Using extensions of the residue theorem for improper integrals computations with CAS

Abstract

The computation of improper integrals of the rst kind (integrals on unbounded domain) are used in di erent applications in Engineering (for example in Kynetic Energy, electric potential, probability density functions, Gamma and Beta functions, Laplace and Fourier Transforms, Di erential Equations, . . . ). Nowadays, Computer Algebra Systems (CAS) are being used for developing such computations. But in many cases, some CAS lack of the appropriate rules for computing some of these improper integrals. In a previous talk in ESCO 2016 and a later extension, we introduced new rules for computing improper integrals of the rst kind using some results from Advanced Calculus Theories (Residue Theorem, Laplace and Fourier Transforms) aimed to improve CAS capabilities on this topic. In this talk, we develop new rules for computing other types of improper integrals using different applications from extended versions of the Residue Theorem. We will show some examples of such improper integrals that current CAS can not compute. Using extensions of the Residue Theorem in Complex Analysis, we will be able to develop new rules schemes for these improper integrals. These new rules will improve the capabilities of CAS, making them able to compute more improper integrals.