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Inverse scattering theory is used to reconstruct profiles of electron density from the analytic representation of the reflection coefficient. The complex reflection coefficient is represented as a rational function of the wavenumber . Using a three-pole approximation for , the one-dimensional inverse scattering theory is applied to obtain a closed-form expression for the electron-density profile function . The integral equation of the inverse scattering theory (Gelfand-Levitan equation) is solved by a differential-operator technique, and several numerical examples are given. Three-pole reflection coefficients are found to be applicable to the reconstruction of relatively thin electron layers which might be generated in the laboratory. Rational reflection coefficients with an increased number of poles are found to be necessary to simulate other electron layers of physical interest. This is demonstrated by comparison of multipole reflection coefficients in the Butterworth approximation with reflection coefficients calculated from Epstein's direct scattering theory for electron layers. A parameter in the Epstein theory, which characterizes the total electron content of the layer, is related to the number of poles needed to reconstruct that layer. Estimates are thus obtained of the number of poles needed to reconstruct ionospheric layers and other plasmas of physical interest.