Our
research is in computer algebra, with a particular emphasis on problems
related to simplification of expressions involving elementary functions
(= rational functions, algebraic functions, trigonometric and inverse
trigonometric
functions, logs, and exponentials) and certain special functions, on algorithms for integration in finite
terms, and for solving differential equations in closed form.
Simplification and Structure of Mathematical Expressions
Simplification is a fundamental problem in symbolic mathematical computation
as it is a mechanism that enables one to determine equality among
expressions and to compute with them effectively. It is also fundamental in
obtaining algorithmic methods for other problems such as integration in
finite terms. Our interest in simplification has ranged from exploring the
boundary between decidable and undecidable problems to effective
simplification algorithms for elementary and special transcendental
functions. We have also considered the much harder problem of simplifying
elementary constants assuming that Schanuel's conjecture holds.
Integration in Finite Terms and
Closed Form Solutions of Differential Equations
Our work on integration in finite terms has ranged from new methods for
determining the minimal algebraic field extensions needed for expressing the
logarithmic part of an elementary integral to extensions of parts of the
Risch algorithm for integration in terms of certain special functions. For
differential equations, we have been interested in a series of problems
related to finding closed form first integrals for first order nonlinear
equations.
Selected Publications