The tri-diagonal determinant for Faraday resonance of a viscous liquid subject to an externally imposed vertical oscillation is expressed in terms of the surface-wave impedance of the liquid and developed as a continued fraction to obtain a systematic sequence of analytical approximations for the threshold acceleration. The impedance is calculated for either a clean or a fully contaminated surface on the assumption that the capillary, gravitational and viscous length scales are small compared with the breadth and depth of the liquid. Limiting approximations for weak and strong viscosity are constructed.