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Operational Calculus

A Theory of Hyperfunctions

  • Book
  • © 1984

Overview

Authors:
  1. K. Yosida

    1. Kamakura 247, Japan

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Part of the book series: Applied Mathematical Sciences (AMS, volume 55)

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Table of contents (11 chapters)

  1. Front Matter

    Pages i-x
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  2. Integration Operator h and Differentiation Operator s (Classes of Hyperfunctions: C and CH)

    1. Introduction of the Operator h Through the Convolution Ring C

      • K. Yosida
      Pages 1-4

    2. Introduction of the Operator s Through the Ring CH

      • K. Yosida
      Pages 5-13

    3. Linear Ordinary Differential Equations with Constant Coefficients

      • K. Yosida
      Pages 14-31

    4. Fractional Powers of Hyperfunctions h, s and \( \frac{I}{{S - \alpha }} \)

      • K. Yosida
      Pages 32-38

    5. Hyperfunctions Represented by Infinite Power Series in h

      • K. Yosida
      Pages 39-46

  3. Linear Ordinary Differential Equations with Linear Coefficients (The Class C/C of Hyperfunctions)

    1. The Titchmarsh Convolution Theorem and the Class C/C

      • K. Yosida
      Pages 47-52

    2. The Algebraic Derivative Applied to Laplace’s Differential Equation

      • K. Yosida
      Pages 53-73

  4. Shift Operator exp(−λs) and Diffusion Operator exp(−λs1/2)

    1. Exponential Hyperfunctions exp(−λs) and exp(−λs1/2)

      • K. Yosida
      Pages 74-105

  5. Applications to Partial Differential Equations

    1. Front Matter

      Pages 106-107
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    2. One Dimensional Wave Equation

      • K. Yosida
      Pages 108-123

    3. Telegraph Equation

      • K. Yosida
      Pages 124-144

    4. Heat Equation

      • K. Yosida
      Pages 145-156

  6. Back Matter

    Pages 157-171
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Keywords

About this book

In the end of the last century, Oliver Heaviside inaugurated an operational calculus in connection with his researches in electromagnetic theory. In his operational calculus, the operator of differentiation was denoted by the symbol "p". The explanation of this operator p as given by him was difficult to understand and to use, and the range of the valid­ ity of his calculus remains unclear still now, although it was widely noticed that his calculus gives correct results in general. In the 1930s, Gustav Doetsch and many other mathematicians began to strive for the mathematical foundation of Heaviside's operational calculus by virtue of the Laplace transform -pt e f(t)dt. ( However, the use of such integrals naturally confronts restrictions con­ cerning the growth behavior of the numerical function f(t) as t ~ ~. At about the midcentury, Jan Mikusinski invented the theory of con­ volution quotients, based upon the Titchmarsh convolution theorem: If f(t) and get) are continuous functions defined on [O,~) such that the convolution f~ f(t-u)g(u)du =0, then either f(t) =0 or get) =0 must hold. The convolution quotients include the operator of differentiation "s" and related operators. Mikusinski's operational calculus gives a satisfactory basis of Heaviside's operational calculus; it can be applied successfully to linear ordinary differential equations with constant coefficients as well as to the telegraph equation which includes both the wave and heat equa­ tions with constant coefficients.

Authors and Affiliations

  • Kamakura 247, Japan

    K. Yosida

Bibliographic Information

  • Book Title: Operational Calculus

  • Book Subtitle: A Theory of Hyperfunctions

  • Authors: K. Yosida

  • Series Title: Applied Mathematical Sciences

  • DOI: https://doi.org/10.1007/978-1-4612-1118-1

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Science+Business Media New York 1984

  • Softcover ISBN: 978-0-387-96047-0Published: 30 July 1984

  • eBook ISBN: 978-1-4612-1118-1Published: 06 December 2012

  • Series ISSN: 0066-5452

  • Series E-ISSN: 2196-968X

  • Edition Number: 1

  • Number of Pages: X, 170

  • Topics: Analysis, Mathematics, general

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