The Aquila Digital Community The Aquila Digital Community An Extension of the Method of Brackets: Part 1 An Extension of the Method of Brackets: Part 1

: 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.


Introduction
The evaluation of definite integrals have a long history dating from the work of Eudoxus of Cnidus (408-355 BC) with the creation of the method of exhaustion. The history of this problem is reported in [17]. A large variety of methods developed for the evaluations of integrals may be found in older Calculus textbooks, such as those by J. Edwards [4,5]. As the number of examples grew, they began to be collected in tables of integrals. The table compiled by I. S. Gradshteyn and I. M. Ryzhik [16] is the most widely used one, now in its 8 th -edition.
The interest of the last author in this topic began with entry 3:248:5 in [14] I D where '.x/ D 1 C 4 3 x 2 .1 C x 2 / 2 : The value =2 p 6 given in the table is incorrect, as a direct numerical evaluation will confirm. Since an evaluation of the integral still elude us, the editors of the table found an ingenious temporary solution to this problem: it does not appear in [15] nor in the latest edition [16]. This motivated an effort to present proofs of all entries in Gradshteyn-Ryzhik. It began with [19] and has continued with several short papers. These have appeared in Revista Scientia, the latest one being [1].
The work presented here deals with the method of brackets. This is a new method for integration developed in [11][12][13] in the context of integrals arising from Feynman diagrams. It consists of a small number of rules that converts the integrand into a collection of series. These rules are reviewed in Section 2, it is important to em-phasize that most of these rules are still not rigorously justified and currently should be considered a collection of heuristic rules.
The success of the method depends on the ability to give closed-form expressions for these series. Some of these heuristic rules are currently being placed on solid ground [2]. The reader will find in [8][9][10] a large collection of examples that illustrate the power and flexibility of this method.
The operational rules are described in Section 2. The method applies to functions that can be expanded in a formal power series f .x/ D 1 X nD0 a.n/x˛n Cˇ 1 ; where˛;ˇ2 C and the coefficients a.n/ 2 C. (The extra 1 in the exponent is for a convenient formulation of the operational rules). The adjective formal refers to the fact that the expansion is used to integrate over OE0; 1/, even though it might be valid only on a proper subset of the half-line.
There is no precise description of the complete class of functions f for which the method can be applied. At the moment, it is a working assumption, that the coefficients a.n/ in (2) are expressions that admit a unique meromorphic continuation to n 2 C. This is required, since the method involves the evaluation of a.n/ for n not a natural number, hence an extension is needed. For example, the Bessel function (3) has˛D 2;ˇD 1 and a.n/ D 1=2 2n nŠ 2 can be written as a.n/ D 1=2 2n 2 .n C 1/ and now the evaluation, say at n D 1 2 , is possible. The same observation holds for the Bessel function The goal of the present work is to produce non-classical series representations for functions f , which do not have expansions like (2). These representations are formally of the type (2) but some of the coefficients a.n/ might be null or divergent. The examples show how to use these representations in conjunction with the method of brackets to evaluate definite integrals. The examples presented here come from the table [16]. This process is, up to now, completely heuristic. These non-classical series are classified according to the following types: 1/ Totally (partially) divergent series. Each term (some of the terms) in the series is a divergent value. For example, . n/x n and 1 X nD0 .n 3/ nŠ x n : 2/ Totally (partially) null series. Each term (some of the terms) in the series vanishes. For example, 1 X nD0 1 . n/ x n and 1 X nD0 1 .3 n/ x n : This type includes series where all but finitely many terms vanish. These are polynomials in the corresponding variable.
