The boundary sheaths of all plasmas are characterized by a gradual transition from unipolarity (electron depletion, n_e n_i) to ambipolarity (quasi-neutrality, n_e ≈ n_i). Capacitively driven sheaths exhibit a transition which is expanded by the RF modulation and smoothed by thermal effects, i.e. by the finiteness of the electron temperature T_e and the Debye length . Sheath models which neglect thermal effects ('step models') are restricted to strongly modulated high voltage sheaths with V_RF T_e/e and fail when this condition is not met. This work presents an improved analysis of the sheath–bulk transition which takes both modulation and thermal effects into account. Based on a previously found asymptotic solution of the Boltzmann–Poisson equation (Brinkmann 2007 J. Appl. Phys. 102 093393), approximate algebraic (i.e. closed) expressions for the phase-resolved electrical field E and electron density n_e in RF sheaths are derived. Under the assumption that the modulation is periodic (not necessarily harmonic) with ω_RF ω_pi, also the phase averages of the field and the electron density can be expressed in closed form. These results—together referred to as the advanced algebraic approximation (AAA)—make it possible to formulate efficient and accurate models for RF driven boundary sheaths for all ratios of V_RF to T_e/e. As an example, a harmonically RF modulated, collision-dominated single species sheath is studied. The outcome is compared both with the numerically constructed exact solution and with the well-known step model approach of Lieberman (1989 IEEE Trans. Plasma Sci. 17 338). It is found that the AAA can reproduce the exact numerical solution within a few per cent for all ratios of V_RF to T_e/e. The step model, in contrast, exhibits strong deviations even for large eV_RF/T_e and fails completely in the case of weak modulation.