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On the Asymptotic Approximation of Integrals
SIAM Journal on Applied Mathematics, 1986This paper is concerned with an attempt to improve the accuracy of certain asymptotic approximations. The basic idea is simply to redefine the large parameter in a particular way. This idea is applied to four examples, including a Laplace integral, the gamma function, and an integral which can be treated by Erdélyi's method.
Reid, W. H., Skates, S. J.
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Asymptotic Integrability of Water Waves
Physical Review Letters, 1996Summary: The asymptotic integrability of the idealized water waves is formally established. Namely, it is shown that in the small amplitude, long wave limit there exists an explicit transformation which maps these equations to a system of two integrable equations.
Fokas, A. S., Liu, Q. M.
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A uniform asymptotic evaluation of integrals
Proceedings of the IEEE, 1986Previous uniform asymptotic evaluations of integrals have been restricted to cases where the integrand singularities are close to the saddle point(s). A method is presented here which allows such uniform evaluations with integrand singularities anywhere near the steepest descent path.
John L. Volakis, Martin I. Herman
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Asymptotic Integration Constants
American Journal of Mathematics, 1946Not ...
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Asymptotic Behavior of Integrals
SIAM Review, 1972An account is given of some developments in the asymptotic evaluation of integrals of a single variable. After a discussion of Laplace integrals and quadrature formulas, estimates are provided of the errors in Laplace-type integrals and the method of steepest descents.
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ON THE ASYMPTOTIC BEHAVIOUR OF AN EXPONENTIAL INTEGRAL
Analysis, 1989Considering measurable functions f: \({\mathbb{R}}^+\to {\mathbb{R}}\), which are locally bounded and satisfy \(q(x)/x\to \infty,\) the author investigates certain asymptotic relations between q and its complementary function \(q^*(x)=\sup_{y>0}\{xy-q(y)\}\) and the following log-Laplace-type transformation \[ \tilde q(s)=\log (s\cdot \int^{\infty}_{0}\
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Asymptotic Expansion of a Multiple Integral
SIAM Journal on Mathematical Analysis, 1987Die Verff. betrachten die asymptotische Entwicklung des Integrals \[ J(s)=\int^{1}_{0}\int^{1}_{0}g(x^ ay^ b/s)x^{\alpha}y^{\beta}f(x,y)dxdy \] für \(s\to +0\). Mittels Mellin Transformation wird J(s) durch das Integral \(J(s)=(1/2\pi i)\int^{c+i\infty}_{c-i\infty}s^{-z}F(z)M[g,-z]dz\) dargestellt.
McClure, J. P., Wong, R.
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On the Asymptotic Periods of Integral Functions
Mathematical Proceedings of the Cambridge Philosophical Society, 1935A period of a function f(z) is defined to be a number ω (≠ 0) such thatis identically zero; and it can be shown that an integral function may either have no periods or else a single sequence kλ (k = ± 1, ± 2, …).
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1999
We describe a test for Asymptotic integrability and apply it to a class of dispersive PDEs. We identify a new equation which is integrable up to order two.
PROCESI, Michela, Degasperis A.
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We describe a test for Asymptotic integrability and apply it to a class of dispersive PDEs. We identify a new equation which is integrable up to order two.
PROCESI, Michela, Degasperis A.
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Asymptotic Evaluation of Multidimensional Integrals
Journal of Mathematical Physics, 1970The asymptotic evaluation of a wide class of multidimensional integrals occurring in mathematical physics is considered. In this class are included integrals of the form 1(2π)N ∫ −∞∞ρ(k)exp[ik·x−σ(k)t]dk. A semiconstructive method is proven and certain classes of integrals are asymptotically evaluated.
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