Asymptotic Expansions of Integrals (Dover Books on Mathematics) Revised Edition
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From the Back Cover
Unabridged, corrected Dover (1986) republication of the edition published by Holt, Rinehart and Winston, New York, 1975.
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The details: Bleistein & Handelsman’s Asymptotic Expansions of Integrals weighs in at a hefty ~425 pages of typical advanced mathematics texts for non-mathematicians, neither heavy on proofs nor heavy on examples. I received a 2nd printing of the book, and the 1st Dover edition (1986). The authors say they have made many corrections since their original 1975 edition. As is typical of these older math offerings, though not universal, the graphics are sub par when compared with modern standards.
The authors assume you are already quite familiar with contour integration, wave equations, Fourier transforms, Laplace transforms, integration by parts, Bessel functions, and the like. An understanding of Mellin transforms would also be helpful.
The book covers background material extensively, and then proceeds to do a survey of kernel types and their asymptotic expansions. The kernels are divided into the general categories: monotonic, nonmonotonic, and oscillatory. These are covered as chapters. Then steepest descent methods, multiple integrals, and issues of uniformity are covered.
On a personal note, I can’t shake the feeling that the authors were working toward some over arching goal or structure about which they have not informed the reader; hence my sense that the book drifts somewhat. Had I been able to discern this putative structure, I might have been substantially more edified.
The book is generally not a bad read, but this is a case of damning with faint praise.
Top international reviews
Of course this is still a very useful book.