  We revisit the problem of the electromagnetic Green function for homogeneous
hyperbolic media, where longitudinal and transverse components of the
dielectric permittivity tensor have different signs. We analyze the dipole
emission patterns for both dipole orientations with respect to the symmetry
axis and for different signs of dielectric constants, and show that the
emission pattern is highly anisotropic and has a characteristic cross-like
shape: the waves are propagating within a certain cone and are evanescent
outside this cone. We demonstrate the coexistence of the cone-like pattern due
to emission of the extraordinary TM-polarized waves and elliptical pattern due
to emission of ordinary TE-polarized waves. We find a singular complex term in
the Green function, proportional to the $\delta-$function and governing the
photonic density of states and Purcell effect in hyperbolic media.