3/ Formally divergent series. This is a classical divergent series: the terms are finite but the sum of the series diverges. For example, 1 X nD0 nŠ 2 .n C 1/ .2n/Š 5 n : In spite of the divergence of these series, they will be used in combination with the method of brackets to evaluate a variety of definite integrals. Examples of these type of series are given next. Some examples of functions that admit non-classical representations are given next. The exponential integral with the partially divergent series The Bessel K 0 -function with totally null representation and the totally divergent one Section 2 presents the rules of the method of brackets. Section 3 shows that the bracket series associated to an integral is independent of the presentation of the integrand. The remaining sections use the method of brackets and non-classical series to evaluate definite integrals. Section 4 contains the exponential integral Ei. x/ in the integrand, Section 5 has the Tricomi function U.a; bI x/ (as an example of the confluent hypergeometric function), Section 6 is dedicated to integrals with the Airy function Ai.x/ and then Section 7 has the Bessel function K .x/, with special emphasis on K 0 .x/. Section 8 gives examples of definite integral whose value contains the Bessel function K .x/.
The final section has a new approach to the evaluation of bracket series, based on a differential equation involving parameters.
The examples presented in the current work have appeared in the literature, where the reader will find proofs of these formulas by classical methods. One of the goals of this work is to illustrate the flexibility of the method of brackets to evaluate these integrals.

The method of brackets
The method of brackets evaluates integrals over the half line OE0; 1/. It is based on a small number of rules reviewed in this section.
is the bracket associated to the (divergent) integral on the right. The symbol is called the indicator associated to the index n. The notation n 1 n 2 n r , or simply 12 r , denotes the product n 1 n 2 n r .
Note 2.2. The indicator n will be used in the series expressions used in the method of brackets. For instance (8) is written as Brought to you by | Cook Library -Serials Authenticated Download Date | 8/13/18 10:29 PM and (11) as In the process of implementing the method of brackets, these series will be evaluated for n 2 C, not necessarily positive integers. Thus the notation for the indices does not include its range of values.

Rules for the production of bracket series
The first part of the method is to associate to the integral a bracket series. This is done following two rules: Then I.f / is assigned the bracket series Note 2.3. The series including the indicator n have indices without limits, since its evaluation requires to take n outside N.
Rule P 2 . For˛2 C, the multinomial power .u 1 C u 2 C C u r /˛is assigned the r-dimension bracket series X n 1 ;n 2 ;:::;n r n 1 n 2 n r u n 1 1 u n r r h ˛C n 1 C C n r i . ˛/ : The integer r is called the dimension of the bracket series.

Rules for the evaluation of a bracket series
The next set of rules associates a complex number to a bracket series.
Rule E 1 . The one-dimensional bracket series is assigned the value X n n a.n/h˛n C bi D 1 j˛j a.n /. n /; where n is obtained from the vanishing of the bracket; that is, n solves an C b D 0.
Note 2.4. The rule E 1 is a version of the Ramanujan's Master Theorem. This theorem requires an extension of the coefficients a.n/ from n 2 N to n 2 C. The assumptions imposed on the function f is precisely for the application of this result. A complete justification of this rule is provided in [2]. Making the remaining rules rigorous is the subject of active research.
The next rule provides a value for multi-dimensional bracket series where the number of sums is equal to the number of brackets.
Rule E 2 . Assume the matrix B D .b ij / is non-singular, then the assignment is X n 1 ;n 2 ; ;n r n 1 n r a.n 1 ; ; n r /hb 11 n 1 C C b 1r n r C c 1 i hb r1 n 1 C C b rr n r C c r i Brought to you by | Cook Library -Serials Authenticated Download Date | 8/13/18 10:29 PM D 1 jdet.B/j a.n 1 ; n r /. n 1 / . n r / where fn i g is the (unique) solution of the linear system obtained from the vanishing of the brackets. There is no assignment if B is singular.
Rule E 3 . 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: In the case of a multi-dimensional bracket series of positive index, the system generated by the vanishing of the coefficients has a number of free parameters. The solution is obtained by computing all the contributions of maximal rank in the system by selecting these free parameters. Series expressed in the same variable (or argument) are added.
