Applications and Methods for Recognition of (Anti)-Symmetric Functions

One of the important advantages held by computer algebra systems (CAS) over purely-numerical computational frameworks is that the CAS can provide a higher-level “symbolic” viewpoint for problem solving. Sometimes this can convert apparently impossible problems to trivial ones. Sometimes the symbolic perspective can provide information about questions which cannot be directly answered, or questions which might be hard to pose. For example, we might be able to analyze the asymptotic behavior of a solution to a differential equation even though we cannot solve the equation. One route to implicitly solving problems is the use of symmetry arguments. We suggest how, through symmetry, one can solve a large class of definite integration problems, including ones that are not currently solved by computer algebra systems. One case of symmetry provides for recognition of periodicity, and this solves additional problems, since removal of periodic components can be important in integration and in asymptotic expansions.