"Definite Integration in Differential Fields"
Definite Integration in Differential Fields
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The general goal of this thesis is to investigate and 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 this 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. A general framework is provided to model those functions in terms of suitable differential fields. Additionally, a generalization of Czichowski's algorithm for computing the logarithmic part of the integral is presented. Moreover, also partial generalizations to include other types of integrands are treated. As subproblems of the integration algorithm one also has to find solutions of linear ordinary differential equations of a certain type. Some contributions are also made to solve those problems in our setting, where the results directly dealing with systems of differential equations have been joint work with Moulay A. Barkatou. For the case of Liouvillian integrands we implemented the algorithm in form of our Mathematica package Integrator. Parts of the implementation also deal with more general functions. Our procedures can be applied to a significant amount of the entries in integral tables, both indefinite and definite integrals.