Example 2.5. A generic bracket series of index 1 has the form X n 1 ; n 2 n 1 ;n 2 C.n 1 ; n 2 /A n 1 B n 2 ha 11 n 1 C a 12 n 2 C c 1 i; where a 11 ; a 12 ; c 1 are fixed coefficients, A; B are parameters and C.n 1 ; n 2 / is a function of the indices. The Rule E 3 is used to generate two series by leaving first n 1 and then n 2 as free parameters. The Rule E 1 is used to assign a value to the corresponding series: n 1 as a free parameter produces Ã AB a 11 =a 12 Á n 1 I n 2 as a free parameter produces The series T 1 and T 2 are expansions of the solution in terms of different parameters x 1 D AB a 11 =a 12 and x 2 D BA a 12 =a 11 : Observe that x 2 D x a 12 =a 11 1 . Therefore the bracket series is assigned the value T 1 or T 2 . If one of the series is a null-series or divergent, it is discarded. If both series are discarded, the method of brackets does not produce a value for the integral that generates the bracket series.
Some special cases will clarify the rules to follow in the use of the series T 1 and T 2 . Suppose a 12 D a 11 , then and and since both series are expansions in the same parameter .AB/, their values must be added to compute the value associated to the bracket series. On the other hand, if a 12 D 2a 11 , then and T 2 D A c 1 =a 11 Splitting the sum in T 1 according to the parity of the indices produces a power series in A 2 B when n 1 D 2n 3 is even and for n 1 odd a second power series in the same argument A 2 B times an extra factor AB 1=2 . Since these are expansions in the same argument, they have to be added to count their contribution to the bracket series.
Brought to you by | Cook Library -Serials Authenticated Download Date | 8/13/18 10:29 PM Note 2.6. 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.
Note 2.7. The extension presented in this work shows how to use these divergent series in the evaluation of definite integrals. Example 9.3 illustrates this procedure.
Rule E 4 . In the evaluation of a bracket series, repeated series are counted only once. For instance, a convergent series appearing repeated in the same region of convergence should be counted only once. The same treatment should be given to null and divergent series.
Any presence of a Pochhammer with a negative index k is transformed by the rule In the special case when a is also a negative integer, the rule holds. This value is justified in [7]. The duplication formula is also used in the simplifications. Many of the evaluations are given as values of the hypergeometric functions p F q Â a 1 ; : : : ; a p b 1 ; : : : with .a/ n as in (26). It is often that the value of 2 F 1 at z D 1 is required. This is given by the classical formula of Gauss: Note 2.10. The extension considered here is to use the method of brackets to functions that do not admit a series representation as described in Rule P 1 . For example, the Bessel function K 0 .x/ has a singular expansion of the form

Independence of the factorization
The evaluation of a definite integral by the method of brackets begins with the association of a bracket series to the integral. It is common that the integrand contains several factors from which the bracket series is generated. This representation is not unique. For example, the integral is associated the bracket series X n 1 ;n 2 n 1 ;n 2 a n 1 2 2n 2 .n 2 C 1/ and rewriting (33) as provides the second bracket series X n 1 ;n 2 ;n 3 n 1 ;n 2 ;n 3 a n 1 Cn 2 2 n 1 Cn 2 C2n 3 .n 3 C 1/ associated to (33). It is shown next that all such bracket series representations of an integral produce the same value.
x/, where f; g and h have expansions as in (2). Then, the method of brackets assigns the same value to the integrals Proof. Suppose that h.x/ D X n 2 n 2 c .n 2 / x˛n 2 Cˇ2 : Then To evaluate the second integral, observe that This yields and matching this with (38) givesˇDˇ1 Cˇ2 and Now, the method of brackets gives and it yields two series as solutions with s D .ˇC 1/=˛. Comparing with (38) shows that I 1 D I 2 is equivalent to .s/a. s/ D X n n .n C s/b.n/c. s n/; that is, a. s/ D X n n .s/ n b.n/c. s n/: The identity (45) is the extension of (41) from n 2 N to s 2 C. This extension is part of the requirements on the functions f explained in Note 1.1. The proof is complete.
It is direct to extend the result to the case of a finite number of factors.
Then the value of the integral, obtained by method of brackets, is the same for both series representations.

