The one‐dimensional Helmholtz equationw″ + k02g(z)w(z) = ,k0 = 2&pgr;/&lgr;0, is solved making use of an extension of Langer's method. The functiong(z)is restricted to the class of Epstein profiles,g(z) = g(0) + [1 − g(0)] tanh2(z/2&lgr;), where&lgr;is the scale length, corresponding to no turning points, a quadratic turning point atz = 0, or a pair of turning points, depending on whetherg(0)is positive, zero, or negative. This method yields uniformly valid solutions in terms of confluent hypergeometric functions. First, a solutionw+(z)is obtained for0 ≤ zwhich, asz → ∞, becomes the transmitted plane wave. Then, applying the method of analytic continuation, a linear combination is obtainedw−(z) = wi(z) + wr(z)valid forz ≤ 0which, asz → − ∞, yields the superposition of the incident and reflected plane waves. As an illustration of this method and as an application to parabolic density profiles, the mean energy density of the electric field is obtained numerically, in the vicinity ofz = 0, when the parameter ratio&lgr;/&lgr;0 ≥ 100, and for the case of a single quadratic turning point and also for the case in which there are two neighboring turning points. With this choice of parameters, the results yield useful information on the tuning criterion for parametric excitation of the ionosphere.