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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(tu)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.
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Bibliographic Information
 Bibliographic Information

 Book Title
 Operational Calculus
 Book Subtitle
 A Theory of Hyperfunctions
 Authors

 Kôsaku Yosida
 Series Title
 Applied Mathematical Sciences
 Series Volume
 55
 Copyright
 1984
 Publisher
 SpringerVerlag New York
 Copyright Holder
 Springer Science+Business Media New York
 eBook ISBN
 9781461211181
 DOI
 10.1007/9781461211181
 Softcover ISBN
 9780387960470
 Series ISSN
 00665452
 Edition Number
 1
 Number of Pages
 X, 170
 Topics