The exponential integral
The exponential integral function is defined by the integral formula Brought to you by | Cook Library -Serials Authenticated Download Date | 8/13/18 10:29 PM (See [16, 8:211:1]). The method of brackets is now used to produce a non-classical series for this function. Start by replacing the exponential function by its power series to obtain and then use the method of brackets to produce Replace this in (47) to obtain Ei. x/ D X n 1 ;n 2 ;n 3 The evaluation of this series by the method of brackets generates two identical terms for Ei. x/: Only one of them is kept, according to Rule E 4 . This is a partially divergent series (from the value at n D 0), written as The next example illustrates how to use this partially divergent series in the evaluation of an integral.
Example 4.1. Entry 6:223 of [16] gives the Mellin transform of the exponential integral as To verify this, use the partially divergent series (50) and the method of brackets to obtain as claimed.
The partially divergent series (50) is now used to establish this formula. First form the bracket series G. ; ;ˇ/ D X n 1 ;n 2 n 1 ;n 2ˇn 1 n 2 Rule E 1 yields two cases from the equation n 1 C n 2 C D 0: Brought to you by | Cook Library -Serials Authenticated Download Date | 8/13/18 10:29 PM Case 1: n 2 D n 1 produces which is discarded since it is partially divergent (due to the term n 1 D 0).
equation (56) becomes The condition j j < jˇj is imposed to guarantee the convergence of the series. Finally, the transformation rule (see entry 9:131: with˛DˇD ; D C 1 and z D =ˇyields (53).
A direct application of the method of brackets using This produces two series for G.a; b/: and The analysis begins with a simplification of T 2 . Use the duplication formula for the gamma function to obtain provided jbj < jaj to guarantee convergence. The form (60) comes from the identity (see 9:121:27 in [16]). The next step is the evaluation of T 1 . Separating the sum (63) into even and odd indices yields and in hypergeometric form and this is the same as (60).
The evaluation of entry 6:232:1 in [16] is obtained in a similar form. Here is the classical Bessel function defined in (4). Therefore The standard procedure using the partially divergent series (49) now gives B.z/ D X n 1 ;n 2 n 1 ;n 2 Brought to you by | Cook Library -Serials Authenticated Download Date | 8/13/18 10:29 PM which gives the convergent series (76) and the series . z/ n 2 .1 n 2 / : Observe that the expression T 2 contains a single non-vanishing term, so it is of the partially null type. An alternative form of T 2 is to write The series 2 F 0 Â a b ˇzÃ diverges, unless one of the parameters a or b is a non-positive integer, in which case the series terminates and it reduces to a polynomial. This is precisely what happens here: only the term for n 2 D 0 is non-vanishing and T 2 reduces to This gives the asymptotic behavior B.z/ 1=z, consistent with the value of T 1 for large z. This phenomena occurs every time one obtains a series of the form p F q .z/ with p q C 2 when the series diverges. The truncation represents an asymptotic approximation of the solution.

The Tricomi function
The confluent hypergeometric function, denoted by 1 , defined in (30), arises when two of the regular singular points of the differential equation for the Gauss hypergeometric function 2 F 1 Â a b cˇz Ã , given by are allowed to merge into one singular point. More specifically, if we replace z by z=b in 2 F 1 Â a b cˇz Ã , then the corresponding differential equation has singular points at 0, b and 1. Now let b ! 1 so as to have infinity as a confluence of two singularities. This results in the function 1 F 1 Â a cˇz Ã so that and the corresponding differential equation zy 00 C .c z/y 0 ay D 0; Brought to you by | Cook Library -Serials Authenticated Download Date | 8/13/18 10:29 PM known as the confluent hypergeometric equation. Evaluation of integrals connected to this equation are provided in [3].
The equation (82) and hypergeometric form A direct application of the method of brackets gives This is a bracket series of index 1 and its evaluation produces three terms: The first two are convergent in the region jxj < 1 and their sum yields (85). The series U 3 is formally divergent, the terms are finite but the series is divergent.
is used in the evaluation of I.a; b;ˇ/. A proof of (87) appears in [3]. The first evaluation of (86) uses the hypergeometric representation (85) and the formula (87). This is a traditional computation. Direct substitution gives The result follows from simplification of the previous expression. The second evaluation of (86) uses the method of brackets and the divergent series U 3 . It produces the result directly. Start with A standard evaluation by the method of brackets now reproduces (88).
to write J.a; bI / D 1 .a/.1 C a b/ X n 1 ;n 2 n 1 ;n 2 n 1 .a C n 2 /.1 C a b C n 2 /hn 1 a n 2 C 1i: This yields the two series In the case j j < 1, both J 1 and J 2 are convergent. Therefore In the case D 1, the series J 2 diverges, so it is discarded. This produces Gauss' value (31) gives In particular, if a is a positive integer, say a D k, then This result is summarized next.
