Barriers bounded by a transition point of order greater than unity

Further exact and approximate investigations are made into the potential-barrier problem. A generalized barrier is considered, bounded at one end by a transition point of odd order greater than unity, namely by a point of inflexion in the barrier profile. A hierarchy of such barriers is investigated analytically based on certain functions without finite singularities expressed in terms of the Whittaker function. The phase-integral method is then developed in an acceptable manner, avoiding the unrigorous speculations of many authors in the past, in order to show the power of the method to produce results corresponding approximately to the exact solution.