Access this book
Tax calculation will be finalised at checkout
Other ways to access
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.
Similar content being viewed by others
Table of contents (11 chapters)
-
Front Matter
-
Integration Operator h and Differentiation Operator s (Classes of Hyperfunctions: C and CH)
-
Linear Ordinary Differential Equations with Linear Coefficients (The Class C/C of Hyperfunctions)
-
Shift Operator exp(−λs) and Diffusion Operator exp(−λs1/2)
-
Applications to Partial Differential Equations
-
Front Matter
-
-
Back Matter
Authors and Affiliations
Accessibility Information
PDF accessibility summary
This PDF is not accessible. It is based on scanned pages and does not support features such as screen reader compatibility or described non-text content (images, graphs etc). However, it likely supports searchable and selectable text based on OCR (Optical Character Recognition). Users with accessibility needs may not be able to use this content effectively. Please contact us at accessibilitysupport@springernature.com if you require assistance or an alternative format.
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