Asymptotic areas between powers of sine and cosine curves

2026/07/27 by Atiratch Laoharenoo, Chanatip Sujsuntinukul
Mathematics · #math.CA

paper · pdf

Abstract

Motivated by Dombrowski and Dresden's work in 2025, we find the exact values of the limits lim_k→∞∫₀^ρπ|sinⁿ(kx)-sinⁿx|dx\quadand lim_k→∞∫₀^ρπ|cosⁿ(kx)-cosⁿx|dx for k,n∈ℕ and ρ≥ 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.

Citations

• Areas Between Cosines

Related

• Areas Between Cosines
• A remark concerning sinc integrals
• lim_{x→ 0}(sin x)/(x) and the definition of π
• Asymptotics of some generalised sine-integrals
• On the derivatives of the powers of trigonometric and hyperbolic sine and cosine