Pith. sign in

REVIEW 3 major objections 5 minor 17 references

Quadratic and quartic integrals using the method of brackets

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The method of brackets gives closed-form hypergeometric evaluations of quadratic and quartic integrals for any positive real exponent, extending table entries that were restricted to integers.

desk verdict A typical method-of-brackets integral application, but all closed forms rest on an unproved free-variable extension of Rule 3; numerical checks and a proof are needed before I would trust it. read the letter →

arxiv 1909.00962 v1 pith:EF3DYQYR submitted 2019-09-03 math-ph hep-phmath.MP

B Ananthanarayan , Sumit Banik , Sudeepan Datta , Tanay Pathak This is my paper · ORCID
classification math-phhep-phmath.MP MSC 33C0533C2033E20
keywords methodofbracketsdefiniteintegralshypergeometricfunctionsquadraticintegralquarticGradshteynandRyzhikRamanujanmastertheoremMellintransform
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper aims to show that the method of brackets turns a family of quadratic and quartic definite integrals—those appearing as entries 3.252 and 2.161 in Gradshteyn and Ryzhik—into closed-form hypergeometric expressions. Where the table gives the quadratic integrals only as repeated derivatives valid for natural-number exponents, the bracket evaluation yields formulas claimed to hold for every positive real exponent for which the integral converges. For quartic integrals, the table offers only recursion relations, and the paper supplies direct hypergeometric evaluations instead. If correct, the results would give a uniform way to evaluate these integrals and would generate new identities relating derivatives and hypergeometric functions.

What carries the argument

The machinery is the method of brackets, a formal operational calculus in which an integral over x is rewritten as a bracket series ⟨a⟩=∫_0^∞ $x^{{a-1}}$dx, and the brackets are then resolved by Rule 3, which assigns a divergent bracket series a finite value via Ramanujan's master theorem: ∑_n φ_n f(n)⟨an+b⟩ = (1/a) f(n*)Γ(-n*), where n* solves an+b=0. The bracket series turns the integrand into sums over nonnegative integers tied by linear equations; choosing different free variables produces different hypergeometric series whose common region of convergence determines which terms are added to give the full integral.

What would settle it

Compare the hypergeometric formula (26) against numerical integration of ∫_0^∞ dx/(a $x^{2}$ + 2 b x + c)^n for a non-integer exponent such as n = 3/4 with, say, a=2, b=1, c=3; any disagreement would show the bracket assignment is not producing the actual integral. Alternatively, test identity (28) at a specific natural number n by computing both sides directly.

Watch

Extended reading notes

Core claim

The central discovery is a set of closed-form evaluations: the integral I(a,b,c;n)=∫_0^∞ dx/(a $x^{2}$ + 2 b x + c)^n is expressed as a combination of a 1F0 and a 2F1 hypergeometric function (equation (26)), valid for all positive real n, and analogous expressions are derived for the numerator-weighted quadratic integral, a generalized quadratic integral ∫_0^∞ x^n dx/(a $x^{2}$ + 2 b x + c)^m, the quartic integral ∫_0^∞ dx/(a $x^{4}$ + 2 b $x^{2}$ + c)^m, and a generalized quartic integral. The paper further claims that by equating these new formulas with the known integer-exponent derivative formulas, one obtains new identities relating nth derivatives to hypergeometric functions.

Load-bearing premise

The derivation assumes that the method-of-brackets rule for assigning a finite value to divergent series (Ramanujan's master theorem) applies to each integrand, without verifying the theorem's analyticity and growth conditions.

Editorial extensions

If this is right

  • Integer-exponent table formulas become limiting cases of hypergeometric expressions that extend to all real positive exponents.
  • The generalized quadratic integral (43) subsumes entries 3.252-1, 3, 4, 7, 8 and 9 as special parameter choices.
  • The quartic evaluation generalizes the known a=c=1 bracket result and provides a closed form where the table has only a recursion.
  • The identities (28), (35), (42) relate nth partial derivatives to specific hypergeometric functions, giving new evaluation formulas for these derivatives.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the method's assignment is valid beyond the verified cases, similar bracket expansions can produce hypergeometric closed forms for other rational integrands whose denominators are sums of two powers, e.g., sixth- or eighth-degree polynomials.
  • The paper does not address the convergence domain at |b^2/(ac)|=1; the hypergeometric series may need analytic continuation there, and the boundary behavior is a natural test case.
  • The claimed validity "for all positive real n" could be checked term-by-term against numerical quadrature; discrepancies would indicate the method's divergent-series assignment diverges from the integral's actual value in some regimes.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper applies the method of brackets to evaluate definite integrals of the forms \int_0^\infty dx/(a x^2 + 2 b x + c)^n, \int_0^\infty x^n dx/(a x^2 + 2 b x + c)^m, and their quartic analogues \int_0^\infty dx/(a x^4 + 2 b x^2 + c)^m and \int_0^\infty dx/[x^n (a x^4 + 2 b x^2 + c)^m]. It claims closed-form hypergeometric expressions for these integrals, including generalizations not found in Gradshteyn and Ryzhik, and derives identities by equating the new expressions with the table's derivative or recursion forms.

