Convergence criteria for Frullani-type integrals involving differences of cosines

Atiratch Laoharenoo; Chanatip Sujsuntinukul

arxiv: 2605.26153 · v1 · pith:FUWFL7HInew · submitted 2026-05-23 · 🧮 math.GM

Convergence criteria for Frullani-type integrals involving differences of cosines

Atiratch Laoharenoo , Chanatip Sujsuntinukul This is my paper

Pith reviewed 2026-06-30 12:07 UTC · model grok-4.3

classification 🧮 math.GM

keywords Frullani integralscosine differencesconvergence criteriaimproper integralstrigonometric expansionscombinatorial identitiesparameter classificationsine differences

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The pith

The integrals of (cos αx - cos βx)^p over x^q from 0 to ∞ converge or diverge according to specific ranges of p, q, α, β, with closed forms supplied when they converge.

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

The paper examines the family of improper integrals that raise the difference of two cosines to a natural-number power p and divide by x to a natural-number power q. It determines the exact ranges of the parameters p, q, α, β where these integrals converge and where they diverge. Explicit closed-form values are obtained for every convergent case. The expansions used in the proof also produce combinatorial identities among the coefficients that appear. The same approach yields parallel results for integrals built from powers of sine differences.

Core claim

We establish a complete classification of the parameter ranges (p, q; α, β) for which the integrals converge or diverge, and we derive explicit closed-form evaluations in all convergent cases. The analysis also reveals a family of combinatorial identities arising naturally from coefficients in the trigonometric power expansions. As a further application of the same method, we study an analogous class of integrals involving powers of sine differences.

What carries the argument

The improper integral of the p-th power of a cosine difference divided by x^q, whose convergence is settled by substituting trigonometric power expansions into known Frullani-type criteria.

If this is right

When the parameters lie in a convergent range the integral equals an explicit algebraic combination of powers of α and β.
Coefficients in the multiple-angle expansion of (cos αx - cos βx)^p satisfy a family of combinatorial identities.
The identical classification and evaluation procedure applies to the family of integrals built from powers of sine differences.
Convergence fails precisely when the exponent q is too small relative to p or when α equals β.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The combinatorial identities extracted from the expansions may be verifiable independently by generating functions or recurrence relations.
The classification supplies an automatic test that could be inserted into symbolic software to decide whether a given parameter set yields a finite value.
The same substitution technique might classify convergence for integrals that mix sine and cosine differences or that involve hyperbolic functions.
Borderline cases near the convergence boundary could be used to test the sharpness of the derived conditions by direct numerical quadrature.

Load-bearing premise

The convergence analysis and evaluation techniques developed for sine-difference integrals apply without modification to the cosine-difference case once the appropriate trigonometric identities are substituted.

What would settle it

Numerically evaluate the integral for the borderline parameters p=2, q=3, α=1, β=0 and compare the result against the closed form predicted by the classification; divergence or mismatch would falsify the claim.

read the original abstract

For $p,q\in\mathbb{N}$ and $\alpha,\beta\in\mathbb{R}$, we investigate the family of improper integrals \[\int_0^\infty\frac{(\cos\alpha x-\cos\beta x)^p}{x^q}dx.\] We establish a complete classification of the parameter ranges $(p, q; \alpha, \beta)$ for which the integrals converge or diverge, and we derive explicit closed-form evaluations in all convergent cases. The analysis also reveals a family of combinatorial identities arising naturally from coefficients in the trigonometric power expansions. As a further application of the same method, we study an analogous class of integrals involving powers of sine differences. This extends the work of Laoharenoo and Boonklurb in 2022.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The cosine classification is new but the convergence thresholds likely need adjustment for the quadratic small-x asymptotics.

read the letter

The paper classifies convergence for integrals of the form ∫ from 0 to ∞ of (cos αx − cos βx)^p / x^q dx and gives closed forms where they converge. It extends the authors' 2022 sine-difference work.

What stands out is the complete parameter classification for natural numbers p, q and real α, β. They also extract combinatorial identities from the coefficients in the power expansions of the trig functions. The closed forms in convergent cases are the main deliverable.

