PhD Defence: Definite Integration in Differential Fields

Definite Integration in Differential Fields

We develop computer algebra tools for the simplification resp. evaluation of definite integrals. One way of finding the value of a definite integral is via the evaluation of an antiderivative of the integrand. If no antiderivative of suitable form is available and the integral depends on additional parameters, then linear relations that are satisfied by the parameter integral of interest may be found based on the principle of parametric integration (often called differentiating under the integral sign or creative telescoping). The main result of the thesis extends results of Risch, Singer, and Bronstein to a complete algorithm for parametric elementary integration for a certain class of integrands covering a majority of the special functions appearing in practice such as orthogonal polynomials, polylogarithms, Bessel functions, etc. As subproblems of the integration algorithm one also has to find solutions of linear ordinary differential equations of a certain type. Our procedures can be applied to a significant amount of the entries in integral tables, both indefinite and definite integrals. In addition, our procedures have been successfully applied to interesting examples of integrals that do not appear in these tables or for which current standard computer algebra systems like Mathematica or Maple do not succeed.