  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.