This is a straightforward extension within the subfield of evaluating difference-of-trigonometric integrals. If the derivations are solid, it provides a reference for this family.

The soft spot is the convergence criteria. Near x=0, the cosine difference expands as a quadratic term, roughly ((β² − α²)/2) x², so the integrand goes like x^{2p − q}. Local integrability at zero therefore needs q < 2p + 1. The sine case has a linear term and a different threshold. The abstract indicates they use the same method after trig identities, which raises the question whether the ranges were adjusted properly. If not, the classification has a gap.

The work is for people already working on Frullani integrals or similar special cases. It won't interest a general analyst, but a reader looking for explicit evaluations of these integrals could use the closed forms.

I would send this to peer review, but the referees should be asked to check the convergence analysis at the origin against the quadratic asymptotics.

Referee Report

2 major / 1 minor

Summary. The manuscript investigates the family of integrals ∫_0^∞ [(cos(αx) − cos(βx))^p / x^q] dx for p, q natural numbers and α, β real. It claims a complete classification of the (p, q; α, β) ranges for convergence versus divergence, together with explicit closed-form evaluations in all convergent cases. The analysis is said to produce combinatorial identities from the coefficients of trigonometric power expansions and is extended to an analogous class of sine-difference integrals, building on the 2022 sine-difference work of Laoharenoo and Boonklurb.

Significance. If the claimed classification and evaluations are correct and the convergence thresholds are properly adjusted for the cosine case, the work would supply a systematic treatment of powered Frullani-type integrals with explicit formulas, which could be useful in analysis. The extraction of combinatorial identities from the expansions would be an additional contribution if they are new.

major comments (2)

[Convergence criteria section] Convergence criteria section: the load-bearing step is the classification of convergence ranges. Near x=0 the cosine difference admits the quadratic expansion cos(αx) − cos(βx) ∼ ((β² − α²)/2) x² (α ≠ β), so the integrand behaves as x^{2p − q} and local integrability at zero requires q < 2p + 1. The manuscript states that the 2022 sine-difference techniques apply after trigonometric substitution; the sine case instead yields the linear small-x behavior x^{p − q} and threshold q < p + 1. The paper must therefore derive and state the cosine-specific thresholds explicitly and verify that its stated ranges match the exponent 2p − q > −1 rather than the sine thresholds. Any mismatch would falsify the claimed complete classification.
[Evaluation of the integrals] Evaluation of the integrals: the explicit closed forms are asserted to hold in all convergent cases. Because the convergence ranges themselves are the load-bearing claim and appear to rest on an unadjusted transfer of the sine analysis, the validity of the closed-form expressions cannot be assessed until the convergence thresholds are corrected and the derivations are shown to respect the proper domains.

minor comments (1)

[Abstract] Abstract: the phrase 'a family of combinatorial identities arising naturally from coefficients in the trigonometric power expansions' is stated without an illustrative example; adding one concrete identity would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and for highlighting the importance of explicitly distinguishing the cosine convergence analysis from the prior sine-difference work. We agree that the cosine-specific small-x behavior must be stated clearly and will revise the manuscript to include a dedicated derivation of the thresholds based on the quadratic expansion. The closed-form evaluations will be confirmed to hold within the properly adjusted domains.

read point-by-point responses

Referee: [Convergence criteria section] Convergence criteria section: the load-bearing step is the classification of convergence ranges. Near x=0 the cosine difference admits the quadratic expansion cos(αx) − cos(βx) ∼ ((β² − α²)/2) x² (α ≠ β), so the integrand behaves as x^{2p − q} and local integrability at zero requires q < 2p + 1. The manuscript states that the 2022 sine-difference techniques apply after trigonometric substitution; the sine case instead yields the linear small-x behavior x^{p − q} and threshold q < p + 1. The paper must therefore derive and state the cosine-specific thresholds explicitly and verify that its stated ranges match the exponent 2p − q > −1 rather than the sine thresholds. Any mismatch would falsify the claimed complete classification.

