An extension of the method of brackets. Part 2

Abstract The method of brackets, developed in the context of evaluation of integrals coming from Feynman diagrams, is a procedure to evaluate definite integrals over the half-line. This method consists of a small number of operational rules devoted to convert the integral into a bracket series. A second small set of rules evaluates this bracket series and produces the result as a regular series. The work presented here combines this method with the classical Mellin transform to extend the class of integrands where the method of brackets can be applied. A selected number of examples are used to illustrate this procedure.


Introduction
The method of brackets is a collection of rules for the evaluation of a definite integral over the half-line [ ∞) 0, . It was developed in the calculation of integrals arising from Feynman diagrams, and its operational rules have appeared in [1][2][3]. These rules are described in Section 3. The method has been used in [4][5][6][7] to compute a variety of definite integrals appearing in [8].
The fundamental object is a bracket series, a formal expression of the form The operational rules for bracket series are described in Section 3. One of these rules associates a value with the sum S. The goal of the work presented here is to connect the method of brackets with the Mellin transform. Section 4 shows how to produce a series for a function starting with an analytic expression for its Mellin transform. Section 5 then uses this procedure to evaluate a variety of definite integrals. Section 6 presents a two-dimensional integral to show that the method applies directly. Finally, Section 7 shows that the method yields an incorrect power series representation of the function − e x but, in spite of this, the formal use of this series yields correct values of integrals. The explanation of this phenomenon is still an open question.

The method of brackets
This is a method that evaluates definite integrals over the half line [ ∞) 0, . The application of the method consists of a small number of rules, deduced in the heuristic form, some of which are placed on solid ground [9].
For ∈ a , the symbol is the bracket associated with the (divergent) integral on the right. The symbol is called the indicator associated with the index n.

Rules for the production of bracket series
The first part of the method is to associate with the integral Then ( ) I f is assigned the bracket series is assigned the r-dimension bracket series 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.

Rules for the evaluation of a bracket series
Rule E 1 . The one-dimensional bracket series is assigned the value where * n is obtained from the vanishing of the bracket, that is, * n solves + = an b 0. This is precisely the Ramanujan Master Theorem.
The next rule provides a value for multi-dimensional bracket series of index 0, that is, the number of sums is equal to the number of brackets.

Rule E 2 . Assuming the matrix = ( )
A a ij is non-singular, then the assignment is ϕ C n n a n a n c a n a n c A C n n n n , , where { } * n i is the (unique) solution of the linear system obtained from the vanishing of the brackets. There is no assignment if A is singular.
Rule E 3 . The value of a multi-dimensional bracket series of positive index is obtained by computing all the contributions of maximal rank by Rule E 2 . These contributions to the integral appear as series in the free indices. Series converging in a common region are added, and divergent/null series are discarded. There is no assignment to a bracket series of negative index. If all the resulting series are discarded, then the method is not applicable.

The generation of series
This section describes how to obtain a series for a function ( ) f x assuming the knowledge of its Mellin transform.
be the Mellin transform of a function ( ) f x . Then, for any choice of ∈ α β , , the function f admits an expansion of the form where the coefficient ( ) C n is given by The result in Theorem 4.1 gives no information about the convergence of the series (5). In particular, examples of functions for which such series do not exist are discussed below. 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].
Then (5) reproduces the Taylor series for ( ) = f x x sin . The Taylor series for x cos is obtained by the same procedure.
The Mellin transform appears as entry 6.561.14 in [8]. The procedure described here gives In order to cancel the term (− ) n Γ in the denominator it is convenient to choose = β ν and = α 2. Then (8) reduces to and this establishes (7).

The evaluation of integrals with an integrand formed by the product of two terms
The goal of this section is to present a procedure to evaluate integrals of the form is a known function. The procedure is described in a sequence of steps. The final expression for I is given in Theorem 5.1.
Step 1. Use the method developed in Section 4 to produce a series for ( ) This is precisely the result given in Theorem 4.1.
Step 2. Replacing (12) in (11) gives The bracket series on the right is now evaluated to obtain This can be expressed as Step 3. Replace the expansions This two-dimensional bracket series now yields solutions, depending on which index, n or k, is kept as the free one. The solutions are as follows.
The results are now summarized as follows.
where the parameters α β , Observe that the term ( / ) k sin π 2 vanishes for k even and (14) is nothing but the Taylor expansion of An interesting point appears in this example. It will be shown that the method of brackets succeeds in the case > b a 2 , and it fails to produce a value when < < b a 0 2 . This problem will be discussed in a future publication. Take 1, and 1 Γ 2 2 . The parameters α β , shows that the sum reduces to the value for = k 0, that is, It 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 < <