The rectangular elastic membrane for Apery’s and Catalan’s constants calculation

Abstract

In this paper, we use the elastic membrane model to calculate Apéry’s constant and others associated series. The de Saint-Venant’s solution of Poisson’s partial differential equation with rectangular boundary is used for the two limit cases of thin and square membranes. Gradient calculations at the boundaries give a relationship involving ratio 𝜋 𝐺⁄𝜁(3) where 𝐺 is the Catalan’s constant and a rapidly convergent series for 𝜁(3) analogous to famous Ramanujan’s series.