Authors: We agree that the cosine case requires its own thresholds derived from the quadratic small-x expansion. Our analysis begins from the Taylor expansion cos(αx) − cos(βx) ∼ ((β² − α²)/2) x² (α ≠ β), yielding the integrand ∼ C x^{2p−q} near zero and the condition q < 2p + 1 for local integrability at the origin (combined with the large-x decay condition). The trigonometric substitution is applied only after this expansion to evaluate the resulting integrals; the parameter ranges in the classification were obtained using the cosine exponent 2p − q > −1, not the sine threshold. To make this explicit, the revised manuscript will contain a new subsection deriving the cosine thresholds from the Taylor series and verifying that the stated convergence ranges satisfy the correct exponent condition. revision: yes
Referee: [Evaluation of the integrals] Evaluation of the integrals: the explicit closed forms are asserted to hold in all convergent cases. Because the convergence ranges themselves are the load-bearing claim and appear to rest on an unadjusted transfer of the sine analysis, the validity of the closed-form expressions cannot be assessed until the convergence thresholds are corrected and the derivations are shown to respect the proper domains.

Authors: The closed-form evaluations are obtained by binomial expansion of the powered cosine difference, followed by term-by-term integration against the resulting multiple-angle cosines, each of which reduces to a Frullani-type integral whose value is known in closed form. These steps are valid precisely when the original integral converges, i.e., inside the cosine-specific domains q < 2p + 1 (near zero) and the corresponding large-x condition. After the explicit derivation of the thresholds is added, we will include a short paragraph confirming that every closed-form expression is stated only for parameter values inside those domains. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to prior sine-difference work; cosine analysis presented as independent extension

full rationale

The paper cites the 2022 Laoharenoo-Boonklurb sine-difference result as the source of the method and states that the cosine case is an extension obtained by substituting trigonometric identities. This is a standard self-citation to prior independent work by overlapping authors. No step reduces the new classification or closed forms to the 2022 inputs by definition, fitting, or algebraic identity; the convergence thresholds and evaluations are derived anew for the cosine integrand. The differing small-x asymptotics (x^{2p-q} vs x^{p-q}) are a potential correctness issue but do not create circularity in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard real-analysis tools for improper-integral convergence and on trigonometric identities; no new free parameters, invented entities, or ad-hoc axioms are introduced.

axioms (2)

standard math Standard tests for convergence of improper integrals at zero and at infinity
Invoked to classify the (p, q; α, β) ranges.
standard math Trigonometric power-reduction and binomial-expansion identities
Used to obtain the combinatorial identities mentioned in the abstract.

pith-pipeline@v0.9.1-grok · 5660 in / 1295 out tokens · 40639 ms · 2026-06-30T12:07:33.212255+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references

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Laoharenoo and R

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[2]

A. M. Ostrowski,On some generalizations of the Cauchy–Frullani integral, Proc. Natl. Acad. Sci. USA35(1949), no. 10, 612–616

1949
[3]

Jung,The Cauchy–Frullani integral formula extended to double integrals, Math

B. Jung,The Cauchy–Frullani integral formula extended to double integrals, Math. Sci.37(2012), no. 2, 83–88

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[4]

Borwein and J

D. Borwein and J. M. Borwein,Some remarkable properties of sinc and related integrals, Ramanujan J.5(2001), no. 1, 73–89

2001
[5]

G. H. Hardy,On the Frullanian integral R ∞ 0 ([ϕ(axm)−ϕ(bx n)]/x)(logp)dx, Quart. J.33(1902), 113–144

1902
[6]

Mˆ aagli and Z

H. Mˆ aagli and Z. Z. El Abidine,Mellin transform of some trigonometric func- tions, Open J. Math. Sci.9(2025), 245–264

2025
[7]

I. S. Gradshteyn and I. M. Ryzhik,Table of Integrals, Series and Products, 7th ed., Academic Press, Cambridge, Massachusetts, 2007

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[8]

