Hardy's Reduction for a Class of Liouville Integrals of Elementary Functions

Abstract

This paper is concerned with a class of integrals whose integrands are the product of a rational function times the exponential of a nonconstant rational function. We call these Liouville integrals. For these integrals, we provide a student-friendly algorithm producing a two-term decomposition with minimum transcendental and maximum elementary components. This decomposition fulfills the conditions of Hardy's reduction theory, determines whether these integrals are elementary functions, and when in the affirmative, finds them. To achieve our goal, we use partial fraction decomposition, simple notions of linear algebra, and a special case of an 1835 theorem of Liouville that we refer to as Liouville's criterion on integration. There is in the literature a complete algorithm to decide if the integral of an elementary function is also elementary. Ours is a gentle alternative for the class of Liouville integrals.