Algorithmic integro-differential algebra

Integrals arise in many areas of science and engineering. Frequently, these integrals are parameter integrals (i.e. the function being integrated depends on parameters), since quantities described by integrals often depend on additional parameters. Such integrals also appear as integral transforms and so-called convolutions of functions. Moreover, many special functions are defined as definite integrals over elementary functions involving parameters. Depending on the task at hand, one usually wants to express the given integral explicitly in terms of known functions or wants to obtain more information about the dependence on the parameters. For this purpose, there are computer algebra algorithms to systematically obtain explicit evaluations of, or linear relations satisfied by, such integrals. In integro-differential algebra, the algebraic aspects of functions, their derivatives, and their integrals are investigated. In the project, we develop and improve algebraic methods and algorithms for exact computations with integrals. We implement these algorithms in dedicated software and apply such computer algebra tools to computational problems from theoretical physics. In particular, we work on integration algorithms in order to make them more efficient and also more powerful. Moreover, we also apply such algorithms in order to find many specialized identities that then can be used for evaluating parameter integrals appearing in applications. The focus is on integrals which involve functions defined by nested integrals, like they arise from computations in quantum field theory. In order to compute with functions defined by nested integrals, we study their algebraic structure and develop algorithms to compute unique representations of such functions. The results of the project not only constitute advances of symbolic computation and integro-differential algebra, but the implementation of the developed algorithms in software will benefit the work of a large variety of users in both fundamental research and applied fields who need to deal with exact computations of integrals. The dedicated methods and software developed in the interaction with experts from quantum field theory, in particular, will serve as a tool to facilitate progress in that field for obtaining a deeper understanding of the world.

As mentioned in the abstract, integrals arise in many areas of science and engineering and often the functions being integrated depend on parameters. In the project, we developed and improved algebraic methods and algorithms for exact computations with integrals and applied such computer algebra tools to computational problems from theoretical particle physics. The main achievements of the project are as follows. We developed a refined version of the so-called Risch-Norman approach to compute integrals exactly. This refinement allows to algorithmically compute integrals that could not be computed with standard versions of the Risch-Norman algorithm, and in certain cases it even allows to prove that no solution of specified form exists. The ideas developed can also be applied for solving linear differential equations. We established several general identities that hold even when functions involved are no longer sufficiently smooth, but can have singularities and discontinuities. For instance, we found a generalized version of the Taylor formula with additional terms that vanish for sufficiently smooth functions. Based on such general identities, we worked out normal forms (i.e. unique representations) of nested integrals as well as their algebraic relations, which are more complicated than for more regular functions. This can be used for simplification of quantities expressed in terms of nested integrals. In collaboration with experts of quantum field theory, we applied computer algebra tools for computing with nested integrals to successfully perform large computations arising in theoretical particle physics that involve thousands of integrals. The results computed in this way allow for a higher precision in the analysis of measurements at particle colliders to determine the mass of certain quarks and bosons. For a certain type of nested integrals arising in this context, we also worked out new transformation formulae dedicated to bring them into simpler form so that the numerical evaluation of the result becomes more efficient. In addition, beyond the original goals of the project, we also developed algebraic frameworks and algorithms for the computer-assisted proof and discovery of identities of linear operators. Based on the general identities involving differentiation and integration of functions with singularities mentioned above, we developed a dedicated framework for integro-differential operators. We also developed a more general framework for statements (of certain form) about equations of arbitrary linear operators and a procedure that can prove all such statements automatically.