A systematization of the saddle point method. Application to the Airy and Hankel functions

AbstractThe standard saddle point method of asymptotic expansions of integrals requires to show the existence of the steepest descent paths of the phase function and the computation of the coefficients of the expansion from a function implicitly defined by solving an inversion problem. This means that the method is not systematic because the steepest descent paths depend on the phase function on hand and there is not a general and explicit formula for the coefficients of the expansion (like in Watson's Lemma for example). We propose a more systematic variant of the method in which the computation of the steepest descent paths is trivial and almost universal: it only depends on the location and the order of the saddle points of the phase function. Moreover, this variant of the method generates an asymptotic expansion given in terms of a generalized (and universal) asymptotic sequence that avoids the computation of the standard coefficients, giving an explicit and systematic formula for the expansion that may be easily implemented on a symbolic manipulation program. As an illustrative example, the well-known asymptotic expansion of the Airy function is rederived almost trivially using this method. New asymptotic expansions of the Hankel function Hn(z) for large n and z are given as non-trivial examples

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

Uniform convergent expansions of the error function in terms of elementary functions

We derive a new analytic representation of the error function erf z in the form of a convergent series whose terms are exponential and rational functions. The expansion holds uniformly in z in the double sector | arg(±z)| < π/4. The expansion is accompanied by realistic error bounds

Asymptotic expansions for Moench’s integral transform of hydrology

Theis’ theory (1935), later improved by Hantush &amp; Jacob (1955) and Moench (1971), is a technique designed to study the water level in aquifers. The key formula in this theory is a certain integral transform H[g](r,t) of the pumping function g that depends on the time t and the relative position r to the pumping point as well as on other physical parameters. Several analytic approximations of H[g](r,t) have been investigated in the literature that are valid and accurate in certain regions of r, t and the mentioned physical parameters. In this paper, the analysis of possible analytic approximations of H[g](r,t) is completed by investigating asymptotic expansions of H[g](r,t) in a region of the parameters that is of interest in practical situations, but that has not yet been investigated. Explicit and/or recursive algorithms for the computation of the coefficients of the expansions and estimates for the remainders are provided. Some numerical examples based on an actual physical experiment conducted by Layne-Western Company in 1953 illustrate the accuracy of the approximations

New series expansions for the H-function of communication theory

The H-function of communication theory plays an important role in the error rate analysis in digital communication with the presence of additive white Gaussian noise (AWGN) and generalized multipath fading conditions. In this paper we investigate several convergent and/or asymptotic expansions of Hp(z,b,η) for some limiting values of their variables and parameters: large values of z, large values of p, small values of η, and values of b→1. We provide explicit and/or recursive algorithms for the computation of the coefficients of the expansions. Some numerical examples illustrate the accuracy of the approximations

Asymptotic behaviour of three-dimensional singularly perturbed convection–diffusion problems with discontinuous data

AbstractWe consider three singularly perturbed convection–diffusion problems defined in three-dimensional domains: (i) a parabolic problem −ϵ(uxx+uyy)+ut+v1ux+v2uy=0 in an octant, (ii) an elliptic problem −ϵ(uxx+uyy+uzz)+v1ux+v2uy+v3uz=0 in an octant and (iii) the same elliptic problem in a half-space. We consider for all of these problems discontinuous boundary conditions at certain regions of the boundaries of the domains. For each problem, an asymptotic approximation of the solution is obtained from an integral representation when the singular parameter ϵ→0+. The solution is approximated by a product of two error functions, and this approximation characterizes the effect of the discontinuities on the small ϵ− behaviour of the solution and its derivatives in the boundary layers or the internal layers

Paleovegetación durante la Edad del Bronce en la Rioja alavesa: Análisis palinológico del yacimiento de Peña Parda ( Laguardia, Álava)

En este trabajo se presenta el estudio palinológico del yacimiento de Peña Parda (Laguardia, Álava). Los resultados obtenidos, en las 13 muestras estudiadas, han aportado interesantes datos sobre el paisaje vegetal existente en la vertiente sur de la Sierra de Cantabria durante la Edad del Bronce, así como sobre las evidencias de antropización

Orthogonal basis for the optical transfer function

We propose systems of orthogonal functions qn to represent optical transfer functions (OTF) characterized by including the diffraction-limited OTF as the first basis function q0 = OTFperfect. To this end, we apply a powerful and rigorous theoretical framework based on applying the appropriate change of variables to well-known orthogonal systems. Here we depart from Legendre polynomials for the particular case of rotationally symmetric OTF and from spherical harmonics for the general case. Numerical experiments with different examples show that the number of terms necessary to obtain an accurate linear expansion of the OTF mainly depends on the image quality. In the rotationally symmetric case we obtained a reasonable accuracy with approximately 10 basis functions, but in general, for cases of poor image quality, the number of basis functions may increase and hence affect the efficiency of the method. Other potential applications, such as new image quality metrics are also discussed.Ministerio de Economía y Competitividad (MINECO) (MTM2014-52859-P, FIS2014-58303-P).Peer Reviewe