Transmission through a real Eckart-Epstein potential barrier treated by means of certain phase-integral approximations

Formulas for the transmission and reflection coefficients for particle transmission through a real potential barrier were given some years ago by Froman and Froman in an explicit and general form, valid for any convenient order of certain phase-integral approximations. To illuminate the great accuracy of those formulas, the relative errors of the transmission and reflection coefficients for a real Eckart-Epstein potential barrier are calculated and graphically displayed up to the eleventh order. It is pointed out that the transmission coefficient recently obtained by Wald and Lu, by means of an extension of the method of Miller and Good to the next order of approximation, yields numerical results essentially equivalent to those obtained from the next lowest order of the Froman and Froman formula. Being explicitly given and containing no divergences, the latter formula is, however, analytically much simpler