Operational Calculus

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.

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.- 1. Convolution Ring.- 2. Operator of Integration h.- II. Introduction of the Operator s Through the Ring CH.- 3. The Ring CH and the Identity Operator I = h/h.- 4. CH as a Class of Generalized Functions of Hyperfunctions.- 5. Operator of Differentiation s and Operator of Scalar Multiplication [?].- 6. The Theorem $$\frac{I}{{s - [\alpha ]}} = {e^{\alpha t}}$$.- III. Linear Ordinary Differential Equations with Constant Coefficients.- 7. The Conversion of the Initial Value Problem for the Differential Equation into a Hyperfunction Equation.- 8. The Polynomial Ring of Polynomials in s has no Zero Factors.- 9. The Partial Fraction Decomposition of a Rational Function of s.- 10. Hyperfunction Solution of the Ordinary Differential Equation (The Operational Calculus).- 11. Boundary Value Problems for Ordinary Differential Equations.- IV. Fractional Powers of Hyperfunctions h, s and $$\frac{I}{{S - \alpha }}$$.- 12. Euler’s Integrals — The Gamma Function and Beta Function.- 13. Fractional Powers of h, of (s-?)?1, and of (s-?).- V. Hyperfunctions Represented by Infinite Power Series in h.- 14. The Binomial Theorem.- 15. Bessel’s Function Jn(t).- 16. Hyperfunctions Represented by 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.- 17. Proof of the Titchmarsh Convolution Theorem.- 18. The Class C/C of Hyperfunctions.- VII. The Algebraic Derivative Applied to Laplace’s Differential Equation.- 19. The Algebraic Derivative.- 20. Laplace’s Differential Equation.- 21. Supplements. I: Weierstrass’ Polynomial Approximation Theorem. II: Mikusi?ski’s Theorem of Moments.- III. Shift Operator exp(??s) and Diffusion Operator exp(??s1/2).- VIII. Exponential Hyperfunctions exp(??s) and exp(??s1/2).- 22. Shift Operator exp(??s) = e??sFunction Space K = K[0,?).- 23 Hyperfunction-Valued Function f(?) and Generalized Derivative $$\frac{d}{{d\lambda }}f\left( \lambda \right) = f'\left( \lambda \right)$$.- 24. Exponential Hyperfunction exp(?s)=e?s.- 25. Examples of Generalized Limit. Power Series in e?s.- $$\int_{0}^{\infty } {{{e}^{{ - \lambda s}}}} f\left( \lambda \right)d\lambda = \left\{ {f\left( t \right)} \right\}For\left\{ {f\left( t \right)} \right\} \in C$$.- 27. Logarithmic Hyperfunction w and Exponential Hyperfunction exp $$ \left( { - \lambda {{s}^{{1/2}}}} \right) = {{e}^{{ - \lambda {{s}^{{{{1} \left/ {2} \right.}}}}}}} $$.- 28. Logarithmic Hyperfunction w and Exponential Hyperfunction exp(?w).- IV. Applications to Partial Differential Equations.- IX. One DimensionaL Wave Equation.- 29. Hyperfunction Equation of the form f?(?) — w2f(?) = g(?), w ? C/C.- 30. The Vibration of a String.- 31. D’Alembert’s Method.- 32. The Vibration of an Infinitely Long String.- X. Telegraph Equation.- 33. The Hyperfunction Equation of the Telegraph Equation.- 34. A Cable With Infinitely Small Loss.- X. (cont.).- 35. Conductance Without Deformation.- 36. The Thomson Cable.- 37. Concrete Representations of exp $$\left( { - \lambda \sqrt {\alpha s + \beta } } \right) $$.- 38. A Cable without Self-Induction.- 39. A Cable without Leak-Conductance.- 40. The Case Where All the Four Parameters Are Positive.- Positive.- XI. Heat Equation.- 41. The Temperature of a Heat-Conducting Bar.- 42. An Infinitely Long Bar.- 43. A Bar Without an Outgoing Flow of Heat.- 44. The Temperature in a Bar with a Given Initial Temperature.- 45. A Heat-Conducting Ring.- 46. Non-Insulated Heat Conduction.- Answers to Exercises.- Formulas and Tables.- References.- Propositions and Theorems in Sections.