Significance. If the results are correct, the paper provides useful closed forms and hypergeometric identities and demonstrates the method of brackets on a class of definite integrals. The paper is self-contained in presenting the method and its formal rules, and the authors clearly identify which table entries are generalized. However, the central derivations rest on an unproved extension of the method's Rule 3 to non-square bracket systems, and the paper does not include machine-checked proofs, numerical verification, or an independent derivation of the new formulas. The significance is therefore contingent on closing this gap.

major comments (3)
  1. [Section 3 (Rule 3 and Note); Eqs. (26)-(27), (44)-(45), (49)-(50), (52)-(53)] The central evaluations are obtained from bracket series with two brackets and three summation indices, a case that Rule 3 as stated in Eq. (13) explicitly excludes because the coefficient matrix is not invertible. The Note in Section 3 extends the rule by selecting free variables, discarding divergent series, and adding series with a common convergence region, but this extension is not derived from Ramanujan's master theorem, and no conditions for its applicability are stated or verified for any of the integrands. Since every closed form in the paper depends on this extension, the main claim is not established; the authors should either prove the extension under explicit hypotheses or provide numerical verification of each formula against direct quadrature.
  2. [Section 4 (Eqs. (26) and (27))] The paper presents Eq. (26) as 'valid for all the values of n \in R+' and Eq. (27) as 'valid for n /\in N', both supposedly evaluating the same integral (23), with overlapping domains. It is not explained whether these two expressions are equal on their common domain, whether one of them represents the full integral while the other is a partial contribution, or how the free-variable choices are combined according to the Note. This ambiguity directly affects the identities (28), (35), and (42) and must be resolved.
  3. [Section 3 (Eq. (15))] Even for the one-dimensional Rule 3, the proof via Ramanujan's master theorem requires the coefficient sequence f(n) to arise from an expansion of the form \sum \phi(k)(-x)^k satisfying the analyticity and growth conditions of the theorem; the paper verifies none of these conditions for the coefficient sequences used in Sections 4 and 5. The authors should either state the precise hypotheses under which the formal manipulations are valid or explicitly label the results as formal and verify them independently.
minor comments (5)
  1. [Section 1] The text contains grammatical slips: 'found it's use' and 'found it's used' should be 'found its use' and 'found its use'.
  2. [Section 3 (Rule 2)] The proof of Rule 2 concludes with 'which is rule 3', but the statement proved is the negative-power instance of Rule 2; the cross-reference is incorrect.
  3. [Section 3 (Eq. (13))] The multidimensional rule contains a typographical error: the last factor should be \Gamma(-n^*_r) rather than f(-n^*_r).
  4. [Section 4 (Eq. (33) and elsewhere)] The conditions of validity are stated as inequalities without precisely specifying the parameter domain, especially which parameters are real and positive; this should be clarified for each formula.
  5. [References] References [1] and [2] are incomplete; full bibliographic information should be provided.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the evaluations are derived from the stated bracket rules rather than assumed, and no fitted parameter or self-citation carries the argument.

full rationale

The paper's derivations start from the method of brackets, with Rule 3 stated as a consequence of Ramanujan's master theorem, and then expand the integrands and apply the bracket rule to obtain hypergeometric closed forms. No parameter is fitted to a subset of data and then renamed a prediction. The claimed identities (28), (35), and (42) are obtained by equating the method-of-brackets evaluation with the Gradshteyn and Ryzhik evaluation of the same integral; this is a synthetic consequence of having two evaluations of one quantity, not a definitional equivalence that makes the hypergeometric expression the input. The paper contains no self-citations by the present authors, and no uniqueness theorem is imported from the authors' prior work. The main limitation visible in the text is that the Note in Section 3 extends Rule 3 to non-square bracket systems without proof and without checking the hypotheses of Ramanujan's master theorem. That is a correctness or rigor concern about an unproved extension, not a circularity concern: the claimed values do not reduce by construction to the assumptions used to derive them. Against the external benchmark of the table entries and the reproduced Gaussian and Feynman--Hibbs examples, the derivation chain is self-contained. Therefore the circularity score is 0.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

The paper does not introduce any new physical entities or mathematical structures beyond the method's brackets. The only postulates are the operational rules of the method itself, which are stated without full proof. The derivations assume that these rules produce unique and correct evaluations.

