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 . 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  is the most widely used one, now in its 8th-edition.
The interest of the last author in this topic began with entry 3.248.5 in  (1)
where φ(x) = 1 + x2(1 + x2)−2. The value 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  nor in the latest edition . This motivated an effort to present proofs of all entries in Gradshteyn-Ryzhik. It began with  and has continued with several short papers. These have appeared in Revista Scientia, the latest one being .
The work presented here deals with the method of brackets. This is a new method for integration developed in [11–13] 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 emphasize 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 . The reader will find in [8–10] 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 (2)
where α, β ∈ ℂ and the coefficients a(n) ∈ 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 [0, ∞), 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 ∈ 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 α = 2, β = 1 and a(n) = 1/22nn!2 can be written as a(n) = 1/22nΓ2(n + 1) and now the evaluation, say at n = , is possible. The same observation holds for the Bessel function (4)
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 . This process is, up to now, completely heuristic. These non-classical series are classified according to the following types:
Totally (partially) divergent series. Each term (some of the terms) in the series is a divergent value. For example, (5)
Totally (partially) null series. Each term (some of the terms) in the series vanishes. For example, (6)
This type includes series where all but finitely many terms vanish. These are polynomials in the corresponding variable.
Formally divergent series. This is a classical divergent series: the terms are finite but the sum of the series diverges. For example, (7)
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 (8)
The Bessel K0-function (9)
with totally null representation (10)
and the totally divergent one (11)
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, b; 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 Kν(x), with special emphasis on K0(x). Section 8 gives examples of definite integral whose value contains the Bessel function Kν(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.
2 The method of brackets
The method of brackets evaluates integrals over the half line [0, ∞). It is based on a small number of rules reviewed in this section.
For a ∈ ℂ, the symbol (12)
is the bracket associated to the (divergent) integral on the right. The symbol (13)
is called the indicator associated to the index n. The notation ϕn1n2…nr, or simply ϕ12…r, denotes the product ϕn1ϕn2…ϕnr.
The indicator ϕn will be used in the series expressions used in the method of brackets. For instance (8) is written as (14)
and (11) as (15)
In the process of implementing the method of brackets, these series will be evaluated for n ∈ 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 (16)
a bracket series. This is done following two rules:
Rule P1. Assume f has the expansion (17)
Then I(f) is assigned the bracket series (18)
The series including the indicator ϕn have indices without limits, since its evaluation requires to take n outside ℕ.
Rule P2. For α ∈ ℂ, the multinomial power (u1 + u2 + … + ur)α is assigned the r-dimension bracket series (19)
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 E1. The one-dimensional bracket series is assigned the value (20)
where n* is obtained from the vanishing of the bracket; that is, n* solves an + b = 0.
The rule E1 is a version of the Ramanujan’s Master Theorem. This theorem requires an extension of the coefficients a(n) from n ∈ ℕ to n ∈ 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 . 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 E2. Assume the matrix B = (bij) is non-singular, then the assignment is
where is the (unique) solution of the linear system obtained from the vanishing of the brackets. There is no assignment if B is singular.
Rule E3. Each representation of an integral by a bracket series has associated an index of the representation via (21)
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.
A generic bracket series of index 1 has the form (22)
where a11, a12, c1 are fixed coefficients, A, B are parameters and C(n1, n2) is a function of the indices.
The Rule E3 is used to generate two series by leaving first n1 and then n2 as free parameters. The Rule E1 is used to assign a value to the corresponding series:
n1 as a free parameter produces
n2 as a free parameter produces
The series T1 and T2 are expansions of the solution in terms of different parameters (23)
Observe that Therefore the bracket series is assigned the value T1 or T2. 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 T1 and T2. Suppose a12 = −a11, then (24)
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 a12 = −2a11, then
Splitting the sum in T1 according to the parity of the indices produces a power series in A2B when n1 = 2n3 is even and for n1 odd a second power series in the same argument A2B times an extra factor AB1/2. Since these are expansions in the same argument, they have to be added to count their contribution to the 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.
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 E4. 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.
A systematic procedure in the simplification of the series has been used throughout the literature: express factorials in terms of the gamma function and the transform quotients of gamma terms into Pochhammer symbols, defined by (26)
Any presence of a Pochhammer with a negative index k is transformed by the rule (27)
In the special case when a is also a negative integer, the rule (28)
holds. This value is justified in . The duplication formula (29)
is also used in the simplifications.
Many of the evaluations are given as values of the hypergeometric functions (30)
with (a)n as in (26). It is often that the value of 2F1 at z = 1 is required. This is given by the classical formula of Gauss: (31)
The extension considered here is to use the method of brackets to functions that do not admit a series representation as described in Rule P1. For example, the Bessel function K0(x) has a singular expansion of the form (32)
(see [21, 10.31.2]). Here I0(x) is the Bessel function given in (3), is the harmonic number and is Euler’s constant. The presence of the logarithm term in (32) does not permit a direct application of the method of brackets. An alternative is presented in Section 7.
