limit point and limit circle boundary conditions
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1: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
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►By Weyl’s alternative
equals either 1 (the limit point case) or 2 (the limit circle case), and similarly for .
… A boundary value for the end point
is a linear form on of the form
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►The above results, especially the discussions of deficiency indices and limit point and limit circle boundary conditions, lay the basis for further applications.
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►The materials developed here follow from the extensions of the Sturm–Liouville theory of second order ODEs as developed by Weyl, to include the limit point and limit circle singular cases.
…See, in particular, the overview Everitt (2005b, pp. 45–74), and the uniformly annotated listing of solved Sturm–Liouville problems in Everitt (2005a, pp. 272–331), each with their limit point, or circle, boundary behaviors categorized.
2: 1.9 Calculus of a Complex Variable
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Continuity
… ►A point is a limit point (limiting point or accumulation point) of a set of points in (or ) if every neighborhood of contains a point of distinct from . …As a consequence, every neighborhood of a limit point of contains an infinite number of points of . Also, the union of and its limit points is the closure of . … ►A function is complex differentiable at a point if the following limit exists: …3: 20.13 Physical Applications
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►In the singular limit
, the functions , , become integral kernels of Feynman path integrals (distribution-valued Green’s functions); see Schulman (1981, pp. 194–195).
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4: 2.1 Definitions and Elementary Properties
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►Let be a point set with a limit point
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As in
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►If is a finite limit point of , then
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►Similarly for finite limit point
in place of .
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►where is a finite, or infinite, limit point of .
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5: 33.18 Limiting Forms for Large
§33.18 Limiting Forms for Large
…6: 1.4 Calculus of One Variable
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►When this limit exists is differentiable at .
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►when the last limit exists.
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►If the limit exists then is called Riemann integrable.
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►when this limit exists.
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►when this limit exists.
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7: 4.31 Special Values and Limits
8: 10.72 Mathematical Applications
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►In regions in which (10.72.1) has a simple turning point
, that is, and are analytic (or with weaker conditions if is a real variable) and is a simple zero of , asymptotic expansions of the solutions for large can be constructed in terms of Airy functions or equivalently Bessel functions or modified Bessel functions of order (§9.6(i)).
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►In regions in which the function has a simple pole at and is analytic at (the case in §10.72(i)), asymptotic expansions of the solutions of (10.72.1) for large can be constructed in terms of Bessel functions and modified Bessel functions of order , where is the limiting value of as .
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9: 2.4 Contour Integrals
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2.4.3
,
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►is seen to converge absolutely at each limit, and be independent of .
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►If this integral converges uniformly at each limit for all sufficiently large , then by the Riemann–Lebesgue lemma (§1.8(i))
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►in which is finite, is finite or infinite, and is the angle of slope of at , that is, as along .
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►The branch of is the one satisfying , where is the limiting value of as from .
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10: 1.10 Functions of a Complex Variable
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►If , analytic in , equals on an arc in , or on just an infinite number of points with a limit point in , then they are equal throughout and is called an analytic continuation of .
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►An isolated singularity is always removable when exists, for example at .
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►If the poles are infinite in number, then the point at infinity is called an essential singularity: it is the limit point of the poles.
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►If the path circles a branch point at , times in the positive sense, and returns to without encircling any other branch point, then its value is denoted conventionally as .
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