Distributional Derivation of an Asymptotic Expansion

Wong, R.

doi:10.2307/2042959

Proceedings of the American Mathematical Society

1980

DOI: 10.2307/2042959

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Distributional Derivation of an Asymptotic Expansion

R. Wong¹

Abstract: Abstract. An alternative derivation of the asymptotic expansion of multiple Fourier transforms is presented. The present approach is based on the use of distributions. With some modifications, this method can also be applied to other integral transforms with oscillatory kernels such as the Hankel transform.

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1982

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Cited by 3 publications

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Finite-part integration of the generalized Stieltjes transform and its dominant asymptotic behavior for small values of the parameter. I. Integer orders

Tica

Galapon

2018

Journal of Mathematical Physics

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The paper addresses the exact evaluation of the generalized Stieltjes transform Sn[f ] = ∞ 0 f (x)(ω + x) −n dx of integral order n = 1, 2, 3, . . . about ω = 0 from which the asymptotic behavior of Sn[f ] for small parameters ω is directly extracted. An attempt to evaluate the integral by expanding the integrand (ω + x) −n about ω = 0 and then naively integrating the resulting infinite series term by term lead to an infinite series whose terms are divergent integrals. Assigning values to the divergent integrals, say, by analytic continuation or by Hadamard's finite part is known to reproduce only some of the correct terms of the expansion but completely misses out a group of terms. Here we evaluate explicitly the generalized Stieltjes transform by means of finite-part integration recently introduced in [E.A. Galapon, Proc. Roy. Soc. A 473, 20160567 (2017)]. It is shown that, when f (x) does not vanish or has zero of order m at the origin such that (n − m) ≥ 1, the dominant terms of Sn[f ] as ω → 0 come from contributions arising from the poles and branch points of the complex valued function f (z)(ω + z) −n . These dominant terms are precisely the terms missed out by naive term by term integration. Furthermore, it is demonstrated how finite-part integration leads to new series representations of special functions by exploiting their known Stieltjes integral representations. Finally, the application of finite part integration in obtaining asymptotic expansions of the effective diffusivity in the limit of high Peclet number, the Green-Kubo formula for the self-diffusion coefficient and the antisymmetric part of the diffusion tensor in the weak noise limit is discussed.

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Finite-part integration of the generalized Stieltjes transform and its dominant asymptotic behavior for small values of the parameter. I. Integer orders

Tica

Galapon

2018

Journal of Mathematical Physics

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Asymptotic expansions of the Appell’s function 𝐹₁

Ferreira¹,

López²

2004

Quart. Appl. Math.

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The first Appell’s hypergeometric function F 1 ( a , b , c , d ; x , y ) {F_1}\left ( a, b, c, d; x, y \right ) is considered for large values of its variables x x and/or y y . An integral representation of F 1 ( a , b , c , d ; x , y ) {F_1}\left ( a, b, c, d; x, y \right ) is obtained in the form of a generalized Stieltjes transform. Distributional approach is applied to this integral to derive four asymptotic expansions of this function in increasing powers of 1 / ( 1 − x ) 1/\left ( 1 - x \right ) and/or 1 / ( 1 − y ) 1/\left ( 1 - y \right ) . For certain values of the parameters a , b , c a, b, c and d d , two of these expansions also involve logarithmic terms in the asymptotic variables 1 − x 1 - x and/or 1 − y 1 - y . Coefficients of these expansions are given in terms of the Gauss hypergeometric function 2 F 1 ( α , β , γ ; x ) _2{F_1}\left ( \alpha , \beta , \gamma ; x \right ) and its derivative with respect to the parameter α \alpha . All of the expansions are accompanied by error bounds for the remainder at any order of the approximation. These error bounds are obtained from the error test and, as numerical experiments show, they are considerably accurate.

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Asymptotic expansions of some integral transforms by using generalized functions

Zayed¹

1982

Trans. Amer. Math. Soc.

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ABSTRACT. The technique devised by Wong to derive the asymptotic expansions of multiple Fourier transforms by using the theory of Schwartz distributions is extended to a large class of integral transforms. The extension requires establishing a general procedure to extend these integral transforms to generalized functions. Wong's technique is then applied to some of these integral transforms to obtain their asymptotic expansions. This class of integral transforms encompasses, among others, the Laplace, the Airy, the K and the Hankel transforms.

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Distributional Derivation of an Asymptotic Expansion

Cited by 3 publications

References 2 publications

Finite-part integration of the generalized Stieltjes transform and its dominant asymptotic behavior for small values of the parameter. I. Integer orders

Finite-part integration of the generalized Stieltjes transform and its dominant asymptotic behavior for small values of the parameter. I. Integer orders

Asymptotic expansions of the Appell’s function 𝐹₁

Asymptotic expansions of some integral transforms by using generalized functions

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