Numerical integration in a symbolic context

Abstract

Techniques for numerical integration within a symbolic computation environment are discussed. The goal is to develop a fully automated numerical integration code that handles infinite intervals of integration and that handles various types of integrand singularities. Such a code should also be able to compute to arbitrarily high precision. For the case of an analytic integrand on a finite interval, a Clenshaw-Curtis quadrature routine is used. A concept of general (non-Taylor) series expansions forms the basis of techniques for identifying transformations that may yield an analytic integrand. For the case when no transformation is successful, the general series expansion is used to represent the integrand and it is directly integrated to move beyond the singular point. The latter technique relies on a powerful symbolic integrator that can express integrals in terms of special functions.