3 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 (33)
is associated the bracket series (34)
and rewriting (33) as (35)
provides the second bracket series (36)
associated to (33). It is shown next that all such bracket series representations of an integral produce the same value.
Assume f(x) = g(x)h(x), where f, g and h have expansions as in (2). Then, the method of brackets assigns the same value to the integrals (37)
with s = (1 + β)/α.
To evaluate the second integral, observe that
This yields (40)
and matching this with (38) gives β = β1 + β2 and (41)
Now, the method of brackets gives (42)
and it yields two series as solutions (43)
with s = (β + 1)/α. Comparing with (38) shows that I1 = I2 is equivalent to (44)
that is, (45)
It is direct to extend the result to the case of a finite number of factors.
Assume f admits a representation of the form Then the value of the integral, obtained by method of brackets, is the same for both series representations.
4 The exponential integral
The exponential integral function is defined by the integral formula (46)
(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 (47)
and then use the method of brackets to produce
Replace this in (47) to obtain (48)
The evaluation of this series by the method of brackets generates two identical terms for Ei(−x): (49)
Only one of them is kept, according to Rule E4. This is a partially divergent series (from the value at n = 0), written as (50)
The next example illustrates how to use this partially divergent series in the evaluation of an integral.
Entry 6.223 of  gives the Mellin transform of the exponential integral as (51)
To verify this, use the partially divergent series (50) and the method of brackets to obtain (52)
Entry 6.228.2 in  is (53)
The partially divergent series (50) is now used to establish this formula. First form the bracket series (54)
Rule E1 yields two cases from the equation n1 + n2 + v = 0:
Case 1: n2 = −n1 − v produces (55)
which is discarded since it is partially divergent (due to the term n1 = 0).
Case 2: n1 = −n2 − v gives (56)
and using (57)
equation (56) becomes (58)
The condition |μ|<|β| is imposed to guarantee the convergence of the series. Finally, the transformation rule (see entry 9. 131. 1 in ) (59)
with α = β = v, y = v + 1 and z = −μ/β yields (53).
The next evaluation is entry 6.232.2 in : (60)
A direct application of the method of brackets using (61)
This produces two series for G(a, b): (63)
The analysis begins with a simplification of T2. Use the duplication formula for the gamma function (65)
and write (66)
to obtain (67)
provided |b| < |a| to guarantee convergence. The form (60) comes from the identity (68)
(see 9.121.27 in ).
The next step is the evaluation of T1. Separating the sum (63) into even and odd indices yields (69)
and in hypergeometric form (70)
and this is the same as (60).
The evaluation of entry 6.232.1 in  (71)
is obtained in a similar form.
Entry 6.782.1 in  is (72)
is the classical Bessel function defined in (4). Therefore (74)
The standard procedure using the partially divergent series (49) now gives (75)
which gives the convergent series (76)
and the series (77)
Observe that the expression T2 contains a single non-vanishing term, so it is of the partially null type. An alternative form of T2 is to write (78)
The series 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 n2 = 0 is non-vanishing and T2 reduces to (79)
This gives the asymptotic behavior B(z)∼−1/z, consistent with the value of T1 for large z. This phenomena occurs every time one obtains a series of the form pFq(z) with p ≥ q+2 when the series diverges. The truncation represents an asymptotic approximation of the solution.
5 The Tricomi function
The confluent hypergeometric function, denoted by defined in (30), arises when two of the regular singular points of the differential equation for the Gauss hypergeometric function given by (80)
are allowed to merge into one singular point. More specifically, if we replace z by z/b in then the corresponding differential equation has singular points at 0, b and ∞. Now let b → ∞ so as to have infinity as a confluence of two singularities. This results in the function so that (81)
and the corresponding differential equation (82)
known as the confluent hypergeometric equation. Evaluation of integrals connected to this equation are provided in .
The equation (82) has two linearly independent solutions: (83)
known as the Kummer function and the Tricomi function with integral representation (84)
and hypergeometric form (85)
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 |x| < 1 and their sum yields (85). The series U3 is formally divergent, the terms are finite but the series is divergent.
The Mellin transform of the Tricomi function is given by (86)
Entry 7.612. 1 of  (87)
The result (88)
follows from simplification of the previous expression.
The second evaluation of (86) uses the method of brackets and the divergent series U3. It produces the result directly. Start with
A standard evaluation by the method of brackets now reproduces (88).
