By Mikhail V. Fedoryuk (auth.)

ISBN-10: 3540548106

ISBN-13: 9783540548102

In this booklet we current the most effects at the asymptotic concept of normal linear differential equations and platforms the place there's a small parameter within the greater derivatives. we're all in favour of the behaviour of options with recognize to the parameter and for giant values of the self sufficient variable. The literature in this query is significant and commonly dispersed, however the tools of proofs are sufficiently comparable for this fabric to be prepare as a reference e-book. we've limited ourselves to homogeneous equations. The asymptotic behaviour of an inhomogeneous equation could be got from the asymptotic behaviour of the corresponding basic procedure of options through utilising equipment for deriving asymptotic bounds at the suitable integrals. We systematically use the concept that of an asymptotic enlargement, info of which could if important be present in [Wasow 2, Olver 6]. via the "formal asymptotic resolution" (F.A.S.) is known a functionality which satisfies the equation to some extent of accuracy. even though this idea isn't accurately outlined, its that means is usually transparent from the context. We additionally be aware that the time period "Stokes line" utilized in the e-book is corresponding to the time period "anti-Stokes line" hired within the physics literature.

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**Extra info for Asymptotic Analysis: Linear Ordinary Differential Equations**

**Example text**

Then A and B must satisfy the system of equations B' -- 1 Q' e2iS A . 4Q If Q( x) is changing slowly for x ~ 1 (for instance if Q( x) --t Qo Q'(x) --t 0) then the coefficients in the right-hand sides of (11) are for x ~ 1. We now introduce the function R(x) = B(x)/A(x) which the role of the reflection coefficient in problems of wave propagation. R( x) satisfies the Riccati equation (11) > 0, small plays Then Chapter 2. Second-Order Equations on the Real Line 28 8. We consider the Dirac-type system J y' + [qll(X) () q12(X)] - A q21 X q22 () X y- y, with real coefficients and q12(X) where T( x) = [c~s 4> - sin 4>] , cos 4> sm 4> = (12) q21(X).

1. The standard assumptions concerning the dependence of the function q on the parameter e are as follows. Let S be a sector in the complex e-plane of the form 0 < lei < eo, -a < arge < {3, where 0 ~ a, {3 ~ 7r. I. q(x,e) E Coo(I X S). II. q(x,e) is holomorphic in e in the sector S for each fixed x E I. III. There is an asymptotic expansion 00 q(x,e) "'Lqk(x)e k , e-+O, eES, k=O which is uniform in x E I in each proper subsector S' of S (that is, for -a < -a' ~ arge ~ {3' < {3). If e takes only real values then in place of S we take the interval J: 0 < e < eo and condition II is discarded.

We will not repeat them but mention only that the turning points of (16) and (20) do not coincide, but are close for >. ~ 1. Let us suppose for simplicity, that the coefficients a and b do not depend on p.. -l a,(x) = o. ) = Xo + O( >. -1). This is a general property: a turning point is not invariant under changes of variables, but it may be asymptotically invariant. 2 Equations in Self-adjoint Form. 2Q(x)y = 0, where P, Q E c oo (1) and P(x) p(x)p2 - Q(x) (21) -I o. The equation =0 is called the characteristic equation, so that the turning points of (21) are the zeros of Q(x).