Even though the theories of operational calculus and integral transforms are centuries old, these topics are constantly developing, due to their use in the fields of mathematics, physics, and electrical and radio engineering. Operational Calculus and Related Topics highlights the classical methods and applications as well as the recent advances in the field.
Combining the best features of a textbook and a monograph, this volume presents an introduction to operational calculus, integral transforms, and generalized functions, the backbones of pure and applied mathematics. The text examines both the analytical and algebraic aspects of operational calculus and includes a comprehensive survey of classical results while stressing new developments in the field. Among the historical methods considered are Oliver Heaviside’s algebraic operational calculus and Paul Dirac’s delta function. Other discussions deal with the conditions for the existence of integral transforms, Jan Mikusiński’s theory of convolution quotients, operator functions, and the sequential approach to the theory of generalized functions.
· Discusses theory and applications of integral transforms
· Gives inversion, complex-inversion, and Dirac’s delta distribution formulas, among others
· Offers a short survey of actual results of finite integral transforms, in particular convolution theorems
Because Operational Calculus and Related Topics provides examples and illustrates the applications to various disciplines, it is an ideal reference for mathematicians, physicists, scientists, engineers, and students.
“This book covers a reasonable portion of topics encountered in the theory and practice of operational calculi for functions of one real variable. It is intended and can be used both as an introductory text or a reference book for mathematicians, physicists and engineers. … chapters are packed with many examples of concrete computations of transforms of selected functions. …”
— In EMS Newsletter, June 2007
Introduction to Operational Calculus
Integral Transforms — Introductory Remarks
The Fourier Transform
The Laplace Transform
The Mellin Transform
The Stieltjes Transform
The Hilbert Transform
The Mehler–Fock Transform
Finite Integral Transforms
The Theorem of Titchmarsh
Bases of the Operator Analysis
Operators Reducible to Functions
Application of Operational Calculus
Generalized Functions — Functional Approach
Generalized Functions — Sequential Approach
Hilbert Transform and Multiplication Forms