Applied Asymptotic Analysis

Miller, Peter D.

doi:10.1090/gsm/075

Cited by 235 publications

(237 citation statements)

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“…A full discussion with many examples is given in texts such as [66,67] and the review [6]. But it is best to illustrate with a concrete example as furnished in the next section.…”

Section: Steepest Descent For Integralsmentioning

confidence: 99%

“…Right panel: exact Chebyshev coefficients for f (x) = exp(−1/x) (circles), the asymptotic coefficients (x's) and the error which is the difference between these curves (dashed) Figure 2 shows the steepest-descent path (left) and the comparison between the exact and asymptotic Chebyshev coefficients on the right. The steepest-descent approximation can be extended to a full asymptotic series as described in [6,66,67].…”

Section: Functions With Trivial Power Series About the Singular Pointmentioning

confidence: 99%

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Large-degree asymptotics and exponential asymptotics for Fourier, Chebyshev and Hermite coefficients and Fourier transforms

Boyd

2008

J Eng Math

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When a function is expanded asfor some set of basis functions φ j (x), its spectral coefficients a n generally have an asymptotic approximation, as n → ∞, in the form of an inverse power series plus terms that decrease exponentially with n. If f (x) is analytic on the expansion interval, then all the coefficients of the inverse power series are zero and the problem becomes one of "beyond-all-orders" or "exponential" asymptotics. The method of steepest descent for integrals and other complexpath integration techniques can successfully connect the exponentially small behavior of the spectral coefficients to the singularities of f (x) off the expansion interval. Many examples are given in both one and two dimensions.

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“…A full discussion with many examples is given in texts such as [66,67] and the review [6]. But it is best to illustrate with a concrete example as furnished in the next section.…”

Section: Steepest Descent For Integralsmentioning

confidence: 99%

Section: Functions With Trivial Power Series About the Singular Pointmentioning

confidence: 99%

Large-degree asymptotics and exponential asymptotics for Fourier, Chebyshev and Hermite coefficients and Fourier transforms

Boyd

2008

J Eng Math

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“…To analyze the Casimir energy and force we will use the zeta function regularization [27,28] and in Section 2 we describe the relevant spectral problem and the geometry of the piston. In Section 3 we find the analytical continuation of the associated zeta function; part of the construction involves the derivation of uniform asymptotics of solutions of initial value problems of an ordinary differential equation using the WKB approximation [29,30]. The analytical continuation of the zeta function is the basis for the Casimir energy and force results in Section 4.…”

Section: Introductionmentioning

confidence: 99%

The Casimir Effect for Generalized Piston Geometries

Fucci

Kirsten

2012

Int. J. Mod. Phys. A

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In this paper we study the Casimir energy and force for generalized pistons constructed from warped product manifolds of the type I × f N where I = [a, b] is an interval of the real line and N is a smooth compact Riemannian manifold either with or without boundary. The piston geometry is obtained by dividing the warped product manifold into two regions separated by the cross section positioned at R ∈ (a, b). By exploiting zeta function regularization techniques we provide formulas for the Casimir energy and force involving the arbitrary warping function f and base manifold N.

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“…A set of particularly effective ways of obtaining the contribution from a special point are the saddle point methods [25,19,8]. Based on Cauchy's integral theorem, the path of integration x=−1 x=1 Figure 1: The contours of the imaginary part of the oscillator g(x) = x 2 in the complex plane and the corresponding paths of steepest descent.…”

Section: Introductionmentioning

confidence: 99%