The procedure of Section V aims at selecting [a.sub.n] to minimize the O([n.sup.-1]) term in the

asymptotic expansion (A2).

It contains all the significant topics of EM wave technology, from the finite element method, boundary element method, point-matching method, mode matching method, the spatial network method, the equivalent source method, the geometrical theory of diffraction, the Wiener-Hopf technique,

asymptotic expansion techniques and beam propagation method to spectral domain method.

The Early Universe models depend on the existence of a preferred energy scale in the

asymptotic expansion of the spectral actions.

In particular, the leading order term [z.sub.[infinity]] in the

asymptotic expansion of z reads

Then the eigenvalue [[lambda].sub.k]([alpha]) obeys an

asymptotic expansionOn each sector [E.sub.p], they share with respect to [epsilon] a common

asymptotic expansion [??](t, z, [epsilon]) = [[summation].sub.n[greater than or equal to]0] [y.sub.n](t, z)[[epsilon].sup.n] which defines a formal series with bounded holomorphic coefficients on T x [H.sub.[beta]].

The moment generating function of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] has the

asymptotic expansionIn [14], we constructed an

asymptotic expansion of the expectation of g([x.sup.[epsilon],[delta]](t), [[alpha].sup.[epsilon]](t)) for some appropriate function g(*, *).

In a similar way, the

asymptotic expansions of [v'.sub.j](r) as r [right arrow] [infinty] can be obtained from (14), and the

asymptotic expansion of [v".sub.j](r) as r [right arrow] [infinity] follows readily from the relation [v".sub.j] = (-[[mu].sup.2.sub.j] + V)[v.sub.j].

This work is based on the relationship between the singular vectors associated with nonzero singular values of a multistatic response (MSR) matrix and

asymptotic expansion formula due to the existence of small inhomogeneities; refer to [23].

Therefore the leading order terms in the

asymptotic expansion of (28), (29) can be represented as a combination of elementary solutions (25) in which, as explained above, we should include the Whittaker function [M.sub.[mu],n/2](-i[chi][tau]).

(2) On such, rescaled domain independent of the perturbation parameter [epsilon], one seeks for the unknowns in the form of the

asymptotic expansion in powers of [epsilon].