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Download Asymptotic Analysis: Linear Ordinary Differential Equations by Mikhail V. Fedoryuk (auth.) PDF

By Mikhail V. Fedoryuk (auth.)

In this booklet we current the most effects at the asymptotic idea of standard linear differential equations and structures the place there's a small parameter within the larger derivatives. we're inquisitive about the behaviour of strategies with appreciate to the parameter and for giant values of the self reliant variable. The literature in this query is substantial and largely dispersed, however the equipment of proofs are sufficiently comparable for this fabric to be prepare as a reference booklet. we've limited ourselves to homogeneous equations. The asymptotic behaviour of an inhomogeneous equation should be got from the asymptotic behaviour of the corresponding basic process of options by way of utilising tools for deriving asymptotic bounds at the suitable integrals. We systematically use the idea that of an asymptotic enlargement, information of that may if beneficial be present in [Wasow 2, Olver 6]. through 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 which 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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Example text

Substituting the expression (2) into this equation, we obtain the identity Equating the powers of ,\-1 to zero, we obtain a recurrence system for the unknown functions a_l(x), ao(x), ... All the computations are of a formal nature. The first of these equations is a:' 1 = q, so that a-I = ±y'q. For a-I = y'q (choosing the branch of the root as shown in paragraph 2) we have (3) Observe that al (x) is the same as in § 1, (9). We obtain the recurrence relation for successive functions ak( x) akH(x) = - ~ (a~(x) 2 q(x) + t j=O aj(x)ak-Ax)) .

The WKB-bounds (6), (8) follow from this and from (13). Moreover lim UI(X) %-+4 = 1, lim U2(X) x-+a = 0, which proves (7). The solution Y2 is constructed in a similar manner. If Q( x) is real the WKB-bounds can be obtained by applying the Liouville transform (§ 1) to equation (1) and then reducing the equation obtained to integral form. 3. The Equation in Self-Adjoint Form. We consider the equation (P(x )y')' - Q(x)y = o. (16) Let P(x), Q(x) E C 2 (I) and suppose that 1) P(x) and Q(x) are nowhere zero in Ij 2) there is a branch of JP(x)/Q(x) of class C 2 (I) such that Re JP(x)/Q(x) ~ 0 in I.

Which proves the above assertion. ). 2. ) be a solution:/ (10) for which the asymptotic series expansion (11) is valid. Since Re (i>. q(x» ~ 0 for 1m >. ~ 0 this asymptotic formula is true as 1>'1 -+ 00, 1m >. ~ O. ). S. )}. S. )} for (10) which has the asymptotic expansion (11) as 1>'1 -+ 00, 1m >. ~ O. S. are not generally the same. 6 More Complicated Dependence on >.. -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.

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