  Using the Oxford Short Wavelength Integral Field specTrograph (SWIFT), we
trace radial variations of initial mass function (IMF) sensitive absorption
features of three galaxies in the Coma cluster. We obtain resolved spectroscopy
of the central 5kpc for the two central brightest-cluster galaxies (BCGs)
NGC4889, NGC4874, and the BCG in the south-west group NGC4839, as well as
unresolved data for NGC4873 as a low-$\sigma_*$ control. We present radial
measurements of the IMF-sensitive features sodium NaI$_{\rm{SDSS}}$, calcium
triplet CaT and iron-hydride FeH0.99, along with the magnesium MgI0.88 and
titanium oxide TiO0.89 features. We employ two separate methods for both
telluric correction and sky-subtraction around the faint FeH feature to verify
our analysis. Within NGC4889 we find strong gradients of NaI$_{\rm{SDSS}}$ and
CaT but a flat FeH profile, which from comparing to stellar population
synthesis models, suggests an old, $\alpha$-enhanced population with a
Chabrier, or even bottom-light IMF. The age and abundance is in line with
previous studies but the normal IMF is in contrast to recent results suggesting
an increased IMF slope with increased velocity dispersion. We measure flat
NaI$_{\rm{SDSS}}$ and FeH profiles within NGC4874 and determine an old,
possibly slightly $\alpha$-enhanced and Chabrier IMF population. We find an
$\alpha$-enhanced, Chabrier IMF population in NGC4873. Within NGC4839 we
measure both strong NaI$_{\rm{SDSS}}$ and strong FeH, although with a large
systematic uncertainty, suggesting a possible heavier IMF. The IMFs we infer
for these galaxies are supported by published dynamical modelling. We stress
that IMF constraints should be corroborated by further spectral coverage and
independent methods on a galaxy-by-galaxy basis.