Arias-de-Reyna,On the theorem of Frullani, Proc

J. Arias-de-Reyna,On the theorem of Frullani, Proc. Amer. Math. Soc.109 (1990), no. 1, 165–175

1990
[9]

J. P. Allouche,Note on an integral of Ramanujan, Ramanujan J.14(2007), no. 1, 39–42

2007
[10]

R. P. Agnew,Mean values and Frullani integrals, Proc. Amer. Math. Soc.2 (1951), no. 2, 237–241

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[11]

Bravo, I

S. Bravo, I. Gonzalez, K. Kohl and V. H. Moll,Integrals of Frullani type and the method of brackets, Open Math.15(2017), no. 1, 1–12

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[12]

Ramanujan,Notebooks, 2 vols., Tata Institute of Fundamental Research, Bom- bay, 1957

S. Ramanujan,Notebooks, 2 vols., Tata Institute of Fundamental Research, Bom- bay, 1957. Atiratch Laoharenoo, Department of Mathematics and Computer Science, Kamnoetvidya Science Academy, Rayong 21210, Thailand, 23 E-mail address:atiratch.l@kvis.ac.th Chanatip Sujsuntinukul, Department of Mathematics, The University of Hong Kong, Pokfulam, Hong Kong, E-ma...

1957

[1] [1]

Laoharenoo and R

A. Laoharenoo and R. Boonklurb,Exact value of integrals involving product of sine or cosine function, N. Z. J. Math.53(2022), 51–61

2022

[2] [2]

A. M. Ostrowski,On some generalizations of the Cauchy–Frullani integral, Proc. Natl. Acad. Sci. USA35(1949), no. 10, 612–616

1949

[3] [3]

Jung,The Cauchy–Frullani integral formula extended to double integrals, Math

B. Jung,The Cauchy–Frullani integral formula extended to double integrals, Math. Sci.37(2012), no. 2, 83–88

2012

[4] [4]

Borwein and J

D. Borwein and J. M. Borwein,Some remarkable properties of sinc and related integrals, Ramanujan J.5(2001), no. 1, 73–89

2001

[5] [5]

G. H. Hardy,On the Frullanian integral R ∞ 0 ([ϕ(axm)−ϕ(bx n)]/x)(logp)dx, Quart. J.33(1902), 113–144

1902

[6] [6]

Mˆ aagli and Z

H. Mˆ aagli and Z. Z. El Abidine,Mellin transform of some trigonometric func- tions, Open J. Math. Sci.9(2025), 245–264

2025

[7] [7]

I. S. Gradshteyn and I. M. Ryzhik,Table of Integrals, Series and Products, 7th ed., Academic Press, Cambridge, Massachusetts, 2007

2007

[8] [8]

Arias-de-Reyna,On the theorem of Frullani, Proc

J. Arias-de-Reyna,On the theorem of Frullani, Proc. Amer. Math. Soc.109 (1990), no. 1, 165–175

1990

[9] [9]

J. P. Allouche,Note on an integral of Ramanujan, Ramanujan J.14(2007), no. 1, 39–42

2007

[10] [10]

R. P. Agnew,Mean values and Frullani integrals, Proc. Amer. Math. Soc.2 (1951), no. 2, 237–241

1951

[11] [11]

Bravo, I

S. Bravo, I. Gonzalez, K. Kohl and V. H. Moll,Integrals of Frullani type and the method of brackets, Open Math.15(2017), no. 1, 1–12

2017

[12] [12]

Ramanujan,Notebooks, 2 vols., Tata Institute of Fundamental Research, Bom- bay, 1957

S. Ramanujan,Notebooks, 2 vols., Tata Institute of Fundamental Research, Bom- bay, 1957. Atiratch Laoharenoo, Department of Mathematics and Computer Science, Kamnoetvidya Science Academy, Rayong 21210, Thailand, 23 E-mail address:atiratch.l@kvis.ac.th Chanatip Sujsuntinukul, Department of Mathematics, The University of Hong Kong, Pokfulam, Hong Kong, E-ma...

1957