Operational Calculus : A Theory of Hyperfunctions 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 Heavisides 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. Mikusinskis operational calculus gives a satisfactory basis of Heavisides 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. I. Integration Operator h and Differentiation Operator s (Classes of Hyperfunctions: C and CH) I. Introduction of the Operator h Through the Convolution Ring C II. Introduction of the Operator s Through the Ring CH III. Linear Ordinary Differential Equations with Constant Coefficients IV. Fractional Powers of Hyperfunctions h s and $$\\frac{I}{{S \\alpha }}$$ V. Hyperfunctions Represented by Infinite Power Series in h II. Linear Ordinary Differential Equations with Linear Coefficients (The Class C C of Hyperfunctions) VI. The Titchmarsh Convolution Theorem and the Class C C VII. The Algebraic Derivative Applied to Laplace’s Differential Equation III. Shift Operator exp(s) and Diffusion Operator exp(s1 2) VIII. Exponential Hyperfunctions exp(s) and exp(s1 2) IV. Applications to Partial Differential Equations IX. One DimensionaL Wave Equation X. Telegraph Equation X. (cont.) XI. Heat Equation Answers to Exercises Formulas and Tables References Propositions and Theorems in Sections.