Improving CAS capabilities: New rules for computing improper integrals

Abstract

Different Engineering applications require dealing with improper integral on unbounded domains (improper integrals of the first kind). The classical way for solving these integrals is by means of elementary Calculus (antiderivatives and limit computations) or using numerical approaches. In both situations different problems can arise. For example: the non-existence of antiderivative, the corresponding limit does not exist or integrals depending on parameters which complicate the use of numerical approaches. In order to solve this situation, Advanced Calculus techniques, such as Laplace or Fourier Transforms and the Residue Theorem can be applied. A brief review of the corresponding theoretical frame is included in this paper. Computer Algebra Systems (Cas) are pieces of software that allow symbolic computations. Nowadays, there are many Cas in the market with an increasing level of sophistication which make them a very powerful tool in Engineering. In this paper, a brief review on the history of Cas evolution is introduced.

In this work, some Advanced Calculus techniques for computing improper integrals of the first kind which cannot be solved with standard procedures are described. These techniques could be easily integrated in almost every Cas. However, we have detected some lacks of these techniques in many widely used Cas. In this paper, some tests involving improper integrals have been developed. These tests have been used to check the capabilities of some Cas and the results have provided a classification of the Cas with respect to this field. One of the main contributions in this work is the generation of different rules to compute improper integrals of the first kind. The rules have been classified in specific rules (those coming from the tests) and general rules (those coming from theoretical frames and generalizations of the specific rules). These rules are easy to include in a Cas increasing the facilities of the Cas in the field of improper integral computation.