In the special case a D k 2 N,

The Airy function
The Airy function, defined by the integral representation satisfies the equation d 2 y dx 2 xy D 0; and the condition y ! 0 as x ! 1. A second linearly independent solution of (99) is usually taken to be Bi.x/ D 1 x n 2 h 2n 1 C n 2 C n 3 i 1 2 n 1 2 2n 1 . 2n 1 /3 n 3 1 Z 0 t 3n 3 Cn 2 dt D X n 1 ;n 2 ;n 3 n 1 ;n 2 ;n 3 x n 2 p . 2n 1 / 1 2 C n 1 2 2n 1 3 n 3 h 2n 1 C n 2 C n 3 i h3n 3 C n 2 C 1i: The usual resolution of this bracket series gives three cases: a totally null series, a partially divergent series (at the index n D 18), and a totally null series, as T 1 was. (108) given as entry 8:432:5 in [16]. Using the representation (61) of cos t as 0 F 1 1 2ˇ t 2 4 ! and using Rule P 2 in Section 2 to expand the binomial in the integrand as a bracket series gives K .x/ D 2 X n 1 ;n 2 ;n 3 n 1 ;n 2 ;n 3 x 2n 3 C 2 2n 1 .
The usual procedure to evaluate this bracket series gives three expressions: The series T 3 is a totally null series for K . In the case 6 2 N, the series T 1 and T 2 are finite and K .x/ D T 1 C T 2 gives the usual expression in terms of the Bessel I function as given in entry 8:485 in [16].
In the case D k 2 N, the series T 1 is partially divergent (the terms n D 0; 1; : : : ; k have divergent coefficients) and the series T 2 is totally divergent (every coefficient is divergent). In the case D 0, both the series T 1 and T 2 become Totally divergent series for K 0 .x/ D 1 2 using Rule E 4 to keep a single copy of the divergent series. This complements the Totally null series for K 0 .x/ D X n n 2 2n . n/ 2 .n C 1 2 /x 2n 1 : The examples presented below illustrate the use of these divergent series in the computation of definite integrals with the Bessel function K 0 in the integrand. Entries in [16] with K 0 as the result of an integral have been discussed in [6].
Example 7.1. Entry 6:511:12 of [16] states that To verify this result, use the totally null representation (113) to obtain The value of the bracket series is X n 1 ;n 2 n 1 n 2 . n 2 / a n 1 b 2n 2 2 2n 2 hn 1 C 2n 2 C 1i: The usual procedure gives two expressions: which is discarded since it is divergent and Separating the series according to the parity of the index n yields The identity [16, 9: The identity a b 2 comes from the Taylor series 2x an equivalent form of (121).
Example 7.4. The next example, appears as entry 6:691 in [16]. The factor sin bx in integrand is expressed as a series: These representation produces two solutions S 1 and S 2 , one per free index, that are identical. The method of brackets rules state that one only should be taken. This is: The result now follows from the identity and the binomial theorem obtaining Brought to you by | Cook Library -Serials Authenticated Download Date | 8/13/18 10:29 PM Example 7.5. The next example in this section evaluates From the representation and the null-series (10) it follows that G.a; b/ D X n 1 ;n 2 n 1 ;n 2 a 2n 1 2 2.n 2 n 1 / 2 .n 2 C 1 2 / .n 1 C 1/. n 2 /b 2n 2 C1 h2n 1 2n 2 i: This bracket series generates two identical series, so only one is kept to produce Here K.z/ is the elliptic integral of the first kind. Using the identity Example 7.6. The next example evaluates Naturally H.a/ D H.1/=a, but it is convenient to keep a as a parameter. The problem is generalized to and H.a/ D H 1 .a; a/. The evaluation uses the totally divergent series (112) as well as the integral representation (see 8:432:6 [16]) and the corresponding bracket series a 2n 1 b 2n 3 . n 1 / 2 2n 1 C2n 3 C2 hn 2 n 3 i h2n 1 C 2n 3 C 1i: The evaluation of this bracket series requires an extra parameter " and to consider H 2 .a; b; "/ D X n 1 ;n 2 ;n 3 n 1 ;n 2 ;n 3 a 2n 1 b 2n 3 . n 1 / 2 2n 1 C2n 3 C2 hn 2 n 3 C "i h2n 1 C 2n 3 C 1i: Evaluating this brackets series produces three values, one divergent, which is discarded, and two others: c " X n n . n "/ 2 ." C n C 1 2 /c n (150) Converting the -factors into Pochhammer symbols produces This yields The case a D b appears in [18].