The evaluation of (89)
is given next. Start with the expansions (90)
This yields the two series
In the case |μ| < 1, both J1 and J2 are convergent. Therefore
In the case |μ| = 1, the series J2 diverges, so it is discarded. This produces (91)
Gauss’ value (31) gives (92)
In particular, if a is a positive integer, say a = k, then (93)
This result is summarized next.
Then, for |μ| < 1, (95)
and for μ = 1, (96)
In the special case a = k ∈ ℕ, (97)
6 The Airy function
The Airy function, defined by the integral representation (98)
satisfies the equation (99)
and the condition y → 0 as x → ∞. A second linearly independent solution of (99) is usually taken to be (100)
Using (61) produces
The usual resolution of this bracket series gives three cases: (101)
a totally null series, (102)
a partially divergent series (at the index n = 18), and (103)
a totally null series, as T1 was.
The series for Ai(x) are now used to evaluate the Mellin transform (104)
This integral is now computed using the three series Tj given above. Using first the value of T1 and the formulas (105)
(these appear as 8.335.1 and 8.335.2 in , respectively), give (106)
Similar calculations, using T2 or T3, give the same result. This result is stated next.
The Mellin transform of the Airy function is given by (107)
7 The Bessel function Kν
This section presents series representations for the Bessel function Kν(x) defined by the integral representation (108)
The usual procedure to evaluate this bracket series gives three expressions: (110)
The series T3 is a totally null series for Kv. In the case v ∉ ℕ, the series T1 and T2 are finite and Kv(x) = T1+T2 gives the usual expression in terms of the Bessel Iv function (111)
as given in entry 8.485 in .
In the case v = k ∈ ℕ, the series T1 is partially divergent (the terms n = 0,1,…, k have divergent coefficients) and the series T2 is totally divergent (every coefficient is divergent). In the case v = 0, both the series T1 and T2 become (112)
using Rule E4 to keep a single copy of the divergent series. This complements the (113)
The examples presented below illustrate the use of these divergent series in the computation of definite integrals with the Bessel function K0 in the integrand. Entries in  with K0 as the result of an integral have been discussed in .
Entry 6.511.12 of  states that (114)
To verify this result, use the totally null representation (113) to obtain (115)
The value of the bracket series is (116)
The Mellin transform (117)
and a direct evaluation of the brackets series using Rule E1 gives (119)
Now using the totally null representation (113) gives the bracket series (120)
One more application of Rule E1 gives (119) again.
Entry 6.611.9 of  is (121)
The usual procedure gives two expressions: (123)
which is discarded since it is divergent and (124)
Separating the series according to the parity of the index n yields (125)
The identity [16, 9. 121. 1] (126)
with gives (127)
The identity (128)
comes from the Taylor series (129)
(See Theorem 7.6.2 in  for a proof). The usual argument now gives (130)
an equivalent form of (121).
The next example, (131)
appears as entry 6.691 in . The factor sin bx in integrand is expressed as a series: (132)
and the Bessel factor is replaced by its totally-null representation (113) (133)
This yields (134)
These representation produces two solutions S1 and S2, one perfree index, that are identical. The method of brackets rules state that one only should be taken. This is: (135)
The result nowfollows from the identity (136)
and the binomial theorem obtaining (137)
The next example in this section evaluates (138)
From the representation (139)
and the null-series (10) it follows that (140)
This bracket series generates two identical series, so only one is kept to produce (141)
Here K(z) is the elliptic integral of the first kind. Using the identity (142)
The next example evaluates (144)
Naturally H(a) = H(1)/a, but it is convenient to keep a as a parameter. The problem is generalized to (145)
and H(a) = H1(a,a) . The evaluation uses the totally divergent series (112) (146)
as well as the integral representation (see 8.432.6 ) and the corresponding bracket series (147)
The evaluation of this bracket series requires an extra parameter ϵ and to consider (149)
Evaluating this brackets series produces three values, one divergent, which is discarded, and two others: (150)
with c = b2/a2. Converting the Γ-factors into Pochhammer symbols produces (151)
Let c → 1 (b → a) and use Gauss’ formula (31) to obtain
and this produces
Expanding H2(a, a, ϵ) in powers of ϵ gives (152)
Letting ϵ → 0 gives (153)
The final example in this section is the general integral (154)
The case a = b appears in .
The evaluation uses the integral representation (155)
appearing in [16, 8.432.6]. This produces the bracket series representation (156)
The second factor uses the totally null representation (10) (157)
Replacing in (154) produces the bracket series (158)
The vanishing of the brackets gives the system of equations (159)
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: n1 free. Then n2 = n1 − v and This gives
Case 2: n2 free. Then n1 = n2 + v and This gives
Case 3: n3 free. Then This produces
This series has the value zero. This proves the next statement:
The integral (160)
is given by
Some special cases of this evaluation are interesting in their own right. Consider first the case a = b. Using Gauss’ theorem (31) it follows that (161)
The integral (163)
is given by
The next special case is to take a = b and λ = v. Then (164)
This proves the next result:
The integral (166)
is given by
The last special case is ρ = 1; that is, the integral (167)
It is shown that the usual application of the method of brackets yield only divergent series, so a new approach is required.
