The Baker-Campbell-Hausdorff formula, in various guises, is often thought of in the context of Lie groups and particle physics though more recently it has found use in quantum mechanics. For example, one could imagine evolving a system first by one Hamiltonian and then another, with the task being to find the Floquet Hamiltonian describing the overall operation. We believe it also occupies a natural space in statistical mechanics, where a transfer matrix may be re-imagined as the exponential of some free-energy operator. The existing formula is essentially a Taylor expansion in two variables which, in the quantum mechanics example, control the time each Hamiltonian is active for. There is a problem however; how should one truncate the expansion? Unless both variables may be considered small this approach may not be fruitful.

I will present recent work (arXiv:1807.07884) which exactly resums one of the variables in this double Taylor series, leaving a power series in the other. Closed form expressions for each coefficient will be found which generalise the first order term found by Campbell in the late 19th century. Practically, this result allows one to have one of the two aforementioned variables be large, opening up regions of phase-space previously inaccessible.