Compatibility of the method of brackets with classical integration rules

: The method of brackets is a symbolic approach to the computation of integrals over (cid:2) n based on a deep result by Ramanujan. Its usefulness to obtain new and di ﬃ cult integrals has been demonstrated many times in the last few years. This article shows that this method is consistent with most classical rules for the computation of integrals, such as the fundamental theorem of calculus, the Laplace transform, the reduc - tion formula for the integration of functions with spherical symmetry, the Cauchy - Schlömilch transforma - tion, and explicit evaluations for multivariate integrals of product of Bessel functions as obtained by Exton and Srivastava. This work is part of a program dedicated to the derivation of solid theoretical grounds for the use of this attractive integration method.


Introduction
The problem of evaluating the definite integral consists of expressing the value of I in terms of a collection of special functions F, evaluated at a set of special values C. Naturally, the sets F and C depend on the real numbers { } a b , and the integrand f . The typical situation is that the integrand f belongs to a class of functions F 0 and one seeks a larger class of functions F F ⊃ 0 whose values produce the integral. (1.5) A simple generalization of this example shows that every polynomial P in the variable ( ) = f x e tx , with ∈ t a fixed parameter, has a primitive of the same form. Therefore, the integral of ( ) P e tx between a and b can be evaluated using Theorem 1.2.
One of the main difficulties in the evaluation of definite integrals is the existence of elementary functions, such as those encountered in elementary courses, that do not have elementary primitives. In general, this is hard to establish. The notion of elementary function is left, for the purposes of this article, undefined. The reader will find in Ritt [1] more information about this topic.
where SinIntegral is the Mathematica notation for the sine integral function defined by: Therefore, (1.7) simply comes from the introduction of a new function. The question of whether Si may be expressed in terms of more elementary functions is more difficult to decide.

The method of brackets
In the study of interaction of elementary particles, R. Feynman introduced his now famous diagrams, from which the physical properties of the reaction can be read. Some parametrizations of these diagrams led to families of quite complicated definite integrals. A Feynman diagram is a graphical method of representing the interactions of elementary particles (see [2,3] for an introduction). For the present exposition, such a diagram is simply a graph G with + E 1 external lines and N internal lines and L loops. The internal lines are sometimes referred to as propagators. All but one of these external lines are assumed to be independent. From its connection to Physics, the internal and external lines represent the particles that transfer momentum among the vertices of the diagram. Each of these particles carries a mass ≥ m 0 i for = … i N 1, , . The vertices represent the interaction of these particles, and conservation of momentum at each vertex assigns the momentum corresponding to the internal lines. A Feynman diagram has an associated integral given by the parametrization of the diagram. The reader will find all the details in [4,5].
Example. The massless sunset diagram is presented in Figure 1. The diagram is parametrized by the integral (in momentum space) where k and q are the D-dimensional vectors. For reasons of convergence, it is traditional to let = − D ε 4 2 and then → + ε 0 . The Schwinger parametrization is a systematic procedure to express the integral G in terms of some parameters (see [4] for details). It produces the integral with P being the square of the norm of the 4-vector p.
Several methods have been developed in order to deal with integrals of this form. The method of brackets used here is an effective variation of the method of negative dimension, one of the most used by the physics community, and consists of a small number of heuristic rules, some of which are in the process of being rigorously established. A complete description of the rules and a comparison with the current methods used to evaluate Feynman diagrams appear in [6].
In order to evaluate the integral, the method proposes a rule to assign a value to a bracket series.
A fundamental (open) question is to give a rigorous definition of this bracket, consistent with the operational rules in the following. where ( ) ∈ C n α β , , , and ϕ n , called the indicator of n, is defined by:  This has been established in [7], and it is a direct consequence of the multinomial theorem.
Up to now, the integral is converted into a bracket series. The evaluation of these series is described next. where * n is obtained from the vanishing of the bracket, that is, * n solves + = an b 0. The sum reduces to a single value.
Remark 2.1. Observe that this evaluation requires the evaluation of the coefficients ( ) C n outside its original domain, that is, for ∉ * n . The conditions on the function f given in Theorem 2.2, guarantee the existence of a unique extension of C. It turns out that Rule 4 is precisely a result of Ramanujan, known these days as Ramanujan's master theorem. This refers to the formal identity stated by S. Ramanujan's in his Quarterly Reports [8, p. 298]. It was widely used by him as a tool in computing definite integrals and infinite series. In fact, as Hardy puts it in [9], he "was particularly fond of them, and used them as one of his commonest tools." An elementary argument adapts this statement to prove (2.11). See [10] for details.
Note. In the multidimensional evaluation of a bracket series, in the special case when the number of sums and brackets is the same, write The multiple sum is declared to be a ij and the values { } * n i are the solutions of the linear system obtained from the vanishing of the brackets: a n a n c a n a n c .

(2.15)
An inductive proof follows directly from the one-dimensional case.

Consistency with the fundamental theorem of calculus
This section shows that in the case, the integrand in (1.1) is a perfect derivative; then, the method of brackets is consistent with the fundamental theorem of calculus.
Theorem 3.1. The value of ( ) computed by the method of brackets gives a consistent result with the analytic procedure, that is, and the last expression can be written as: , and it gives The system coming from the vanishing of the brackets gives ( Taking k and ℓ as free variables, it gives Simplifying the previous summand gives The proof is complete. The solutions corresponding to other combination of free variables yield the same result. □

The Laplace transform
A common variety of definite integrals appear as the Laplace transform of a function. This is defined by: 3) The next statement shows that (4.3) can be obtained directly by the method of brackets. Proof. The method of brackets gives and ≤ < ∞ r 0 and The last integral is π 2 , and the integrals in the product are evaluated using the classical formula [11, 3.261.5] The change of variables = t r 2 gives, with the notation = / n m 2, The computation of I by the method of brackets begins with the expansion:   The associated linear system coming from the vanishing of the brackets is

Cauchy-Schlömilch transformation
In this section, we show that the method of brackets is consistent with the Cauchy-Schlömilch transformation.
Theorem 6.1. The method of brackets is consistent with the Cauchy-Schlömilch transformation. That is, using the method of brackets to evaluate the integrals, we have be the even and odd parts of f , respectively. Then, ( ) , and we have Similarly, Thus, we have reduced (6.1) to the identity Now applying the method of brackets, we find  The vanishing of the brackets produces three series, one for each choice of free index. These are the usual procedure of the method of brackets evaluated the right-hand side as and this is equivalent to In order to evaluate the left-hand side of (6.23), start with Replacing in (6.27) and using the standard rule for the method of brackets lead to three expressions for the integral ℓ I on the left-hand side.
Index k is free. This yields The integrals were introduced by David and Borwein in [12] and are somewhat famous for exhibiting consistency patterns that eventually break down. As an example, the following evaluations hold: The correct evaluation of the integral is off from / π 2 by roughly × − 2.31 10 11 . It turns out that whenever , the integral will evaluate to / π 2. We will now show this with the method of brackets. We start from (7.1) and expand the ( ) b x sin k terms. Denoting the integral by I , we have where ( ) x n denotes the Pochhammer symbol. According to [13], any presence of a Pochhammer symbol with a negative index must be treated, in the application of the method of brackets, using the rule: 1 .
m m (7.14) Replacing in (7.13)  . This inequality fails at the eighth integral, in which a factor of ( ) ( ) / / / x x sin 15 15 appears. Note that the method of brackets does not produce an evaluation for the cases where this inequality fails.

Extension to Bessel functions
The result on Borwein integrals can be extended to the integral of a product of Bessel functions. Indeed, recall the Bessel function of the first kind is given by: