  We derive a general exact form of the phase space distribution function and
the thermal expectation values of local operators for the free quantum scalar
field at equilibrium with rotation and acceleration in flat space-time without
solving field equations in curvilinear coordinates. After factorizing the
density operator with group theoretical methods, we obtain the exact form of
the phase space distribution function as a formal series in thermal vorticity
through an iterative method and we calculate thermal expectation values by
means of analytic continuation techniques. We separately discuss the cases of
pure rotation and pure acceleration and derive analytic results for the
stress-energy tensor of the massless field. The expressions found agree with
the exact analytic solutions obtained by solving the field equation in suitable
curvilinear coordinates for the two cases at stake and already - or implicitly
- known in literature. In order to extract finite values for the pure
acceleration case we introduce the concept of analytic distillation of a
complex function. For the massless field, the obtained expressions of the
currents are polynomials in the acceleration/temperature ratios which vanish at
$2\pi$, in full accordance with the Unruh effect.
