asymptotic behavior of coefficients
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11: 2.11 Remainder Terms; Stokes Phenomenon
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►with
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►However, to enjoy the resurgence property (§2.7(ii)) we often seek instead expansions in terms of the -functions introduced in §2.11(iii), leaving the connection of the error-function type behavior as an implicit consequence of this property of the -functions.
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►in which
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12: 2.7 Differential Equations
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►is one at which the coefficients
and are analytic.
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►Formal solutions are
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►Note that the coefficients in the expansions (2.7.12), (2.7.13) for the “late” coefficients, that is, , with large, are the “early” coefficients
, with small.
…See §2.11(v) for other examples.
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►For irregular singularities of nonclassifiable rank, a powerful tool for finding the asymptotic behavior of solutions, complete with error bounds, is as follows:
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13: 2.3 Integrals of a Real Variable
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►For the Fourier integral
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►Other types of singular behavior in the integrand can be treated in an analogous manner.
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(b)
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►Then
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As
2.3.14
and the expansion for is differentiable. Again and are positive constants. Also (consistent with (a)).
§2.3(vi) Asymptotics of Mellin Transforms
…14: Bibliography K
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Pascal program for generating tables of Clebsch-Gordan coefficients.
Comput. Phys. Comm. 85 (1), pp. 82–88.
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Asymptotic behavior of the solutions of the Painlevé equation of the first kind.
Differ. Uravn. 24 (10), pp. 1684–1695 (Russian).
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Asymptotic approximations for the first incomplete elliptic integral near logarithmic singularity.
J. Comput. Appl. Math. 205 (1), pp. 186–206.
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Clebsch-Gordan coefficients for and Hahn polynomials.
Nieuw Arch. Wisk. (3) 29 (2), pp. 140–155.
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Asymptotic solution of Maxwell’s equations near caustics.
Izv. Vuz. Radiofiz. 7, pp. 1049–1056.
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15: 3.5 Quadrature
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►The are also known as Christoffel coefficients or Christoffel numbers and they are all positive.
The remainder is given by
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►In practical applications the weight function is chosen to simulate the asymptotic behavior of the integrand as the endpoints are approached.
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►Below we give for the classical orthogonal polynomials the recurrence coefficients
and in (3.5.30).
These also immediately yield the recurrence coefficients in (3.5.30_5).
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16: 18.2 General Orthogonal Polynomials
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►Then, with the coefficients (18.2.11_4) associated with the monic OP’s , the orthonormal recurrence relation for takes the form
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►The monic and orthonormal OP’s, and their determination via recursion, are more fully discussed in §§3.5(v) and 3.5(vi), where modified recursion coefficients are listed for the classical OP’s in their monic and orthonormal forms.
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►Alternatives for numerical calculation of the recursion coefficients in terms of the moments are discussed in these references, and in §18.40(ii).
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►This says roughly that the series (18.2.25) has the same pointwise convergence behavior as the same series with , a Chebyshev polynomial of the first kind, see Table 18.3.1.
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►for certain coefficients
with independent of .
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17: 30.11 Radial Spheroidal Wave Functions
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§30.11(iii) Asymptotic Behavior
… ►For asymptotic expansions in negative powers of see Meixner and Schäfke (1954, p. 293). … ►where ►
30.11.10
even,
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30.11.11
odd.
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18: 18.15 Asymptotic Approximations
§18.15 Asymptotic Approximations
►§18.15(i) Jacobi
… ►§18.15(ii) Ultraspherical
… ►The leading coefficients are given by … ►The asymptotic behavior of the classical OP’s as with the degree and parameters fixed is evident from their explicit polynomial forms; see, for example, (18.2.7) and the last two columns of Table 18.3.1. …19: 25.11 Hurwitz Zeta Function
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§25.11(xii) -Asymptotic Behavior
… ►As in the sector , with and fixed, we have the asymptotic expansion … ►Similarly, as in the sector , ►
25.11.44
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25.11.45
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