Uniform and Quasi-Uniform Asymptotic Expansions of Incomplete Diffraction Integrals

Ciarkowski, A.

doi:10.1137/0148074

Cited by 3 publications

(5 citation statements)

References 9 publications

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“…I w ant to em phasize th a t my modified expansion is still of the Bleistein type, because th e rem ainder in any p artial sum of th e expansion is the same in both the original and my procedure. Sim ilar sim plification was obtained by myself (Ciarkowski 1988 We su b stitu te this into (3.13), let r 0 tend to ooe~l3,t/4 and m ake use of (3.3) and (3.10). In this m anner we obtain for any yh and yp, In this section I assume th a t the critical points = and = in (1.1) are well separated, i.e.…”

Section: )mentioning

confidence: 99%

Uniform asymptotic expansion of an integral with a saddle point, a pole and a branch point

Ciarkowski

1989

Proc. R. Soc. Lond. A

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The paper is concerned with an asymptotic behaviour of a contour integral with three critical points: a saddle point, a pole and a branch point. Two leading terms of a uniform asymptotic expansion of the integral have been effectively constructed. The expansion remains valid as its critical points approach one another or coalesce. In the special case that the saddle point is bounded away from one of the remaining critical points, or from both, the expansion reduces to simpler in form quasi-uniform and non-uniform expansions, respectively. With the aid of the non-uniform expansion the integral has been interpreted as describing three different waves, each one associated with a corresponding critical point.

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Section: )mentioning

confidence: 99%

Uniform asymptotic expansion of an integral with a saddle point, a pole and a branch point

Ciarkowski

1989

Proc. R. Soc. Lond. A

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“…See also [2] for an application with a more general phase function. In this case, we have a saddle point at the origin that can coalesce with the pole at t = −v if v is small and with the endpoint t = u if u is small.…”

Section: Related and Generalized Functions And Integralsmentioning

confidence: 99%

“…In this case the solution y (2) n (z) is minimal for increasing n and −π < ph(β − α)z < π, see [4,Sec. 4].…”

Section: Minimal and Dominant Solutionsmentioning

confidence: 99%

“…In this case, we have a saddle point at the origin that can coalesce with the pole at t = −v if v is small and with the endpoint t = u if u is small. These two situations can be analysed separately using standard methods in uniform asymptotics (in particular with the complementary error function, see [2]). An alternative form for this function is…”

Section: Related and Generalized Functions And Integralsmentioning

confidence: 99%

“…Corresponding series for the minimal solution of the homogeneous recursion (that is, the function y (2) n (z) given in (3.9)) are 12) with for λ = m − c (m = 0, 1, 2, . .…”

Section: Normalization Relationsmentioning

confidence: 99%

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Analytical and numerical aspects of a generalization of the complementary error function

Deaño

Temme

2010

Applied Mathematics and Computation

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In this paper we discuss analytical and numerical properties of the function V ν,µ (α, β, z) = ∞ 0 e −zt (t + α) ν (t + β) µ dt, with α, β, ℜz > 0, which can be viewed as a generalization of the complementary error function, and in fact also as a generalization of the Kummer U −function. The function V ν,µ (α, β, z) is used for certain values of the parameters as an approximant in a singular perturbation problem. We consider the relation with other special functions and give asymptotic expansions as well as recurrence relations. Several methods for its numerical evaluation and examples are given.2000 Mathematics Subject Classification: 33B20, 33C15, 41A60, 65D20, 65Q05.

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Uniform asymptotic expansions of integrals: a selection of problems

Temme

1995

Journal of Computational and Applied Mathematics

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On the occasion of the conference we mention examples of Stieltjes' work on asymptotics of special functions. The remaining part of the paper gives a selection of asymptotic methods for integrals, in particular on uniform approximations. We discuss several "standard" problems and examples, in which known special functions (error functions, Airy functions, Bessel functions, etc.) are needed to construct uniform approximations. Finally, we discuss the recent interest and new insights in the Stokes phenomenon. An extensive bibliography on uniform asymptotic methods for integrals is given, together with references to recent papers on the Stokes phenomenon for integrals and related topics.

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Uniform and Quasi-Uniform Asymptotic Expansions of Incomplete Diffraction Integrals

Cited by 3 publications

References 9 publications

Uniform asymptotic expansion of an integral with a saddle point, a pole and a branch point

Uniform asymptotic expansion of an integral with a saddle point, a pole and a branch point

Analytical and numerical aspects of a generalization of the complementary error function

Uniform asymptotic expansions of integrals: a selection of problems

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