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.

     

On the extraction technique in boundary integral equations

In this paper we develop and analyze a bootstrapping algorithm for the extraction of potentials and arbitrary derivatives of the Cauchy data of regular three-dimensional second order elliptic boundary value problems in connection with corresponding boundary integral equations. The method rests on the derivatives of the generalized Green's representation formula, which are expressed in terms of singular boundary integrals as Hadamard's finite parts. Their regularization, together with asymptotic pseudohomogeneous kernel expansions, yields a constructive method for obtaining generalized jump relations. These expansions are obtained via composition of Taylor expansions of the local surface representation, the density functions, differential operators and the fundamental solution of the original problem, together with the use of local polar coordinates in the parameter domain. For boundary integral equations obtained by the direct method, this method allows the recursive numerical extraction of potentials and their derivatives near and up to the boundary surface.

     

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 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.     

Generalized Abel type integral equations with localized fractional integrals and derivatives

Generalized Abel type integral equations with Gauss, Kummer's and Humbert's confluent hypergeometric functions in the kernel and generalized Abel type integral equations with localized fractional integrals are considered. The left-hand sides of these equations are inversed by using generalized fractional derivatives. Explicit solutions of the equations in the class of locally summable functions are obtained. They are represented in terms of hypergeometric functions. Asymptotic power exponential type expansions of the generalized and localized fractional integrals are obtained. The base solutions of the generalized Abel type integral equation are given in the form of asymptotic series.     

A Systematic “Saddle Point Near a Pole” Asymptotic Method with Application to the Gauss Hypergeometric Function

In recent works [ 1 ] and [ 2 ], we have proposed more systematic versions of the Laplace’s and saddle point methods for asymptotic expansions of integrals. Those variants of the standard methods avoid the classical change of variables and give closed algebraic formulas for the coefficients of the expansions. In this work we apply the ideas introduced in [ 1 ] and [ 2 ] to the uniform method “saddle point near a pole.” We obtain a computationally more systematic version of that uniform asymptotic method for integrals having a saddle point near a pole that, in many interesting examples, gives a closed algebraic formula for the coefficients. The asymptotic sequence is given, in general, in terms of exponential integrals of fractional order (or incomplete gamma functions). In particular, when the order of the saddle point is two, the basic approximant is given in terms of the error function (as in the standard method). As an application, we obtain new asymptotic expansions of the Gauss Hypergeometric function ₂F₁(a, b, c; z) for large b and c with c > b + 1 .     

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.     

Error Bounds for the Large-Argument Asymptotic Expansions of the Hankel and Bessel Functions

In this paper, we reconsider the large-argument asymptotic expansions of the Hankel, Bessel and modified Bessel functions and their derivatives. New integral representations for the remainder terms of these asymptotic expansions are found and used to obtain sharp and realistic error bounds. We also give re-expansions for these remainder terms and provide their error estimates. A detailed discussion on the sharpness of our error bounds and their relation to other results in the literature is given. The techniques used in this paper should also generalize to asymptotic expansions which arise from an application of the method of steepest descents.     

An Explicit Formula for the Coefficients in Laplace’s Method

Laplace’s method is one of the fundamental techniques in the asymptotic approximation of integrals. The coefficients appearing in the resulting asymptotic expansion arise as the coefficients of a convergent or asymptotic series of a function defined in an implicit form. Due to the tedious computation of these coefficients, most standard textbooks on asymptotic approximations of integrals do not give explicit formulas for them. Nevertheless, we can find some more or less explicit representations for the coefficients in the literature: Perron’s formula gives them in terms of derivatives of an explicit function; Campbell, Fröman and Walles simplified Perron’s method by computing these derivatives using an explicit recurrence relation. The most recent contribution is due to Wojdylo, who rediscovered the Campbell, Fröman and Walles formula and rewrote it in terms of partial ordinary Bell polynomials. In this paper, we provide an alternative representation for the coefficients that contains ordinary potential polynomials. The proof is based on Perron’s formula and a theorem of Comtet. The asymptotic expansions of the gamma function and the incomplete gamma function are given as illustrations.     

Asymptotic Expansions of Generalized Stieltjes Transforms of Algebraically Decaying Functions

Asymptotic expansions of Stieltjes and generalized Stieltjes transforms of functions having an asymptotic expansion in negative integer powers of their variable have been exhaustively investigated by R. Wong. In this article, we extend this analysis to Stieltjes and generalized Stieltjes transforms of functions having an asymptotic expansion in negative rational powers of their variable. Distributional approach is used to derive asymptotic expansions of the Stieltjes and generalized Stieltjes transforms of this kind of functions for large values of the parameter(s) of the transformation. Error bounds are obtained at any order of the approximation for a large family of integrands. The asymptotic approximation of an integral involved in the calculation of the mass renormalization of the quantum scalar field and of the third symmetric elliptic integral are given as illustrations.     

Asymptotic Expansions of Generalized Stieltjes Transforms of Algebraically Decaying Functions

Asymptotic expansions of Stieltjes and generalized Stieltjes transforms of functions having an asymptotic expansion in negative integer powers of their variable have been exhaustively investigated by R. Wong. In this article, we extend this analysis to Stieltjes and generalized Stieltjes transforms of functions having an asymptotic expansion in negative rational powers of their variable. Distributional approach is used to derive asymptotic expansions of the Stieltjes and generalized Stieltjes transforms of this kind of functions for large values of the parameter(s) of the transformation. Error bounds are obtained at any order of the approximation for a large family of integrands. The asymptotic approximation of an integral involved in the calculation of the mass renormalization of the quantum scalar field and of the third symmetric elliptic integral are given as illustrations.     

