly the spiric curve as the intersection of a plane with a self-intersecting torus.

Sequences of definite integrals, infinite series and Stirling numbers

We compute closed forms for some parametric integrals. The tools used are MacLaurin developments and other infinite series. Finally, Stirling numbers appear.

The importance of ‘low-level’ CAS commands in teaching engineering mathematics

When a professional engineer solves a mathematical problem using technology, he/she generally uses a single one-step ‘high-level’ command of a readily available computer package to obtain the solution immediately. This should not happen during his/her learning cursus. In an engineering mathematics course, the educator should decompose the solution process into elementary steps and reinforce each step by a ‘low-level’ usage of a CAS. This approach is absolutely essential in order to provide the future engineer with the conceptual insight into the solution process. We illustrate this with the teaching of the solution of ordinary differential equations using Laplace transforms.