Asymptotic Approximation Of Integrals Research Articles

Late-order terms of second order ODEs in terms of pre-factors

Factorial over a power approach is one of the fundamental techniques for deriving the late-order terms in the asymptotic approximation of integrals and differential equations. To our knowledge, although many differential equations depending on small or large parameters are addressed thoroughly and intensively by this approach in the literature to date, no explicit formula of the general representation of singularly-perturbed second order inhomogeneous ODEs in the form of this paper has yet been discussed generally in terms of their pre-factors. In this paper, we obtain a leading order asymptotic formula of the general asymptotic expansions suitable for the particular type of ODE by its pre-factors.

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.

An Explicit Formula for the Coefficients of the Saddle Point Method

Most standard textbooks about asymptotic approximations of integrals do not give explicit formulas for the coefficients of the asymptotic methods of Laplace and saddle point. In these techniques, those coefficients arise as the Taylor coefficients of a function defined in an implicit form, and the coefficients are not given by a closed algebraic formula. Despite this fact, we can extract from the literature some formulas of varying degrees of explicitness for those coefficients: Perron’s method (in Sitzungsber. Bayr. Akad. Wissensch. (Munch. Ber.), 191–219, 1917) offers an explicit computation in terms of the derivatives of an explicit function; in (de Bruijn, Asymptotic Methods in Analysis. Dover, New York, 1950) we can find a similar formula for the Laplace method which uses derivatives of an explicit function. Dingle (in Asymptotic Expansions: Their Derivation and Interpretation, Academic Press, New York, 1973) gives the coefficients of the saddle point method in terms of a contour integral. Perron’s method is rediscovered in (Campbell et al., Stud. Appl. Math. 77:151–172, 1987), but they also go farther and compute the above mentioned derivatives by means of a recurrence. The most recent contribution is (Wojdylo, SIAM Rev. 48(1):76–96, 2006), which rediscovers the Campbell, Froman and Walles’ formula and rewrites it in terms of Bell polynomials (in the Laplace method) using new ideas of combinatorial analysis which efficiently simplify and systematize the computations. In this paper we continue the research line of these authors. We combine the more systematic version of the saddle point method introduced in (Lopez et al., J. Math. Anal. Appl. 354(1):347–359, 2009) with Wojdylo’s idea to derive a new and more explicit formula for the coefficients of the saddle point method, similar to Wojdylo’s formula for the coefficients of the Laplace method. As an example, we show the application of this formula to the Bessel function.

Asymptotic expansions for a class of q-integral transforms

Using the q-Mellin transform studied in [A. Fitouhi, N. Bettaibi, K. Brahim, The Mellin transform in quantum calculus, Constructive Approximation 23 (3) (2006) 305–323], we provide an asymptotic expansion for a class of q-integral transforms having the form ∫ 0 ∞ f ( t ) h ( xt ) d q t . In particular, we are interested with the q-Laplace, q-Fourier-cosine and q-Hankel transforms. We adopt the method of Handelsman and Lew [R. Wong, Asymptotic Approximations of integrals, Computer Science and Scientific Computing, Academic Press, New York, 1989].

Asymptotic expansions of Mellin convolutions by means of analytic continuation

We present a method for deriving asymptotic expansions of integrals of the form ∫ 0 ∞ f ( t ) h ( xt ) d t for small x based on analytic continuation. The expansion is given in terms of two asymptotic sequences, the coefficients of both sequences being Mellin transforms of h and f . Many known and unknown asymptotic expansions of important integral transforms are derived trivially from the approach presented here. This paper reconsiders earlier work of McClure and Wong [Explicit error terms for asymptotic expansions of Stieltjes transforms, J. Inst. Math. Appl. 22 (1978) 129–145; Exact remainders for asymptotic expansions of fractional integrals, J. Inst. Math. Appl. 24 (1979) 139–147] and Asymptotic approximations of integrals, Academic Press, New York, 1989. Chaps. 5, where elements of distribution theory are used, and Wong [Explicit error terms for asymptotic expansions of Mellin convolutions, J. Math. Anal. Appl. 72(2) (1979) 740–756], where, as in the present paper, the asymptotic expansions are obtained without the use of distributions. In this paper we re-derive the expansions given in Wong [Explicit error terms for asymptotic expansions of Mellin convolutions, J. Math. Anal. Appl. 72(2) (1979) 740–756] by using a different approach and we obtain new results which are not present in Wong [Explicit error terms for asymptotic expansions of Mellin convolutions, J. Math. Anal. Appl. 72(2) (1979) 740–756]: a proof of the asymptotic character of the expansions and accurate error bounds.

