Analytic calculation of the nonzero fast wave reflection coefficient from the tunneling equation with absorption

An analytic series expression of the nonzero fast wave reflection coefficient has been found from the tunneling equation which models the wave propagation in weakly inhomogeneous media near a back-to-back resonance-cutoff pair. Explicit calculations are done for both second ion cyclotron and second electron cyclotron harmonic cases. Numerical values of this expression are compared to those calculated by solving an integral equation iteratively. They agree well with each other whenever both the series and the iteration converge. This result is based on a new method of evaluating integrals derived from the integral equation without having to solve the equation itself. This new method may have more applications and deserves further research and development