A theorem on integration based on the digital expansion
Abstract
The binary radix expansion of a real number can be used to code the outcome of any series of coin tosses, a fact that provides an intriguing link between number theory, measure theory and statistical physics. Inspired by this fact, a general result is established for the definite integral of a differentiable function of a single variable that allows any such integral to be exactly written in terms of a double series. The theorem can be directly applied to a wide variety of integrals of physical interest and to derive new series expansions of real numbers and real-valued functions. We apply the theorem to the integration of the equation of motion in one dimension of classical Hamiltonian systems, focusing in the analysis of the nonlinear pendulum.
Cite
@article{arxiv.2003.04762,
title = {A theorem on integration based on the digital expansion},
author = {Vladimir García-Morales and Javier Cervera and José A. Manzanares},
journal= {arXiv preprint arXiv:2003.04762},
year = {2020}
}
Comments
12 pages, 2 figures, submitted for publication