Uniform Asymptotic Expansions of Symmetric Elliptic Integrals

Symmetric standard elliptic integrals are considered when two or more parameters are larger than the others. The distributional approach is used to derive seven expansions of these integrals in inverse powers of the asymptotic parameters. Some of these expansions also involve logarithmic terms in the asymptotic variables. These expansions are uniformly convergent when the asymptotic parameters are greater than the remaining ones. The coefficients of six of these expansions involve hypergeometric functions with less parameters than the original integrals. The coefficients of the seventh expansion again involve elliptic integrals, but with less parameters than the original integrals. The convergence speed of any of these expansions increases for an increasing difference between the asymptotic variables and the remaining ones. All the expansions are accompanied by an error bound at any order of the approximation. January 31, 2000. Date revised: May 18, 2000. Date accepted: August 4, 2000.     

A Method for Incorporating Transcendentally Small Terms into the Method of Matched Asymptotic Expansions

The essential ideas behind a method for incorporating exponentially small terms into the method of matched asymptotic expansions are demonstrated using an Ackerberg–O'Malley resonance problem and a spurious solutions problem of Carrier and Pearson. One begins with the application of the standard method of matched asymptotic expansions to obtain at least the leading terms in outer and inner (Poincaré-type) expansions; some, although not all, matching can be carried out at this stage. This is followed by the introduction of supplementary expansions whose gauge functions are transcendentally small compared to those in the standard expansions. Analysis of terms in these expansions allows the matching to be completed. Furthermore, the method allows for the inclusion of globally valid transcendentally small contributions to the asymptotic solution; it is well known that such terms may be numerically significant.     

Asymptotic expansions of integrals of two Bessel functions via the generalized hypergeometric and Meijer functions

Asymptotic expansions of certain finite and infinite integrals involving products of two Bessel functions of the first kind are obtained by using the generalized hypergeometric and Meijer functions. The Bessel functions involved are of arbitrary (generally different) orders, but of the same argument containing a parameter which tends to infinity. These types of integrals arise in various contexts, including wave scattering and crystallography, and are of general mathematical interest being related to the Riemann—Liouville and Hankel integrals. The results complete the asymptotic expansions derived previously by two different methods — a straightforward approach and the Mellin-transform technique. These asymptotic expansions supply practical algorithms for computing the integrals. The leading terms explicitly provide valuable analytical insight into the high-frequency behavior of the solutions to the wave-scattering problems.     

Derivation of transitional asymptotic expansions from integral representations

A method for deriving transitional asymptotic expansions from integral representations is described and applied to Anger function and modified Hankel function. The method consists in deriving asymptotic expansions of the function considered as well as its first derivativeat the transition point using conventional methods such as Laplace’s method or the method of steepest descents. Since both the functions considered satisfy a second order linear differential equation, it is possible to obtain asymptotic expansions of higher order derivatives of the functions from the first two expansions. Thus asymptotic expressions for all the derivatives at the transition point are known and a Taylor expansion of the function in the neighbourhood of the transition point can be written. The method is also applicable to the generalized exponential integral, Weber’s parabolic cylinder function and Poiseuille function.     

Asymptotic solutions of inhomogeneous differential equations having a turning point

Asymptotic solutions are derived for inhomogeneous differential equations having a large real or complex parameter and a simple turning point. They involve Scorer functions and three slowly varying analytic coefficient functions. The asymptotic approximations are uniformly valid for unbounded complex values of the argument, and are applied to inhomogeneous Airy equations having polynomial and exponential forcing terms. Error bounds are available for all approximations, including new simple ones for the well-known asymptotic expansions of Scorer functions of large complex argument.     

Some solutions (linear in the spatial variables) and generalized Chapman–Enskog functions basic to the linearized Boltzmann equations for a binary mixture of rigid spheres

A Legendre expansion of the (matrix) scattering kernel relevant to the (vector- valued) linearized Boltzmann equation for a binary mixture of rigid spheres is used to define twelve solutions that are linear in the spatial variables {x, y, z}. The twelve (asymptotic) solutions are expressed in terms of three vector-valued functions A ⁽¹⁾(c), A⁽²⁾(c), and B(c). These functions are generalizations of the Chapman–Enskog functions used to define asymptotic solutions and viscosity and heat conduction coefficients for the case of a single-species gas. To provide evidence that the three Chapman–Enskog vectors exist as solutions of the defining linear integral equations, numerical results developed in terms of expansions based on Hermite cubic splines and a collocation scheme are reported for two binary mixtures (Ne-Ar and He-Xe) with various molar concentrations.     

Some solutions (linear in the spatial variables) and generalized Chapman–Enskog functions basic to the linearized Boltzmann equations for a binary mixture of rigid spheres

A Legendre expansion of the (matrix) scattering kernel relevant to the (vector- valued) linearized Boltzmann equation for a binary mixture of rigid spheres is used to define twelve solutions that are linear in the spatial variables {x, y, z}. The twelve (asymptotic) solutions are expressed in terms of three vector-valued functions A ⁽¹⁾(c), A⁽²⁾(c), and B(c). These functions are generalizations of the Chapman–Enskog functions used to define asymptotic solutions and viscosity and heat conduction coefficients for the case of a single-species gas. To provide evidence that the three Chapman–Enskog vectors exist as solutions of the defining linear integral equations, numerical results developed in terms of expansions based on Hermite cubic splines and a collocation scheme are reported for two binary mixtures (Ne-Ar and He-Xe) with various molar concentrations.     