The argument begins with converting the brackets series in (158) to (168)
A routine application of the method of brackets gives three series
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 ϵ → 0 gives (170)
In the special case λ = v, it follows that (171)
8 An example with an integral producing the Bessel function
The evaluation of integrals in Section 7 contain the Bessel function Kv in the integrand. This section uses the method developed in the current work to evaluate some entries in  where the answer involves K0.
The first example is entry 6.532.4 in  (172)
The analysis begins with the series (173)
Rule P2 gives (174)
The method of brackets produces three series as candidates for solutions, one perfree index n1, n2, n3: (176)
The fact that T1 = T3 and using Rule E4 shows that only one of these series has to be counted. Since T1 and T2 are non-classical series of distinct variables, both are representations of the value of the integral. Observe that T2 is the totally null representation of K0 (ab) given in (11). This confirms (172). The fact that T3 is also a value for the integral gives another totally divergent representation for K0: (177)
The bracket series is evaluated using Rule E1 to confirm (114).
Entry 6.226.2 in  is (179)
The evaluation starts with the partially divergent series (50) (180)
and this yields (181)
The method of brackets gives two series. The first one (182)
using (11). The second series is (183)
Now shift the index by m = n2+1 to obtain
This is the same sum as T1 in the second line of (182). Recall that the summation indices are placed after the conversion of the indicator ϕn2 to its expression in terms of the gamma function. According to Rule E4, the sum T2 is discarded. This establishes (179).
9 A new use of the method of brackets
This section introduces a procedure to evaluate integrals of the form (184)
Differentiating with respect to the parameters leads to (185)
Integration by parts produces (186)
A direct extension to many parameters leads to the following result.
The integral (189)
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) (190)
is obtained directly from (90) (191)
Solving for n1 in the equation coming from the vanishing of the bracket gives n1 = −2n2−1, which yields (193)
The identity A direct calculation shows that the series obtained from solving for n2 yields the same solution, so it is discarded. Therefore (195)
The evaluation of this integral using Theorem 9.1 begins with checking that the boundary terms vanish. This comes from the asymptotic behavior J0(x) ∼ 1 as x → 0 and as x → ∞. The term (196)
This generates two series (197)
which yields the two series
Since the boundary terms vanish, the relation (186) gives (200)
The form T2 + is simplified by converting them to hypergeometric form to produce (201)
This gives (203)
The option T1 + gives the same result.
Entry 6.222 in  is (204)
In particular (205)
The usual procedure requires the relation n1 + n2 + 1 = 0 and taking n1 as the free parameter gives (207)
and when n2 as free parameter one obtains the series (208)
These two series correspond to different expansions: the first one in x = a1/a2 and the second one in x−1 = a2/a1. Both series are partially divergent, so the Rule E3 states that these sums must be discarded. The usual method of brackets fails for this problem.
using (206) to compute the partial derivatives. The method of brackets gives two series for each of the sums S1 and S2: (210) (211) (212) (213)
the series T1,1 and T1,2 come from the first sum S1 and T2,1, T2,2 from S2. Rule E3 indicates that the value of the integral is either (214)
the first form is an expression in a1/a2 and the second one in a2/a1.
The series T1, 1 is convergent when |a1|<|a2| and it produces the function (215)
and T2,2 is also convergent and is gives (216)
Observe that, according to (214) to complete the evaluation of I(a1, a2), 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 T2,1 as a partially divergent series attached to the function g(a1, a2). Therefore, the sum in (214), the term T2, 1 is replaced by g(a1, a2) to produce (217)
and this confirms (204). A similar interpretation of T1,2 + T2,2 gives the same result.
The method of brackets consists of a small number of heuristic rules used for the evaluation of definite integrals on [0, +∞). The original formulation of the method applied to functions that admit an expansion of the form a(n)xαn+β−1. The results presented here extend this method to functions, like the Bessel function Kv and the exponential integral Ei, where the expansions have expansions of the form Γ(−n)xn (where all the coefficients are divergent) or (where all the coefficients vanish). A variety of examples illustrate the validity of this formal procedure.
The authors wish to thank a referee for a careful reading of the original version of the paper. The first author thanks the support of the Centro de Astrofísica de Valparaiso. The last author acknowledges the partial support of NSF-DMS 1112656.
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About the article
Published Online: 2017-09-27
Citation Information: Open Mathematics, Volume 15, Issue 1, Pages 1181–1211, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2017-0100.
© 2017 Gonzalez et al.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0