archives@tulane.edu / Many symbolic integration methods and algorithms have been developed to deal with definite integrals such as those of interest to physicists, theoretical chemists and engineers. These have been implemented into mathematical softwares like Maple and Mathematica to give closed forms of definite integrals. The work presented here introduces and analytically investigates an algorithm called Method of brackets. Method of brackets consists of a small number of rules to transform the evaluation of a definite integral into a problem of solving a system of linear equations. These rules are heuristic so justification is needed to make this method rigorous. Here we use contour integrals to justify the evaluation given by the algorithm. / 1 / Tri Ngo

archives@tulane.edu / In this work, five algorithms of negative dimensional integration (NDIM) are compared in several examples of Feynman diagram calculations, and the resulting solutions are compared. The methods used are the Ricotta method without parametrization, the Ricotta method with Schwinger parametrization, the Suzuki method, the Anastasiou method, and the method of brackets. It is found that for one-loop diagrams, the method of brackets gives the same solution as the other methods, but without requiring analytic continuation of the gamma factors in the solution. For multi-loop diagrams, the method of brackets gives solutions in a simpler form than the other methods, and often gives fewer possible solutions as well. In addition to its use in the evaluation of Feynman diagrams, the method of brackets is also useful when extended to the evaluation of definite integrals over the positive real numbers. This extended method of brackets is applied to several examples of definite integrals, and the five NDIM methods are also used to evaluate these examples when possible. In particular, it is shown that the method of brackets is the only method of NDIM which may be extended to the evaluation of a large class of definite integrals over the positive real numbers. / 1 / Kristina E. VanDusen