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 $$ I(λ)=\int_{0}^{π/2} \arccos\!\left(\frac{\cosθ}{1+λ\cosθ}\right)\,dθ, $$ and differentiating with respect to the parameter \(λ\), we show that the derivative $I'(λ)$ can be expressed as an elliptic-type integral. Integrating $I'(λ)$ between 0 and 2 yields the identity $$ \int_0^2 \int_0^{π/2} \frac{\cos^2θ} {(1+s\cosθ)\sqrt{(1+s\cosθ)^2-\cos^2θ}} \,dθ\, ds=A-B=\frac{π^2}{12}, $$ where $A$ and $B$ are the first two so-called Coxeter integrals $$ A = \int_0^{π/2} \arccos\!\left(\frac{\cosθ}{1+2\cosθ}\right) dθ, $$ and $$ B = \int_0^{π/2} \arccos\!\left(\frac{1}{1+2\cosθ}\right) dθ. $$ The derivative $I'(λ)$ can be expressed in terms of incomplete elliptic integrals of the first kind $F$ and of the third kind $Π$. 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.