181 research outputs found
A numerical method for oscillatory integrals with coalescing saddle points
The value of a highly oscillatory integral is typically determined
asymptotically by the behaviour of the integrand near a small number of
critical points. These include the endpoints of the integration domain and the
so-called stationary points or saddle points -- roots of the derivative of the
phase of the integrand -- where the integrand is locally non-oscillatory.
Modern methods for highly oscillatory quadrature exhibit numerical issues when
two such saddle points coalesce. On the other hand, integrals with coalescing
saddle points are a classical topic in asymptotic analysis, where they give
rise to uniform asymptotic expansions in terms of the Airy function. In this
paper we construct Gaussian quadrature rules that remain uniformly accurate
when two saddle points coalesce. These rules are based on orthogonal
polynomials in the complex plane. We analyze these polynomials, prove their
existence for even degrees, and describe an accurate and efficient numerical
scheme for the evaluation of oscillatory integrals with coalescing saddle
points
An asymptotic expansion of the hyberbolic umbilic catastrophe integral
We obtain an asymptotic expansion of the hyperbolic umbilic catastrophe integral Ψ(H) (x,y,z) := ∫∞−∞∫∞−∞exp(i(s3+t3+zst +yt+xs))ds dt
for large values of |x| and bounded values of |y| and |z|. The expansion is given in terms of Airy functions and inverse powers of x. There is only one Stokes ray at argx=Ï€
. We use the modified saddle point method introduced in (López et al. J Math Anal Appl 354(1):347–359, 2009). The accuracy and the asymptotic character of the approximations are illustrated with numerical experiments.This research was supported by the Universidad Pública de Navarra, research grant PRO-UPNA (6158) 01/01/2022
Uniform Asymptotic Methods for Integrals
We give an overview of basic methods that can be used for obtaining
asymptotic expansions of integrals: Watson's lemma, Laplace's method, the
saddle point method, and the method of stationary phase. Certain developments
in the field of asymptotic analysis will be compared with De Bruijn's book {\em
Asymptotic Methods in Analysis}. The classical methods can be modified for
obtaining expansions that hold uniformly with respect to additional parameters.
We give an overview of examples in which special functions, such as the
complementary error function, Airy functions, and Bessel functions, are used as
approximations in uniform asymptotic expansions.Comment: 31 pages, 3 figure
Three routes to the exact asymptotics for the one-dimensional quantum walk
We demonstrate an alternative method for calculating the asymptotic behaviour
of the discrete one-coin quantum walk on the infinite line, via the Jacobi
polynomials that arise in the path integral representation. This is
significantly easier to use than the Darboux method. It also provides a single
integral representation for the wavefunction that works over the full range of
positions, including throughout the transitional range where the behaviour
changes from oscillatory to exponential. Previous analyses of this system have
run into difficulties in the transitional range, because the approximations on
which they were based break down here. The fact that there are two different
kinds of approach to this problem (Path Integral vs. Schr\"{o}dinger wave
mechanics) is ultimately a manifestation of the equivalence between the
path-integral formulation of quantum mechanics and the original formulation
developed in the 1920s. We discuss how and why our approach is related to the
two methods that have already been used to analyse these systems.Comment: 25 pages, AMS preprint format, 4 figures as encapsulated postscript.
Replaced because there were sign errors in equations (80) & (85) and Lemma 2
of the journal version (v3
Asymptotic approximation of a highly oscillatory integral with application to the canonical catastrophe integrals
We consider the highly oscillatory integral () ∶= ∫ ∞ −∞ (+2+) () for large positive values of , − < ≤ , and positive integers with 1 ≤ ≤ , and () an entire function. The standard saddle point method is complicated and we use here a simplified version of this method introduced by López et al. We derive an asymptotic approximation of this integral when → +∞ for general values of and in terms of elementary functions, and determine the Stokes lines. For ≠1, the asymptotic behavior of this integral may be classified in four different regions according to the even/odd character of the couple of parameters and ; the special case =1 requires a separate analysis. As an important application, we consider the family of canonical catastrophe integrals Ψ(1, 2,…,) for large values of one of its variables, say , and bounded values of the remaining ones. This family of integrals may be written in the form () for appropriate values of the parameters , and the function (). Then, we derive an asymptotic approximation of the family of canonical catastrophe integrals for large ||. The approximations are accompanied by several numerical experiments. The asymptotic formulas presented here fill up a gap in the NIST Handbook of Mathematical Functions by Olver et al
Mixed powers of generating functions
Given an integer m>=1, let || || be a norm in R^{m+1} and let S denote the
set of points with nonnegative coordinates in the unit sphere with respect to
this norm. Consider for each 1<= j<= m a function f_j(z) that is analytic in an
open neighborhood of the point z=0 in the complex plane and with possibly
negative Taylor coefficients. Given a vector n=(n_0,...,n_m) with nonnegative
integer coefficients, we develop a method to systematically associate a
parameter-varying integral to study the asymptotic behavior of the coefficient
of z^{n_0} of the Taylor series of (f_1(z))^{n_1}...(f_m(z))^{n_m}, as ||n||
tends to infinity. The associated parameter-varying integral has a phase term
with well specified properties that make the asymptotic analysis of the
integral amenable to saddle-point methods: for many directions d in S, these
methods ensure uniform asymptotic expansions for the Taylor coefficient of
z^{n_0} of (f_1(z))^{n_1}...(f_m(z))^{n_m}, provided that n/||n|| stays
sufficiently close to d as ||n|| blows up to infinity. Our method finds
applications in studying the asymptotic behavior of the coefficients of a
certain multivariable generating functions as well as in problems related to
the Lagrange inversion formula for instance in the context random planar maps.Comment: 14 page
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