Aims: We present an analytic approximate seismic inversion scheme for damped transverse coronal loop oscillations based on the thin tube and thin boundary approximation for computing the period and the damping time. Methods: Asymptotic expressions for the period and damping rate are used to illustrate the process of seismological inversion in a simple and easy to follow manner. The inversion procedure is formulated in terms of two simple functions, which are given by simple closed expressions. Results: The analytic seismic inversion shows that an infinite amount of 1-dimensional equilibrium models can reproduce the observed periods and damping times. It predicts a specific range of allowable values for the Alfvén travel time and lower bounds for the density contrast and the inhomogeneity length scale. When the results of the present analytic seismic inversion are compared with those of a previous numerical inversion, excellent agreement is found up to the point that the analytic seismic inversion emerges as a tool for validating results of numerical inversions. Actually it helped us to identify and correct inaccuracies in a previous numerical investigation.