Mathematical identities for the electrostatic potential of a uniformly charged ring: A rigorous proof

We show a rigorous mathematical proof of some mathematical identities that were uncovered during studies of the electrostatic potential generated by a uniformly charged ring. The authors in Ciftja et al., (2009) presented two independent expressions for this potential on the plane of the ring: one derived using complete elliptic integrals and the other one obtained through an integral involving Bessel functions. By asserting the equivalence of these two expressions they proposed a remarkable identity that connects integrals of products of Bessel functions with complete elliptic integrals of the first kind. However, they were unable to provide a formal mathematical proof of these results. The goal of this paper is to fill this gap by presenting a rigorous and detailed derivation of these mathematical identities. To accomplish this task, we employed standard transformations from the theory of special functions. The key steps in the proof include rewriting the Bessel integral representation of the potential in terms of the Legendre function of the second kind and subsequently expressing it in terms of the complete elliptic integral via known transformations. By establishing the exact equivalence between the Bessel-based and complete elliptic integral-based formulations of the problem, we not only can prove the proposed identities, but also reinforce their relevance for future applications in electrostatics and mathematical physics. These results demonstrate the deep connections that exist among various special functions and highlight the importance of rigorous analysis in validating assumptions made in earlier theoretical work.