arXiv subjects

Atiratch Laoharenoo

Publications and source records attributed to Atiratch Laoharenoo.

2 recordsLinked to original sources

arXiv2026

Convergence criteria for Frullani-type integrals involving differences of cosines

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.

math.GM↗

arXiv2026

Asymptotic areas between powers of sine and cosine curves

Motivated by Dombrowski and Dresden's work in 2025, we find the exact values of the limits \[\lim_{k\to\infty}\int_0^{ρπ}|\sin^n(kx)-\sin^nx|dx\quad\text{and}\quad \lim_{k\to\infty}\int_0^{ρπ}|\cos^n(kx)-\cos^nx|dx\] for $k,n\in\mathbb{N}$ and $ρ\ge 0$. In addition, we provide several simple recursive formulas which relate these integrals together. The key technique is to locate the zeros of the integrands explicitly, which allows removal of the absolute value and reduces the problem to evaluating limits of telescoping trigonometric sums via asymptotic analysis.

math.CA↗