Asymptotic expansions andL ∞-error estimates for mixed finite element methods for second order elliptic problems

Summary Asymptotic expansions for mixed finite element approximations of the second order elliptic problem are derived and Richardson extrapolation can be applied to increase the accuracy of the approximations. A new procedure, which is called the error corrected method, is presented as a further application of the asymptotic error expansion for the first order BDM approximation of the scalar field. The key point in deriving the asymptotic expansions for the error is an establishment ofL ¹-error estimates for mixed finite element approximations for the regularized Green's functions. As another application of theL ¹-error estimates for the regularized Green's functions, we shall present maximum norm error estimates for mixed finite element methods for second order elliptic problems.     

Asymptotic and convergent expansions for solutions of third-order linear differential equations with a large parameter

In previous papers [6-8,10], we derived convergent and asymptotic expansions of solutions of second order linear differential equations with a large parameter. In those papers we generalized and developed special cases not considered in Olver"s theory [Olver, 1974]. In this paper we go one step forward and consider linear differential equations of the third order: $y"+a\Lambda^2 y"+b\Lambda^3y=f(x)y"+g(x)y$, with $a,b\in\mathbb{C}$ fixed, $f"$ and $g$ continuous, and $\Lambda$ a large positive parameter. We propose two different techniques to handle the problem: (i) a generalization of Olver"s method and (ii) the transformation of the differential problem into a fixed point problem from which we construct an asymptotic sequence of functions that converges to the unique solution of the problem. Moreover, we show that this second technique may also be applied to nonlinear differential equations with a large parameter. As an application of the theory, we obtain new convergent and asymptotic expansions of the Pearcey integral $P(x,y)$ for large $|x|$.     

A Note on Asymptotic Expansions to Approximate Eigenvalues of Integral Operators with Green's Kernels using the Iterated Galerkin Method

In this article, we consider approximation of eigenvalues of integral operators with Green's function-type kernels using the iterated Galerkin method. We obtain asymptotic expansions for approximate eigenvalues. The Richardson extrapolation is used to obtain eigenvalue approximations of higher order. A numerical example is considered in order to illustrate our theoretical results.     

Error Bounds for Uniform Asymptotic Expansions—Modified Bessel Function of Purely Imaginary Order

The authors modify a method of Olde Daalhuis and Temme for representing the remainder and coefficients in Airy-type expansions of integrals. By using a class of rational functions, they express these quantities in terms of Cauchy-type integrals; these expressions are natural generalizations of integral representations of the coefficients and the remainders in the Taylor expansions of analytic functions. By using the new representation, a computable error bound for the remainder in the uniform asymptotic expansion of the modified Bessel function of purely imaginary order is derived.     

Numerical algorithms for uniform Airy-type asymptotic expansions

Airy-type asymptotic representations of a class of special functions are considered from a numerical point of view. It is well known that the evaluation of the coefficients of the asymptotic series near the transition point is a difficult problem. We discuss two methods for computing the asymptotic series. One method is based on expanding the coefficients of the asymptotic series in Maclaurin series. In the second method we consider auxiliary functions that can be computed more efficiently than the coefficients in the first method, and we do not need the tabulation of many coefficients. The methods are quite general, but the paper concentrates on Bessel functions, in particular on the differential equation of the Bessel functions, which has a turning point character when order and argument of the Bessel functions are equal.     

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 for Approximate Eigenvalues of Integral Operators with Nonsmooth Kernels

We consider approximation of eigenvalues of integral operators with Green's function kernels using the Nyström method and the iterated collocation method and obtain asymptotic expansions for approximate eigenvalues. We show that the Richardson extrapolation is applicable to find eigenvalue approximations of higher order and illustrate our results by numerical examples.     

Superinterpolation in Highly Oscillatory Quadrature

Asymptotic expansions for oscillatory integrals typically depend on the values and derivatives of the integrand at a small number of critical points. We show that using values of the integrand at certain complex points close to the critical points can actually yield a higher asymptotic order approximation to the integral. This superinterpolation property has interesting ramifications for numerical methods based on exploiting asymptotic behaviour. The asymptotic convergence rates of Filon-type methods can be doubled at no additional cost. Numerical steepest descent methods already exhibit this high asymptotic order, but their analyticity requirements can be significantly relaxed. The method can be applied to general oscillators with stationary points as well, through a simple change of variables.     

Algorithms for the computation of Hankel functions of complex order

Hankel functions of complex order and real argument arise in the study of wave propagation and many other applications. Hankel functions are computed using, for example, Chebyshev expansions, recursion relations and numerical integration of the integral representation. In practice, approximation of these functions is required when the order and the argumentz are large.When andz are large, the Chebyshev series expansion of the Hankel function is of limited use. The situation is remedied by the use of appropriate asymptotic expansions. These expansions are normally expressed in terms of coefficients which are defined recursively involving derivatives and integrals of polynomials. The applicability of these expansions in both numerical and symbolic software is discussed with illustrative examples.     

Mellin-type differential equations and associated sampling expansions

This paper is devoted to a self-contained approach to Mellin-type differential equations and associated ssampling expansions. Here the first order differential operator is not the normal d/dx but D_M,c=xd/dx+c,c E R being connected with the definition of the Mellin transform. Existence and uniqueness theorems are established for a system of first order Mellin equations and the properties of nth order linear equations are investigated. Then self adjoint Mellin-type second order Sturm-Liouville eigenvalue problems are considered and properties of the eigenvalues, eigenfunctions and Green's functions are derived. As applications. sampling representations for two classes of integral transforms arising from the eigenvalue problem are introduced. In the first class the kernesl are solutions of the problem and in the second they are expressed in terms of green's function.