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 \(λ\), we show that the derivative $I^{\prime }(\lambda )$ can be expressed as an elliptic-type integral. Integrating $I^{\prime }(\lambda )$ 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 $I^{\prime }(\lambda )$ 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.