Convergence criteria for Frullani-type integrals involving differences of cosines
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917532969992192 |
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| author | Laoharenoo, Atiratch Sujsuntinukul, Chanatip |
| author_facet | Laoharenoo, Atiratch Sujsuntinukul, Chanatip |
| contents | For $p,q\in\mathbb{N}$ and $α,β\in\mathbb{R}$, we investigate the family of improper integrals
\[\int_0^\infty\frac{(\cosαx-\cosβx)^p}{x^q}dx.\] 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. This extends the work of Laoharenoo and Boonklurb in 2022. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_26153 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Convergence criteria for Frullani-type integrals involving differences of cosines Laoharenoo, Atiratch Sujsuntinukul, Chanatip General Mathematics 05A19, 26A06, 26A09 For $p,q\in\mathbb{N}$ and $α,β\in\mathbb{R}$, we investigate the family of improper integrals \[\int_0^\infty\frac{(\cosαx-\cosβx)^p}{x^q}dx.\] 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. This extends the work of Laoharenoo and Boonklurb in 2022. |
| title | Convergence criteria for Frullani-type integrals involving differences of cosines |
| topic | General Mathematics 05A19, 26A06, 26A09 |
| url | https://arxiv.org/abs/2605.26153 |