A new functional equation in the plasma inverse problem and its analytic properties

In the one-dimensional form of the plasma inverse problem, reflection of transverse electromagnetic waves is used to determine the electron density in a cold, collisionless, unmagnetized plasma. We extend the applicable Gel’fand–Levitan integral equation so that it is valid for all times. Laplace transformation of the extended equation gives a linear functional equation containing the complex reflection coefficient. We solve the functional equation analytically in special cases, and classify reflection coefficients by their analytic properties.