The vanishing of the brackets gives the system of equations The matrix of coefficients is of rank 2, so it produces three series as candidates for values of the integral, one per free index.
Case 1: n 1 free. Then n 2 D n 1 and n 3 D 1 2 C n 1 . This gives Case 2: n 2 free. Then n 1 D n 2 C and n 3 D C 1 2 C n 2 . This gives Case 3: n 3 free. Then n 2 D n 3 C C1 2 and n 1 D n 3 C C C1 2 . This produces This series has the value zero. This proves the next statement: Some special cases of this evaluation are interesting in their own right. Consider first the case a D b. Using Gauss' theorem (31) it follows that and Proposition 7.9. The integral is given by The next special case is to take a D b and D . Then and This proves the next result: is given by The last special case is D 1; that is, the integral M.a; bI ; / D 1 Z 0 K .ax/K .bx/ dx: Brought to you by | Cook Library -Serials Authenticated Download Date | 8/13/18 10:29 PM It is shown that the usual application of the method of brackets yield only divergent series, so a new approach is required.
Proceeding as before produces a null series that is discarded and also In the limit as b ! a, these become Passing to the limit as " ! 0 gives In the special case D , it follows that

An example with an integral producing the Bessel function
The evaluation of integrals in Section 7 contain the Bessel function K in the integrand. This section uses the method developed in the current work to evaluate some entries in [16] where the answer involves K 0 .
Example 8.1. The first example is entry 6:532:4 in [16] The analysis begins with the series Rule P 2 gives 1 x 2 C b 2 D X n 2 ;n 3 n 2 ;n 3 x 2n 2 b 2n 3 h1 C n 2 C n 3 i: (174) The method of brackets produces three series as candidates for solutions, one per free index n 1 ; n 2 ; n 3 : The fact that T 1 D T 3 and using Rule E 4 shows that only one of these series has to be counted. Since T 1 and T 2 are non-classical series of distinct variables, both are representations of the value of the integral. Observe that T 2 is the totally null representation of K 0 .ab/ given in (11). This confirms (172). The fact that T 3 is also a value for the integral gives another totally divergent representation for K 0 : To test its validity, the integral in Example 7.1 is evaluated again, this time using (177): The evaluation starts with the partially divergent series (50) Ei and this yields ! e x dx D X n 1 ;n 2 n 1 n 2 a 2n 1 n 2 n 1 2 2n 1 hn 2 n 1 C 1i: The method of brackets gives two series. The first one using (11). The second series is This is the same sum as T 1 in the second line of (182). Recall that the summation indices are placed after the conversion of the indicator n 2 to its expression in terms of the gamma function. According to Rule E 4 , the sum T 2 is discarded. This establishes (179).

