An exact power series representation of the Baker-Campbell-Hausdorff formula

TL;DR

This paper presents an exact power series representation of the Baker-Campbell-Hausdorff formula in one variable, with coefficients expressed via hyperbolic functions, enabling better approximations when only one variable is small.

Contribution

It introduces a novel exact power series in one variable for the BCH formula with closed-form hyperbolic function coefficients, improving upon previous methods.

Findings

Provides an exact power series in one variable for BCH formula

Coefficients expressed in terms of hyperbolic functions

Potential for improved approximations in physical problems

Abstract

An exact representation of the Baker-Campbell-Hausdorff formula as a power series in just one of the two variables is constructed. Closed form coefficients of this series are found in terms of hyperbolic functions, which contain all of the dependence of the second variable. It is argued that this exact series may then be truncated and expected to give a good approximation to the full expansion if only the perturbative variable is small. This improves upon existing formulae, which require both to be small. As such this may allow access to larger phase spaces in physical problems which employ the Baker-Campbell-Hausdorff formula, along with enabling new problems to be tackled.