  A detailed study of the analytic structure of 1-loop self energy graphs for
neutral and charged $\rho$ mesons is presented at finite temperature and
arbitrary magnetic field using the real time formalism of thermal field theory.
The imaginary part of the self energy is obtained from the discontinuities of
these graphs across the Unitary and Landau cuts, which is seen to be different
for $\rho^0$ and $\rho^\pm$. The magnetic field dependent vacuum contribution
to the real part of the self energy, which is usually ignored, is found to be
appreciable. A significant effect of temperature and magnetic field is seen in
the self energy, spectral function, effective mass and dispersion relation of
$\rho^0$ as well as of $\rho^\pm$ relative to its trivial Landau shift.
However, for charged $\rho$ mesons, on account of the dominance of the Landau
term, the effective mass appears to be independent of temperature. The trivial
coupling of magnetic moment of $\rho^\pm$ with external magnetic field, when
incorporated in the calculation, makes the $\rho^\pm$ to condense at high
magnetic field.
