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Applied Hyperfunction Theory : Mathematics and its Applications - Isao Imai

Applied Hyperfunction Theory

Mathematics and its Applications

By: Isao Imai

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Generalized functions are now widely recognized as important mathematical tools for engineers and physicists. But they are considered to be inaccessible for non-specialists. To remedy this situation, this book gives an intelligible exposition of generalized functions based on Sato's hyperfunction, which is essentially the 'boundary value of analytic functions'. An intuitive image -- hyperfunction = vortex layer -- is adopted, and only an elementary knowledge of complex function theory is assumed. The treatment is entirely self-contained. The first part of the book gives a detailed account of fundamental operations such as the four arithmetical operations applicable to hyperfunctions, namely differentiation, integration, and convolution, as well as Fourier transform. Fourier series are seen to be nothing but periodic hyperfunctions. In the second part, based on the general theory, the Hilbert transform and Poisson-Schwarz integral formula are treated and their application to integral equations is studied. A great number of formulas obtained in the course of treatment are summarized as tables in the appendix. In particular, those concerning convolution, the Hilbert transform and Fourier transform contain much new material. For mathematicians, mathematical physicists and engineers whose work involves generalized functions.

Series Editor's Preface
Introductionp. 1
Operations on Hyperfunctionsp. 11
Basic Hyperfunctionsp. 25
Hyperfunctions Depending on Parametersp. 53
Fourier Transformationp. 83
Fourier Transformation of Power-Type Hyperfunctionsp. 101
Upper (Lower)-Type Hyperfunctionsp. 115
Fourier Transforms-Existence and Regularityp. 133
Fourier Transform-Asymptotic Behaviourp. 147
Periodic Hyperfunctions and Fourier Seriesp. 165
Analytic Continuation and Projection of Hyperfunctionsp. 205
Product of Hyperfunctionsp. 225
Convolution of Hyperfunctionsp. 249
Convolution of Periodic Hyperfunctionsp. 283
Hilbert Transforms and Conjugate Hyperfunctionsp. 303
Poisson-Schwarz Integral Formulaep. 327
Integral Equationsp. 357
Laplace Transformsp. 381
Epiloguep. 393
Referencesp. 395
Appendicesp. 397
Indexp. 437
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780792315070
ISBN-10: 0792315073
Series: Mathematics and its Applications
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 438
Country of Publication: US
Dimensions (cm): 23.5 x 15.5  x 2.6
Weight (kg): 0.85