Multi-point Taylor expansions of analytic functions

Taylor expansions of analytic functions are considered with respect to several points, allowing confluence of any of them. Cauchy-type formulas are given for coefficients and remainders in the expansions, and the regions of convergence are indicated. It is explained how these expansions can be used in deriving uniform asymptotic expansions of integrals. The method is also used for obtaining Laurent expansions in several points as well as Taylor-Laurent expansions.

     

A method for constructing asymptotic expansions of bisingularly perturbed problems

In this paper we propose an analog of the method of boundary functions for constructing uniform asymptotic expansions of solutions to bisingularly perturbed problems. With the help of this method we construct uniform asymptotic expansions of solutions to the Dirichlet problem for bisingularly perturbed ordinary differential equations and elliptic equations of the second order. By the use of the maximum principle we obtain estimates for the remainder terms.     

On binary search tree recursions with monomials as toll functions

We consider distributional recursions which appear in the study of random binary search trees with monomials as toll functions. This extends classical parameters as the internal path length in binary search trees. As our main results we derive asymptotic expansions for the moments of the random variables under consideration as well as limit laws and properties of the densities of the limit distributions. The analysis is based on the contraction method.     

Asymptotic Solutions of Two-Dimensional Hartree-Type Equations Localized in the Neighborhood of Line Segments

We consider the eigenvalue problem for the two-dimensional Schrödinger equation containing an integral Hartree-type nonlinearity with an interaction potential having a logarithmic singularity. Global asymptotic solutions localized in the neighborhood of a line segment in the plane are constructed using the matching method for asymptotic expansions. The Bogoliubov and Airy polarons are used as model functions in these solutions. An analogue of the Bohr–Sommerfeld quantization rule is established to find the related series of eigenvalues.     

Asymptotic Analysis of Symmetric Functions

In this paper we consider asymptotic expansions for a class of sequences of symmetric functions of many variables. Applications to classical and free probability theory are discussed.     

轴对称圆环壳的一个新解

指出了现有渐近解的不足之处.本文统一用广义Airy函数表示齐解和非齐特解的完全渐近展开,而现有的渐近解是用Besel或Airy函数表示齐解,用Lommer函数表示非齐特解的.本文所得到的新解是全域一致有效的,达到了薄壳的理论精度,且齐解和特解之间满足变动参数关系.事实上,本文给出了三个特解,其中之一正好与Tumarkin(1959)和Clark(1963)的解相同.     

Two-Point Taylor Expansions of Analytic Functions

Taylor expansions of analytic functions are considered with respect to two points. Cauchy-type formulas are given for coefficients and remainders in the expansions, and the regions of convergence are indicated. It is explained how these expansions can be used in deriving uniform asymptotic expansions of integrals. The method is also used for obtaining Laurent expansions in two points.     

Extension of Edgeworth type expansion of the distribution of the sums of I.I.D. random variables in non-regular cases

The asymptotic expansions of the distributions of the sums of independent identically distributed random variables are given by Edgeworth type expansions when moments do not necessarily exist, but when the density can be approximated by rational functions. Supported in part by the Sakkokai Foundation.     

Distribution theory for some tests of independence of seemingly unrelated regressions

Series expansions and second-order asymptotic expansions are obtained for the characteristic functions of three statistics useful for testing the independence of two multivariate regression equations with different design matrices.     

The Bergström–Grigelionis asymptotic expansions

We consider a triangular array of independent identically distributed discrete random variables. We assume that the probability distribution of sums satisfies the necessary and sufficient conditions for the weak convergence to the compound Poisson distribution. The first known result (the case where random variables take only integer values) is due to B. Grigelionis, who estimated the convergence rate to the compound Poisson distribution. We extend the summation of random variables by including the variables taking discrete values and by using the Grigelionis ideas to obtain “lengthy” asymptotic expansions. These expansions are based on the well-known Bergstrom identity [H. Bergstrom, On asymptotic expansions of probability functions, Scand. Actuarial J., 34(1):1–33, 1951].     

The Second Appell Function for one Large Variable

We consider the Mellin convolution integral representation of the second Appell function given in [8]. Then, we apply the asymptotic method designed in [12] for this kind of integrals to derive new asymptotic expansions of the Appell function F ₂ for one large variable in terms of hypergeometric functions. For certain values of the parameters, some of these expansions involve logarithmic terms in the asymptotic variables. The accuracy of the approximations is illustrated with numerical experiments.     

Asymptotic expansions based on smooth functions in the central limit theorem

Summary Stein's method is used to derive asymptotic expansions for expectations of smooth functions of sums of independent random variables, together with Lyapounov estimates of the error in the approximation.     

Laplace's method on a computer algebra system with an application to the real valued modified Bessel functions

We examine a Maple implementation of two distinct approaches to Laplace's method used to obtain asymptotic expansions of Laplace-type integrals. One algorithm uses power series reversion, whereas the other expands all quantities in Taylor or Puiseux series. These algorithms are used to derive asymptotic expansions for the real valued modified Bessel functions of pure imaginary order and real argument that mimic the well-known corresponding expansions for the unmodified Bessel functions.