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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.

Inhaltsverzeichnis

Frontmatter

Integration Operator h and Differentiation Operator s (Classes of Hyperfunctions: C and CH)

Chapter I. Introduction of the Operator h Through the Convolution Ring C

Abstract
The totality of complex-valued continuous functions a(t), b(t), f(t) and so forth defined on the interval [0,∞) will play a particularly important role in the operational calculus; we shall denote the class of those functions by C[0,∞) or simply by the letter C. The convolution of two functions a = a(t) and b = b(t) of C is defined by
$$ (a*b)(t) = a * b(t) = \int_{0}^{t} {a(t - u)b(u)du} (0 \mathbin{\lower.3ex\hbox{$\buildrel<\over {\smash{\scriptstyle=}\vphantom{_x}}$}} t < \infty ), $$
(1.1)
and we have PROPOSITION 1. a*b belongs to C; i.e., a*b(t) is a continuous function defined on [0, ∞).
K. Yosida

Chapter II. Introduction of the Operator s Through the Ring CH

Abstract
Let
$$H = \{ k:k = {{h}^{n}}(n = 1,2,...)\}$$
. then,for any k = hn∈H and f ∈c,we have
$$ kf = 0{\text{ }}implies f = 0, where 0 = \{ 0\} \in c $$
(3.1)
.
K. Yosida

Chapter III. Linear Ordinary Differential Equations with Constant Coefficients

Abstract
Let α01,...αn be complex numbers (αn≠ 0) and f ∈ C = C[0,∞). Consider the equation
$${{\alpha }_{n}}{{y}^{{\left( n \right)}}} + {{\alpha }_{{n - 1}}}{{y}^{{\left( {n - 1} \right)}}} + \cdot \cdot \cdot + {{\alpha }_{1}}{{y}^{1}} + {{\alpha }_{0}}y = f$$
(7.1)
together with the initial condition at t = 0:
$$ y\left( 0 \right) = {\gamma _0}, y'\left( 0 \right) = {\gamma _1},...,{y^{\left( {n - 1} \right)}}\left( 0 \right) = {\gamma _{n - 1}}. $$
(7.2)
K. Yosida

Chapter IV. Fractional Powers of Hyperfunctions h, s and

Abstract
These functions are respectively defined by Euler’s* integrals:
$$\Gamma \left( \lambda \right) = \int_{0}^{\infty } {{{t}^{{\lambda - 1}}}{{e}^{{ - t}}}dt(Re\lambda > 0)**,}$$
(1)
$$ B(\lambda ,\mu ) = {\text{ }}\int_0^1 {{t^{\lambda - 1}}{{(1 - t)}^{\mu - 1}}dt (Re \lambda > 0, Re \mu > 0).} $$
(12.2)
K. Yosida

Chapter V. Hyperfunctions Represented by Infinite Power Series in h

Abstract
THE DEFINITION OF (1+z)α. Let α be any complex number. For any complex number z with |z| < 1, we define
$$ \left\{ {\begin{array}{*{20}{c}} {{{\left( {1 + z} \right)}^\alpha } = {e^{\alpha \log \left( {1 + z} \right)}},where we take the branch of the} \\ {function log\left( {1 + z} \right)determined by log\left( {1 + 0} \right) = 0.} \end{array}} \right. $$
(14.1)
K. Yosida

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

Chapter VI. The Titchmarsh Convolution Theorem and the Class C/C

Abstract
THEOREM 16. (THE TITCHMARSH CONVOLUTION THEOREM). If the convolution fg of two functions f and g ∈ C is 0, then either f = 0 or g = 0 must be true.
K. Yosida

Chapter VII. The Algebraic Derivative Applied to Laplace’s Differential Equation

Abstract
Pierre Simon Laplace (1749–1827) in his treatise “Théorie analytique des probabilités” of 1817 considered a differential equation which now carries his name and which may be written as
$$ \left( {{a_2}t + {b_2}} \right)y''\left( t \right) + \left( {{a_1}t + {b_1}} \right)y'(t) + \left( {{a_0}t + b} \right)y\left( t \right) = 0, $$
(19.1)
where the a’s and b’s are given complex numbers with a2≠ 0.
K. Yosida

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

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

Without Abstract
K. Yosida

Applications to Partial Differential Equations

Frontmatter

Chapter IX. One Dimensional Wave Equation

Without Abstract
K. Yosida

Chapter X. Telegraph Equation

Abstract
We shall discuss the voltage and the current in a long cable consisting of two parallel wires stretched along the λ-axis. Let U(λ,t) and I(λ,t) respectively be the voltage and the current at the point of the cable with coordinate λ at the instant t. Then the following relations hold between the functions U and I:
$$ {U_\lambda } = - L{I_t} - RI, {I_\lambda } = - C{U_t} - GU, $$
(33.1)
where R denotes resistance, L self-inductance, G leak-conductance and C capacitance; these quantities are measured per unit length of the cable.
K. Yosida

Chapter XI. Heat Equation

Abstract
Let us imagine that a bar of length λ0 is placed along the λ-axis, the abcissa of the left end of the bar being λ = 0 and the right end λ = λ0. Let k denote the heat conductivity, c the specific heat, and ρ the mass density of the bar. Furthermore, let the lateral surface of the bar be perfectly insulated so that heat can flow in and flow out only through the ends of the bar. If we denote by z(λ,t) the temperature at the point of the bar at abcissa λ at the instant t, then the heat equation in the bar is
$$ {z_{\lambda \lambda }}\left( {\lambda ,t} \right) = {\alpha ^2}{z_t}\left( {\lambda ,t} \right) \left( {\alpha = \sqrt {c\rho /k} } \right). $$
(41.1)
K. Yosida

Backmatter

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