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arXiv · 2605.26153

Convergence criteria for Frullani-type integrals involving differences of cosines

Atiratch Laoharenoo·Chanatip Sujsuntinukul
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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.

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Atiratch Laoharenoo, Chanatip Sujsuntinukul. 2026-05-23. Convergence criteria for Frullani-type integrals involving differences of cosines. https://arxiv.org/abs/2605.26153

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