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 $2pi$, in full accordance with the Unruh effect.
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