A new use of the method of brackets
This section introduces a procedure to evaluate integrals of the form Differentiating with respect to the parameters leads to a 1 @I.a 1 ; a 2 / @a 1 C a 2 @I.a 1 ; a 2 / @a 2 D Integration by parts produces I.a 1 ; a 2 / D xf 1 .a 1 x/f 2 .a 2 x/ˇ1 0 Â a 1 @I.a 1 ; a 2 / @a 1 C a 2 @I.a 1 ; a 2 / @a 2 Ã : A direct extension to many parameters leads to the following result. ; a n / D f .a j x/ dx: Then I.a 1 ; ; a n / D x a j @I.a 1 ; ; a n / @a j : Example 9.2. The integral is evaluated first by a direct application of the method of brackets and then using Theorem 9.1. The bracket series for I.a; b/ I.a; b/ D X n 1 ;n 2 n 1 ;n 2 a n 1 b 2n 2 2 2n 2 .n 2 C 1/ is obtained directly from (90) e ax D X n 1 n 1 a n 1 x n 1 and Solving for n 1 in the equation coming from the vanishing of the bracket gives n 1 D 2n 2 1, which yields . 1/ n 2 n 2 Š a 2n 2 1 b 2n 2 2 2n 2 .2n 2 C 1/ .n 2 C 1/ : To simplify this sum transform the gamma factors via (26) and use the duplication formula (29) to produce The A direct calculation shows that the series obtained from solving for n 2 yields the same solution, so it is discarded. Therefore The evaluation of this integral using Theorem 9.1 begins with checking that the boundary terms vanish. This comes from the asymptotic behavior J 0 .x/ 1 as x ! 0 and J 0 .x/ r 2 x cos x as x ! 1. The term a @I.a; b/ @a D X n 1 ;n 2 n 1 n 2 n 1 a n 1 b 2n 2 2 2n 2 .n 2 C 1/ hn 1 C 2n 2 C 1i: This generates two series @I.a; b/ @b D 2 X n 1 ;n 2 n 1 ;n 2 n 2 a n 1 b 2n 2 2 2n 2 .n 2 C 1/ hn 1 C 2n 2 C 1i (199) which yields the two series Since the boundary terms vanish, the relation (186) gives The form T 2 C Q T 2 is simplified by converting them to hypergeometric form to produce This gives The option T 1 C Q T 1 gives the same result.
The evaluation of this integral by the method of brackets begins with the partially divergent series for Ei. x/ which yields (using (14) D (50)): I.a 1 ; a 2 / D X n 1 ;n 2 n 1 ;n 2 a n 1 1 a n 2 2 n 1 n 2 hn 1 C n 2 C 1i: The usual procedure requires the relation n 1 C n 2 C 1 D 0 and taking n 1 as the free parameter gives . 1/ n 2 n 2 .n 2 C 1/ Â a 2 a 1 Ã n 2 : (208) These two series correspond to different expansions: the first one in x D a 1 =a 2 and the second one in x 1 D a 2 =a 1 . Both series are partially divergent, so the Rule E 3 states that these sums must be discarded. The usual method of brackets fails for this problem. The solution using Theorem 9.1 is described next. An elementary argument shows that xEi. x/ ! 0 as x ! 0 or 1. Then (186) becomes (213) the series T 1;1 and T 1;2 come from the first sum S 1 and T 2;1 ; T 2;2 from S 2 . Rule E 3 indicates that the value of the integral is either I.a 1 ; a 2 / D T 1;1 C T 2;1 or I.a 1 ; a 2 / D T 1;2 C T 2;2 I (214) the first form is an expression in a 1 =a 2 and the second one in a 2 =a 1 . The series T 1;1 is convergent when ja 1 j < ja 2 j and it produces the function f .a 1 ; a 2 / D 1 a 1 log and T 2;2 is also convergent and is gives g.a 1 ; a 2 / D 1 a 2 log Observe that, according to (214) to complete the evaluation of I.a 1 ; a 2 /, some of the series required are partially divergent series. The question is how to make sense of these divergent series. The solution proposed here is, for instance, to interpret T 2;1 as a partially divergent series attached to the function g.a 1 ; a 2 /. Therefore, the sum in (214), the term T 2;1 is replaced by g.a 1 ; a 2 / to produce I.a 1 ; a 2 / D f .a 1 ; a 2 / C g.a 1 ; a 2 /

Conclusions
The method of brackets consists of a small number of heuristic rules used for the evaluation of definite integrals on OE0; C1/. The original formulation of the method applied to functions that admit an expansion of the form