free parameters (1)
  • a, b, c, n, m, x-power
    These are the original variables in the integrands. They are not fitted to data; they are parameters of the integrals. No free parameters are introduced by the derivations.
assumptions (3)
  • domain assumption Ramanujan's master theorem
    The method of brackets relies on Ramanujan's master theorem to assign a value to bracket series. The theorem is a standard result in complex analysis, but the paper does not verify that the integrands satisfy the theorem's conditions (e.g., the series expansion converges in a suitable domain and the function has appropriate growth).
  • domain assumption The method of brackets rules (Rules 1-3) are valid for the integrals considered
    The paper states the rules and proves Rule 3 under the assumption of Ramanujan's master theorem, but it does not rigorously justify the transition from divergent bracket series to finite evaluations for these specific integrands. The rule that involves summing over free variables and discarding divergent series is an operational assumption.
  • ad hoc to paper The bracket integral ⟨a⟩ = ∫_0^∞ x^{a-1} dx acts as a delta function inside summations
    The bracket is defined as a divergent integral, but the method treats it as a delta function when summed. The paper states this as a formal rule without present a rigorous justification. This is a necessary assumption for the method to produce finite results.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Quadratic and quartic integrals using the method of brackets." pith.science (2026). https://pith.science/paper/EF3DYQYR

@misc{pith2026190900962,
  author       = {Pith},
  title        = {Pith review of: Quadratic and quartic integrals using the method of brackets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EF3DYQYR}},
  note         = {Machine review of arXiv:1909.00962}
}
read the original abstract

We use the method of brackets to evaluate quadratic and quartic type integrals. We recall the operational rules of the method and give examples to illustrate its working. The method is then used to evaluate the quadratic type integrals which occur in entries 3.251.1,3,4 in the table of integrals by Gradshteyn and Ryzhik and obtain closed form expressions in terms of hypergeometric functions. The method is further used to evaluate the quartic integrals, entry 2.161.5 and 6 in the table. We also present generalization of both types of integrals with closed form expression in terms of hypergeometric functions.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

17 extracted references · 15 canonical work pages

  1. [1]

    Journal of Ramanujan Mathematical Society, Volume 29, I ssue 2, Page 37-43

    work page

  2. [2]

    Open Mathematics, Volume 15, Issue 1, Pages 112, ISSN (On line) 2391-5455, DOI: https://doi.org/10.1515/math-2017-0001

    work page doi:10.1515/math-2017-0001 2017

  3. [3]

    Ryzhik, Table of Integrals, Seri es and Products 7th Edition

    I.S Gradshteyn and I.M. Ryzhik, Table of Integrals, Seri es and Products 7th Edition

    work page

  4. [4]

    I. G. Halliday and R. M. Ricotta, Phys. Lett. B 193, 241 (1987). doi:10.1016/0370- 2693(87)91229-9

    doi:10.1016/0370- 1987

  5. [5]

    A. T. Suzuki, arXiv:1408.4064 [math-ph]

    work page Pith review arXiv

  6. [6]

    A. T. Suzuki, A. G. M. Schmidt and J. D. Bolzan, J. Phys. Soc . Jap. 81 (2012) 044101. doi:10.1143/JPSJ.81.044101

    work page doi:10.1143/jpsj.81.044101 2012

  7. [7]

    The method of brackets. Part 2: examples and applications

    I. Gonzalez, V. H. Moll and A. Straub, arXiv:1004.2062 [m ath-ph]

    work page Pith review arXiv

  8. [8]

    Gonzalez and V

    I. Gonzalez and V. H. Moll, Advances in Applied Mathemati cs https://doi.org/10.1016/j.aam.2009.11.003

    work page doi:10.1016/j.aam.2009.11.003 2009

Show all 17 references
  1. [9]

    Gonzalez, K

    I. Gonzalez, K. Kohl, I. Kondrashuk, V. H. Moll and D. Sali nas, SIGMA 13 (2017) 001 doi:10.3842/SIGMA.2017.001 [arXiv:1505.06274 [math-ph ]]

    2017 arXiv

  2. [10]

    Feynm an and A.R

    Quantum mechanics and path integrals, Richard P. Feynm an and A.R. Hibbs

  3. [11]

    & Moll, V.H

    Amdeberhan, T. & Moll, V.H. Ramanujan J (2009) 18: 91. ht tps://doi.org/10.1007/s11139- 007-9041-9

    2009 doi

  4. [12]

    George Boros, Victor H Moll Jour. Comp. Appl. Math. 106, 1999, 361368

    1999

  5. [13]

    Gonzalez, V

    I. Gonzalez, V. H. Moll and I. Schmidt, arXiv:1103.0588 [math-ph]

    arXiv

  6. [14]

    A. T. Suzuki, arXiv:math-ph/0407032

    arXiv

  7. [15]

    Amdeberhan, T., Espinosa, O., Gonzalez, I. et al. Raman ujan J (2012) 29: 103. https://doi.org/10.1007/s11139-011-9333-y

    2012 doi

  8. [16]

    Open Mathematics, Volume 15, Issue 1, Pages 11811211, I SSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2017-0100

    2017 doi

  9. [17]

    Gonzalez, Nucl

    I. Gonzalez, Nucl. Phys. Proc. Suppl. 205-206 (2010) 141 doi:10.1016/j.nuclphysbps.2010.08.033 [arXiv:1008.2148 [hep-th]]. 16

    2010 arXiv

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.