Asymptotic Methods for Integrals

Temme, Nico

doi:10.1142/9195

Cited by 111 publications

(144 citation statements)

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“…To simplify this integral, we apply the now-standard procedure [21][22][23] of transforming the integration variable K to a new variable Q, replacing the exponent by a simpler function with the same qualitative behaviour. In this case the behaviour is linear+cubic, so we choose the transformation…”

Section: Uniform Approximation For Profilementioning

confidence: 99%

Minimal model for tidal bore revisited

Berry

2019

New J. Phys.

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This develops a recent analysis of gentle undular tidal bores (2018 New J. Phys. 20 053066) and corrects an error. The simplest linear-wave superposition, of monochromatic waves propagating according to the shallow-water dispersion relation, leads to a family of profiles satisfying natural tidal bore boundary conditions, involving initial smoothed steps with different shapes. These profiles can be uniformly approximated to high accuracy in terms of the integral of an Airy function with deformed argument. For the long times corresponding to realistic bores, the profiles condense asymptotically onto the previously obtained integral-Airy function with linear argument: as the bore propagates, it forgets the shape of the initial step. The integral-Airy profile expands slowly, as the cube root of time, rather than advancing rigidly. This 'minimal model' leads to simple formulas for the main properties of the profile: amplitude, maximum slope, 'wavelength', and steepness; and an assumption about energy loss suggests how the bore weakens as it propagates.

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Section: Uniform Approximation For Profilementioning

confidence: 99%

Minimal model for tidal bore revisited

Berry

2019

New J. Phys.

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“…We thank the anonymous referee for useful feedback. RBP thanks Nico Temme for providing the manuscript of his monograph [24]. VV thanks Don Dawson who introduced him to the branching particle systems and provided a very useful feedback on Section 4 of this paper, Tom Salisbury for his advice, and is grateful to the Fields Institute, University of Toronto and York University for their hospitality.…”

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confidence: 97%

Branching particle systems and compound Poisson processes related to Pólya-Aeppli distributions

Paris¹,

Vinogradov

2015

COSA

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ABSTRACT. We establish numerous new refined local limit theorems for a class of compound Poisson processes with Pólya-Aeppli marginals, and for a particular family of the branching particle systems which undergo critical binary branching and can be approximated by the backshifted Feller diffusion. To this end, we also derive new results for the families of Pólya-Aeppli and Poisson-exponential distributions. We relate a few of them to properties of certain special functions some of which were previously unknown.

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“…where μ = √ 2n + 1, t = x/μ, and g(μ) is the function with asymptotic expansion given in (11). For ζ we have the definition…”

Section: Expansions In Terms Of Airy Functionsmentioning

confidence: 99%

Asymptotic Approximations to the Nodes and Weights of Gauss–Hermite and Gauss–Laguerre Quadratures

2018

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Asymptotic approximations to the zeros of Hermite and Laguerre polynomials are given, together with methods for obtaining the coefficients in the expansions. These approximations can be used as a stand-alone method of computation of Gaussian quadratures for high enough degrees, with Gaussian weights computed from asymptotic approximations for the orthogonal polynomials. We provide numerical evidence showing that for degrees greater than 100, the asymptotic methods are enough for a double precision accuracy computation (15-16 digits) of the nodes and weights of the Gauss-Hermite and Gauss-Laguerre quadratures.

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Asymptotic Methods for Integrals

Cited by 111 publications

References 0 publications

Minimal model for tidal bore revisited

Minimal model for tidal bore revisited

Branching particle systems and compound Poisson processes related to Pólya-Aeppli distributions

Asymptotic Approximations to the Nodes and Weights of Gauss–Hermite and Gauss–Laguerre Quadratures

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