An extension of the method of brackets. Part 1

Abstract

Abstract:The method of brackets is an efficient method for the evaluation of a large class of definite integrals on the half-line. It is based on a small collection of rules, some of which are heuristic. The extension discussed here is based on the concepts of null and divergent series. These are formal representations of functions, whose coefficients a n have meromorphic representations for n 2 C, but might vanish or blow up when n 2 N. These ideas are illustrated with the evaluation of a variety of entries from the classical table of integrals by Gradshteyn and Ryzhik.

Integrals of Frullani type and the method of brackets

“…The heuristic rules are currently being placed on solid ground [2]. The reader will find in [5,7,8] a large collection of evaluations of definite integrals that illustrate the power and flexibility of this method.…”

Pochhammer symbol with negative indices. A new rule for the method of brackets

“…The heuristic rules are currently being made rigorous in [2] and [14]. The reader will find in [7][8][9] a large collection of evaluations of definite integrals that illustrate the power and flexibility of this method.…”

“…Each representation of an integral by a bracket series has associated an index of the representation via index D number of sums number of brackets: (8) It is important to observe that the index is attached to a specific representation of the integral and not just to integral itself. The experience obtained by the authors using this method suggests that, among all representations of an integral as a bracket series, the one with minimal index should be chosen.…”

An extension of the method of brackets. Part 2

“…These include series where all the coefficients vanish (the so-called null series) and also those for which all the coefficients blow up (the divergent series). The use of these formal series in the process of integration has been presented in [10].…”

“…are arbitrary. These are chosen here as = This is a null series, in the sense of[10], where every coefficient vanishes.The series I 2 has the value The presence of the factor ( − ) k Γ 1shows that the sum reduces to the value for = k 0, that is,= bIt is curious that none of the techniques developed for the method of brackets is able to produce the value of this integral for the case < < -dimensional problemThe method described here also applies to multidimensional integrals. An example illustrating this is presented next.…”