By Nanny Fröman, Per Olof Fröman (auth.)

The potency of the phase-integral process built by way of the current au­ thors has been proven either analytically and numerically in lots of publica­ tions. With the inclusion of supplementary amounts, heavily with regards to new Stokes constants and bought via comparability equation process, very important sessions of difficulties within which transition issues may perhaps method one another develop into available to exact analytical therapy. The exposition during this monograph is of a mathematical nature yet has very important actual purposes, a few examples of that are present in the adjoined papers. therefore, we wish to stress that, even though we objective at mathematical rigor, our remedy is made basically with actual wishes in brain. To introduce the reader into the heritage of this booklet, we begin by way of de­ scribing the phase-integral approximation of arbitrary order generated from an unspecified base functionality. this is often performed in bankruptcy 1, that's reprinted, after minor alterations, from a evaluation article. bankruptcy 2 is the results of re­ seek paintings that was once pursued in the course of greater than 20 years, interrupted from time to time. It begun within the sixties, once we have been nonetheless utilizing a phase-integral approximation, which in our current terminology corresponds to a distinct number of the bottom functionality. on the time our fundamental objective used to be to derive expressions for the supplementary amounts wanted that allows you to receive a correct connection formulation for a true capability barrier, while the power lies locally of the pinnacle of the barrier.

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In Chapter 6. The final analytic formulas for the supplementary quantities are expressed in terms of phase-integrals in which the contributions q(1) (z), q(3)(z), ... to the function q( z) in successive orders of approximation are to be integrated along contours enclosing a pair of two transition points. It should be emphasized that comparison equation technique allows us only to show that there exists a solution that is approximated by the resulting formulas. As has been stressed by N. Froman on pp.

2b) h(z) = q-l/2(z) exp ( -i (see Chapter 1). 6) and (= JZ Q(z) dz. 7) All the functions Y2n can be expressed in terms of co and the derivatives of co with respect to (. Thus, the choice of the base function Q(z) affects cO and ( but not the analytic form of Y2n , expressed in terms of co and derivatives of co with respect to (. 2. 1). , of Q2(z). With the aid of comparison equation technique the wave function can be obtained in the neighborhood of such transition points. This may be necessary, for instance, if there is a boundary condition close to a transition point; but in general one does not need the wave function in the neighborhood of transition points.

That I is independent of z. 12), where a, ,6, and, are constants uniquely determined 30 N. O. 11). 11). A corresponding formula has been given by Floyd (1986). 6 Phase-Integral Method In this section we shall briefly mention a few items that, beside the phaseintegral approximation described in detail above and the F-matrix method mentioned in passing, are essential parts of a phase-integral method, which has proved to be very efficient and accurate in a great variety of important applications in various fields of theoretical physics, as well as in applied mathematics.

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