  We have performed systemmatic local linear stability analysis on a radially
stratified infinite self-gravitating cylinder of rotating plasma under the
influence of magnetic field. In order to render the system analytically
tractable, we have focussed solely on the axisymmetric modes of perturbations.
Using cylindrical coordinate system, we have derived the critical linear mass
density of a non-rotating filament required for gravitational collapse to ensue
in the presence of azimuthal magnetic field. Moreover, for such filaments
threaded by axial magnetic field, we show that the growth rates of the modes
having non-zero radial wavenumber are reduced more strongly by the magnetic
field than that of the modes having zero radial wavenumber. More importantly,
our study contributes to the understanding of the stability property of
rotating astrophysical filaments that are more often than not influenced by
magnetic fields. In addition to complementing many relevant numerical studies
reported the literature, our results on filaments under the influence of
magnetic field generalize some of the very recent analytical works
(e.g.,~\citet{jog2014}, etc.). For example, here we prove that even a weak
magnetic field can play a dominant role in determining stability of the
filament when the rotation timescale is larger than the free fall timescale. A
filamentary structure with faster rotation is, however, comparatively more
stable for the same magnetic field. The results reported herein, due to strong
locality assumption, are strictly valid for the modes for which one can ignore
the radial variations in the density and the magnetic field profiles.
