Areas Between Cosines

TL;DR

This paper investigates the limiting area between powers of cosine functions as the frequency ratio increases, revealing a connection to a specific exponential generating function related to arcsine.

Contribution

It introduces a novel analysis of the asymptotic area between cosine powers and links it to a known generating function, expanding understanding of these trigonometric integrals.

Findings

Derived the limit of the area between $ ext{cos}^p x$ and $ ext{cos}^p nx$ as $n$ approaches infinity.

Established a connection between these limits and the exponential generating function for $ ext{arcsin} x/(1-x)$.

Linked the results to OEIS sequence A296726.

Abstract

We find the area between $\cos^p x$ and $\cos^p nx$ as $n$ heads to infinity, and we establish a connection between these limiting values and the exponential generating function for $\arcsin x/(1-x)$ at sequence number A296726 on the OEIS.