ASYMPTOTIC BEHAVIOUR AS λ→∞ OF THE SOLUTION OF THE EQUATIONw″(z)−p(z, λ)w(z) = 0 IN THE COMPLEXz-PLANE

Evgrafov, M A; Fedoryuk, M. V.

doi:10.1070/rm1966v021n01abeh004145

Cited by 69 publications

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References 14 publications

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“…Some of our results appear in Stengle [5]. We also note that Evgrafov and Fedoryuk [6] study the leading term of our expansion in a function-theoretic context. The author is indebted to Professor N. Kazarinoff for calling his attention to problem (1-6).…”

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confidence: 86%

“…Since these values of t in general depend on e, the secondary turning points typically lie on a curve in (i, e) space, that is they lie on a subvariety of (£, e) space (which in our case is usually an algebraic subvariety). We do not wish to perpetuate the distinction between turning point and secondary turning point since the point of [6] and of this paper is that the secondary turning points, properly defined, are in fact the "true" turning points of (1-1). Ideally the best terminology would simply use the term "turning point" in an extended sense.…”

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confidence: 99%

“…Examples (Concluded). We complete our analysis of (1- 6) êy" + (<p + e)y = 0 by computing connection formulas relating the classical asymptotic forms (see Section 5) valid for negative t to the similar forms valid for positive t. We accomplish this by reducing the uniform formulas given by (11-1) to two different classical forms, depending on whether t is positive or negative. Let Ylt and Fa± denote the solution pairs obtained from (11-2) by choosing r = -1 and r -+1, respectively, as the limit of integration in the exponential.…”

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confidence: 99%

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Uniform Asymptotic Solution of Second Order Linear Differential Equations without Turning Varieties

Stengle¹

1969

Mathematics of Computation

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Abstract. The purpose of this paper is to initiate the study of a new kind of asymptotic series expansion for solutions of differential equations containing a parameter. We obtain uniform asymptotic solutions for certain equations of the form ê»y" = ait, e)y ,where n is a positive integer, t and e are real variables ranging over \t\ S U, 0 < ê €o, and a is a function infinitely differentiable on the closure of this domain. We require that a(i, e) satisfy conditions which can be regarded as generalized nonturning-point conditions. These conditions imply the absence of secondary turning points, and reduce in the simplest case to the condition a(¿, 0) 5¿ 0, but also include cases (the interesting ones) in which a(0, 0) = 0. | 1. Introduction. The purpose of this paper is to initiate the study of a new kind of asymptotic series expansion for solutions of differential equations containing a parameter. We shall obtain uniform asymptotic solutions for certain equations of

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Uniform Asymptotic Solution of Second Order Linear Differential Equations without Turning Varieties

Stengle¹

1969

Mathematics of Computation

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“…An interesting contribution to this kind of question appears in [May work of Evgrafov and Fedorjuk [13] who study admissible domains for problems of the form w'\z)-X2p{z)w = 0 where p is either a polynomial or an entire function of a certain special kind. Their object is to exploit the formula w ~ p 1/4 exp (»J>*) for its power to represent solutions asymptotically in A uniformly on unbounded z-domains.…”

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confidence: 99%

A new basis for uniform asymptotic solution of differential equations containing one or several parameters

Stengle¹

1971

Trans. Amer. Math. Soc.

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Introduction. The purpose of this paper is to put forward a ring-ideal theoretic basis for the theory of uniform asymptotic expansions, and to study a new elementary technique for the asymptotic solution of differential equations containing one or more parameters. In Part 1 we formulate a new notion of asymptotic expansion and ancillary concepts of formal convergence and formal series. Because we consider this formulation to have significance beyond the uses exhibited here, we make it at a level of generality which seems to us to best reveal its essential nature. In Part 2, we use the notions of Part 1 to frame an investigation of some turning point problems for linear equations of arbitrary order containing a single parameter. Our analysis is based on a new classification of such problems depending on a refinement of the notion of a turning point. In Part 3, we study a second order equation containing two parameters. This analysis contains, as a special case, results for Bessel's equation as both the independent variable and the parameter tend to infinity. We obtain uniform asymptotic results under weak relative growth restrictions on the two parameters which differ qualitatively from any previously known to us.This investigation is an outgrowth of a previous study [1] of turning point problems for second order linear differential equations. We are indebted to Professor Wolfgang Wasow for reading preliminary versions of Part 2 and to the late

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“…Problems of this type are of interest in the study of particle scattering and wave mechanics. Sibuya [4], Weinberg [6], and Evgrafov and Fedoryuk [1] have studied the asymptotic distribution of large positive eigenvalues for problems (1.1), (1.2). However, essentially they all assumed ihatp(x) can only have pairs of first order zeroes on the real axis.…”

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Distribution of eigenvalues in the presence of higher order turning points

Leung¹

1977

Trans. Amer. Math. Soc.

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Abstract. This article is concerned with the eigenvalue problem u"(x) -A^j(jc)u(jc) = 0, u(x) E ¿2(-oo, oo), where p(x) is real, analytic and possesses zeroes of arbitrary orders. Under certain conditions on p(x), approximate formulas for the eigenvalues are found. The problem considered is of interest in the study of particle scattering and wave mechanics. The formula is analogous to the quantum rule of Bohr-Sommerfeld.

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ASYMPTOTIC BEHAVIOUR AS λ→∞ OF THE SOLUTION OF THE EQUATIONw″(z)−p(z, λ)w(z) = 0 IN THE COMPLEXz-PLANE

Cited by 69 publications

References 14 publications

Uniform Asymptotic Solution of Second Order Linear Differential Equations without Turning Varieties

Uniform Asymptotic Solution of Second Order Linear Differential Equations without Turning Varieties

A new basis for uniform asymptotic solution of differential equations containing one or several parameters

Distribution of eigenvalues in the presence of higher order turning points

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