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arXiv · 2003.04762

A theorem on integration based on the digital expansion

Vladimir García-Morales·Javier Cervera·José A. Manzanares
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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.

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Vladimir García-Morales, Javier Cervera, José A. Manzanares. 2020-03-09. A theorem on integration based on the digital expansion. https://arxiv.org/abs/2003.04762

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