# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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 ${e}^{x}cos\left({e}^{x}\right)$ is a tempered distribution. (ii) Construct an asymptotic expansion for the integral ${\int }_{0}^{\infty }{t}^{n}{\left(logt\right)}^{m}\left({e}^{-t}/t+x\right)dt$ as $x\to {0}^{+}$, where n and m are nonnegative integers. (iii) Show that, if A is a real, symmetric and positive definite matrix, ${\int }_{{ℝ}^{n}}exp\left({x}^{T}Ax\right)dx={\pi }^{n/2}/{\left(detA\right)}^{1/2}·$ 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.

##### MSC:
 41-01 Textbooks (approximations and expansions) 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.) 44A15 Special transforms (Legendre, Hilbert, etc.)