##
**Asymptotic approximations of integrals.**
*(English)*
Zbl 0679.41001

Computer Science and Scientific Computing. Boston, MA etc.: Academic Press, Inc. xiii, 543 p. $ 69.95 (1989).

The book focuses on the classical techniques-Laplace method, Perron method, the method of steepest decents, Darboux method (Chapters I and II) as well as the recent techniques, namely, the Mellin transform, the Hankel transform, the Stieltjes transform and the Hilbert transform (Chapters III, IV and VI) in the asymptotic evaluation of integrals. Chapter V is devoted to theory of distributions. Chapter VII deals with integrals which depend on auxiliary parameters in addition to the asymptotic variable. Multidimensional integrals are studied in Chapters VIII and IX. The following exercises, among many, indicate the scope of the book: (i) show that \(e^ x \cos (e^ x)\) is a tempered distribution. (ii) Construct an asymptotic expansion for the integral \(\int^{\infty}_{0}t^ n(\log t)^ m(e^{-t}/t+x)dt\) as \(x\to 0^+\), where n and m are nonnegative integers. (iii) Show that, if A is a real, symmetric and positive definite matrix, \(\int_{{\mathbb{R}}^ n}\exp (x^ TAx)dx=\pi^{n/2}/(\det A)^{1/2}.\) The get-up is nice. The book is highly recommended for students, and researchers, whose interests impinge on asymptotic approximation of integrals, as it is expertly written. Supplementary Notes, Bibliography, Symbol Index and Subject Index are given to help the reader. The purpose of the book is to provide an up-to-date account of methods used in asymptotic approximation of integrals.

Reviewer: K.Chandrasekhara Rao

### MSC:

41-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to approximations and expansions |

41A60 | Asymptotic approximations, asymptotic expansions (steepest descent, etc.) |

44A15 | Special integral transforms (Legendre, Hilbert, etc.) |

### Keywords:

method of steepest decents; Darboux method; Mellin transform; Hankel transform; Stieltjes transform; Hilbert transform; Multidimensional integrals; asymptotic approximation of integrals
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\textit{R. Wong}, Asymptotic approximations of integrals. Boston, MA etc.: Academic Press, Inc. (1989; Zbl 0679.41001)

### Digital Library of Mathematical Functions:

§1.14(iv) Mellin Transform ‣ §1.14 Integral Transforms ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods§1.14(vii) Tables ‣ §1.14 Integral Transforms ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods

§1.15(iv) Definitions for Integrals ‣ §1.15 Summability Methods ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods

§1.16(iii) Dirac Delta Distribution ‣ §1.16 Distributions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods

§1.16(ii) Derivatives of a Distribution ‣ §1.16 Distributions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods

§1.16(i) Test Functions ‣ §1.16 Distributions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods

§1.16(iv) Heaviside Function ‣ §1.16 Distributions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods

§1.16(viii) Fourier Transforms of Special Distributions ‣ §1.16 Distributions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods

§1.16(vii) Fourier Transforms of Tempered Distributions ‣ §1.16 Distributions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods

§1.16(vi) Distributions of Several Variables ‣ §1.16 Distributions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods

§1.16(v) Tempered Distributions ‣ §1.16 Distributions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods

§14.26 Uniform Asymptotic Expansions ‣ Complex Arguments ‣ Chapter 14 Legendre and Related Functions

Chapter 14 Legendre and Related Functions

Example ‣ §2.10(iv) Taylor and Laurent Coefficients: Darboux’s Method ‣ §2.10 Sums and Sequences ‣ Areas ‣ Chapter 2 Asymptotic Approximations

§2.3(ii) Watson’s Lemma ‣ §2.3 Integrals of a Real Variable ‣ Areas ‣ Chapter 2 Asymptotic Approximations

§2.3(iii) Laplace’s Method ‣ §2.3 Integrals of a Real Variable ‣ Areas ‣ Chapter 2 Asymptotic Approximations

§2.3(i) Integration by Parts ‣ §2.3 Integrals of a Real Variable ‣ Areas ‣ Chapter 2 Asymptotic Approximations

§2.3(v) Coalescing Peak and Endpoint: Bleistein’s Method ‣ §2.3 Integrals of a Real Variable ‣ Areas ‣ Chapter 2 Asymptotic Approximations

§2.4(ii) Inverse Laplace Transforms ‣ §2.4 Contour Integrals ‣ Areas ‣ Chapter 2 Asymptotic Approximations

§2.4(i) Watson’s Lemma ‣ §2.4 Contour Integrals ‣ Areas ‣ Chapter 2 Asymptotic Approximations

§2.4(iv) Saddle Points ‣ §2.4 Contour Integrals ‣ Areas ‣ Chapter 2 Asymptotic Approximations

§2.4(vi) Other Coalescing Critical Points ‣ §2.4 Contour Integrals ‣ Areas ‣ Chapter 2 Asymptotic Approximations

§2.4(v) Coalescing Saddle Points: Chester, Friedman, and Ursell’s Method ‣ §2.4 Contour Integrals ‣ Areas ‣ Chapter 2 Asymptotic Approximations

§2.5(iii) Laplace Transforms with Small Parameters ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations

§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations

§2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations

§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations

§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations

Example ‣ §2.5(iii) Laplace Transforms with Small Parameters ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations

§2.6(iii) Fractional Integrals ‣ §2.6 Distributional Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations

§2.6(ii) Stieltjes Transform ‣ §2.6 Distributional Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations

§2.6(i) Divergent Integrals ‣ §2.6 Distributional Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations

§2.6(ii) Stieltjes Transform ‣ §2.6 Distributional Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations

§2.6(i) Divergent Integrals ‣ §2.6 Distributional Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations

§2.6(iv) Regularization ‣ §2.6 Distributional Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations

§2.6(iv) Regularization ‣ §2.6 Distributional Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations

Example ‣ §2.6(iii) Fractional Integrals ‣ §2.6 Distributional Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations

Chapter 2 Asymptotic Approximations

§7.20(i) Asymptotics ‣ §7.20 Mathematical Applications ‣ Applications ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals

Chapter 8 Incomplete Gamma and Related Functions

Chapter 9 Airy and Related Functions

Profile Roderick S. C. Wong ‣ About the Project