Computing the Coefficients in Laplace's Method

Laplace's method is a preeminent technique in the asymptotic approximation of integrals. Its utility was enhanced enormously in 1956 when Erdélyi found a way to apply Watson's lemma and thereby obtain an infinite asymptotic expansion valid, in principle, for any integral of Laplace type. Erdélyi's formulation requires tedious computation of coefficients $c_{s}$ for each specific application of the method, and traditionally this has involved reverting a series. Recently, it was shown that the coefficients $c_{s}$ can be computed via a simple, explicit expression that is probably computationally optimal, which avoids the reversion approach altogether. The formula is made possible by recognizing the central role of Faà di Bruno's formula, alongside Watson's lemma, in Erdélyi's formulation of Laplace's classical method. Laplace's method can now be implemented cleanly and relatively quickly, provided one has the luck and the patience to get to the point where implementation becomes automatic. The present paper outlines the recent discovery of the role of Faà di Bruno's formula in Laplace's method, gives examples of the application of the explicit expression for the coefficients $c_{s},$ and provides grounds for a possible generalization of the result.

On Some Nonstandard Asymptotic Approximation of Integrals and Series

The aim of this paper is to use some concepts of nonstandard analysis given by Robinson, A. and axiomataized by Nelson, E. to prove some theorems concerning the approximation of integrals and the convergence of sequences and series.

On the coefficients that arise from Laplace's method

Laplace's method is one of the best-known techniques in the asymptotic approximation of integrals. The salient step in the technique's historical development was Erdélyi's use of Watson's Lemma to obtain an infinite asymptotic expansion valid for any Laplace-type integral, published in 1956. Erdélyi's expansion contains coefficients c s that must be calculated in each application of Laplace's method, a tedious process that has traditionally involved the reversion of a series. This paper shows that the coefficients c s in fact have a very simple general form. In effect, we extend Erdélyi's theorem. Our results greatly simplify calculation of the c s in any particular application and clarify the theoretical basis of Erdélyi's expansion: it turns out that Faà di Bruno's formula has always played a central role in it. We prove or derive the following: • The correct dimensionless groups. Erdélyi's expansion is properly expressed in terms of scaled coefficients c s * . • Two explicit expressions for c s * in terms of combinatorial objects called partial ordinary Bell polynomials. This form is probably computationally optimal and makes checking for correctness a relatively straightforward process. • A recursive expression for c s * . • Each coefficient c s * can be expressed as a polynomial in ( α + s ) / μ , where α and μ are quantities in Erdélyi's formulation. The main insight that emerges is that the traditional approach to Laplace's method, involving reversion of a series, is less efficient and need only be invoked if one is interested in the role of the reversion coefficients in Erdélyi's expansion—a point which Erdélyi himself alluded to. We consider as an example an integral that occurs in a variational approach to finding the binding energy of helium dimers. We also present a three-line computer code to generate the coefficients c s * exactly in the general case. In a sequel paper (to be published in SIAM Review), a new representation for the gamma function is obtained, and the link with Faà di Bruno's formula is explained.

Book Review: Asymptotic approximations of integrals

Book Review: Asymptotic approximations of integrals

Book Review: Asymptotic approximations of integrals

Book Review: Asymptotic approximations of integrals

Asymptotic Approximation of Integrals.

Asymptotic Approximation of Integrals.

Asymptotic approximations for surface scattering integrals

Asymptotic approximations of integrals with the saddle point method depend on the power series and their inversions utilized. This dependence is evident in the second and subsequent terms in the asymptotic series. Modifications for double integrals and poles are also indicated.