Modified equations and the Basel problem

Abstract

Discretizations of differential equations are often studied through their modified equation. This is a differential equation, usually obtained as a power series, with solutions that exactly interpolate the discretization. By comparing the Störmer-Verlet discretization of the harmonic oscillator with its modified equation, we obtain a relatively simple derivation of the expansion \[ \left( \arcsin \frac{h}{2} \right)^2 = \frac{1}{2} \sum_{k=1}^\infty \frac{(k-1)!^2}{(2k)!} h^{2k}, \] which can be used to show that $ζ(2) = \frac{π^2}{6}$.