On the convergence and optimization of the Baker–Campbell–Hausdorff formula

In this paper the problem of the convergence of the Baker–Campbell–Hausdorff series for Z=log(eXeY) is revisited. We collect some previous results about the convergence domain and present a new estimate which improves all of them. We also provide a new expression of the truncated Lie presentation of the series up to sixth degree in X and Y requiring the minimum number of commutators. Numerical experiments suggest that a similar accuracy is reached with this approximation at a considerably reduced computational cost.