Brownian aerosol coagulation was studied theoretically using the moment method of log-normal size distribution functions. An analytic solution to the size distribution of a coagulating aerosol was derived. In order to cover the entire size range the harmonic mean of the near-continuum and the free-molecule coagulation coefficient were applied. Therefore, the analytic solution is valid for the entire particle size range, i.e. covering from the free-molecule regime, via both the transition and the near-continuum regimes, to the continuum regime; The present work represents the first analytical solution to the Brownian aerosol coagulation problem that addresses the entire particle size range.