Applied Mathematical Sciences

# Operational Calculus

## A Theory of Hyperfunctions

Authors: Yosida, Kôsaku

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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.

• Introduction of the Operator h Through the Convolution Ring C

Pages 1-4

Yosida, K.

• Introduction of the Operator s Through the Ring CH

Pages 5-13

Yosida, K.

• Linear Ordinary Differential Equations with Constant Coefficients

Pages 14-31

Yosida, K.

• Fractional Powers of Hyperfunctions h, s and $\frac{I}{{S - \alpha }}$

Pages 32-38

Yosida, K.

• Hyperfunctions Represented by Infinite Power Series in h

Pages 39-46

Yosida, K.

eBook $84.99 price for USA in USD • ISBN 978-1-4612-1118-1 • Digitally watermarked, DRM-free • Included format: PDF • ebooks can be used on all reading devices • Immediate eBook download after purchase Softcover$109.99
price for USA in USD ## Recommended for you ## Bibliographic Information

Bibliographic Information
Book Title
Operational Calculus
Book Subtitle
A Theory of Hyperfunctions
Authors
Series Title
Applied Mathematical Sciences
Series Volume
55
1984
Publisher
Springer-Verlag New York