Algorithms for integrals of holonomic functions over domains defined by polynomial inequalities

TL;DR

This paper introduces an algorithm to compute holonomic systems for definite integrals of holonomic functions over domains defined by polynomial inequalities, extending the holonomic approach to parameterized integrals involving distributions.

Contribution

It presents a novel algorithm that computes holonomic systems for integrals over polynomial inequality domains, including cases with parameters and distributions.

Findings

Algorithm successfully computes holonomic systems for complex integrals.

Handles integrals with parameters and distributions.

Extends holonomic methods to broader classes of integrals.

Abstract

We present an algorithm for computing a holonomic system for a definite integral of a holonomic function over a domain defined by polynomial inequalities. If the integrand satisfies a holonomic difference-differential system including parameters, then a holonomic difference-differential system for the integral can also be computed. In the algorithm, holonomic distributions (generalized functions in the sense of L. Schwartz) are inevitably involved even if the integrand is a usual function.