# Applied Hyperfunction Theory

Part of the Mathematics and Its Applications (Japanese Series) book series (MAJA, volume 8)

Advertisement

Part of the Mathematics and Its Applications (Japanese Series) book series (MAJA, volume 8)

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.

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.

Fourier series analytic function differential equation distribution generalized function integral integral equation integration special function transform theory

- DOI https://doi.org/10.1007/978-94-011-2548-2
- Copyright Information Kluwer Academic Publishers 1992
- Publisher Name Springer, Dordrecht
- eBook Packages Springer Book Archive
- Print ISBN 978-94-010-5125-5
- Online ISBN 978-94-011-2548-2
- Series Print ISSN 0924-4913
- Buy this book on publisher's site