Elliptic integral identities derived from Coxeter's integrals

TL;DR

This paper derives new elliptic integral identities by embedding Coxeter's classical integrals into a parameterized family and analyzing their derivatives, revealing connections between Coxeter integrals and elliptic functions.

Contribution

The paper introduces a novel method linking Coxeter integrals to elliptic integrals through parameter differentiation, providing new identities and analytic insights.

Findings

Derived elliptic integral identities from Coxeter's integrals.

Expressed derivatives in terms of incomplete elliptic integrals of the first and third kind.

Established a direct connection between Coxeter integrals and elliptic functions.

Abstract

We revisit the classical integrals introduced by Coxeter, not to recalculate their well-known exact values, but to use them as a tool to derive elliptic integral identities. By embedding Coxeter's first integral into a one-parameter family $$ I(\lambda)=\int_{0}^{\pi/2} \arccos\!\left(\frac{\cos\theta}{1+\lambda\cos\theta}\right)\,d\theta, $$ and differentiating with respect to the parameter \(\lambda\), we show that the derivative $I'(\lambda)$ can be expressed as an elliptic-type integral. Integrating $I'(\lambda)$ between 0 and 2 yields the identity $$ \int_0^2 \int_0^{\pi/2} \frac{\cos^2\theta} {(1+s\cos\theta)\sqrt{(1+s\cos\theta)^2-\cos^2\theta}} \,d\theta\, ds=A-B=\frac{\pi^2}{12}, $$ where $A$ and $B$ are the first two so-called Coxeter integrals $$ A = \int_0^{\pi/2} \arccos\!\left(\frac{\cos\theta}{1+2\cos\theta}\right) d\theta, $$ and $$ B = \int_0^{\pi/2} \arccos\!\left(\frac{1}{1+2\cos\theta}\right) d\theta. $$ The derivative $I'(\lambda)$ can be expressed in terms of incomplete elliptic integrals of the first kind $F$ and of the third kind $\Pi$. This approach establishes a direct connection between classical Coxeter integrals and elliptic functions. The method highlights how well-known trigonometric integrals can serve as a bridge to explore properties and relations of elliptic integrals, offering new analytic insights beyond the original Coxeter evaluations.