Elliptic integral identities derived from Coxeter's integrals

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.