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Operational Calculus and Related Topics by H.J. Glaeske, K.A. Skornik



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Operational Calculus and Related Topics by H.J. Glaeske , Friedrich-Schiller University, Jena, Germany and A.P.Prudnikov (Deceased) and K.A. Skornik , Institute of Mathematics, Polish Academy of Sciences, Katowice, Poland.

Operational Calculus and Related Topics by H.J. Glaeske, K.A. Skornik cover the following topics.

  • Preface
    List of Symbols

  • 1. Integral Transforms
    1.1 Introduction to Operational Calculus
    1.2 Integral Transforms – Introductory Remarks
    1.3 The Fourier Transform
    1.3.1 Definition and Basic Properties
    1.3.2 Examples
    1.3.3 Operational Properties
    1.3.4 The Inversion Formula
    1.3.5 Applications
    1.4 The Laplace Transform
    1.4.1 Definition and Basic Properties
    1.4.2 Examples
    1.4.3 Operational Properties
    1.4.4 The Complex Inversion Formula
    1.4.5 Inversion Methods
    1.4.6 Asymptotic Behavior
    1.4.7 Remarks on the Bilateral Laplace Transform
    1.4.8 Applications
    1.5 The Mellin Transform
    1.5.1 Definition and Basic Properties
    1.5.2 Operational Properties
    1.5.3 The Complex Inversion Formula
    1.5.4 Applications
    1.6 The Stieltjes Transform
    1.6.1 Definition and Basic Properties
    1.6.2 Operational Properties
    1.6.3 Asymptotics
    1.6.4 Inversion and Application
    1.7 The Hilbert Transform
    1.7.1 Definition and Basic Properties
    1.7.2 Operational Properties
    1.7.3 Applications
    1.8 Bessel Transforms
    1.8.1 The Hankel Transform
    1.8.2 The Meijer (K-) Transform
    1.8.3 The Kontorovich–Lebedev Transform
    1.8.4 Application
    1.9 The Mehler–Fock Transform
    1.10 Finite Integral Transforms
    1.10.1 Introduction
    1.10.2 The Chebyshev Transform
    1.10.3 The Legendre Transform
    1.10.4 The Gegenbauer Transform
    1.10.5 The Jacobi Transform
    1.10.6 The Laguerre Transform
    1.10.7 The Hermite Transform

  • 2. Operational Calculus
    2.1 Introduction
    2.2 Titchmarsh’s Theorem
    2.3 Operators
    2.3.1 Ring of Functions
    2.3.2 The Field of Operators
    2.3.3 Finite Parts of Divergent Integrals
    2.3.4 Rational Operators
    2.3.5 Laplace Transformable Operators
    2.3.6 Examples
    2.3.7 Periodic Functions
    2.4 Bases of the Operator Analysis
    2.4.1 Sequences and Series of Operators
    2.4.2 Operator Functions
    2.4.3 The Derivative of an Operator Function
    2.4.4 Properties of the Continuous Derivative of an Operator Function
    2.4.5 The Integral of an Operator Function
    2.5 Operators Reducible to Functions
    2.5.1 Regular Operators
    2.5.2 The Realization of Some Operators
    2.5.3 Efros Transforms
    2.6 Application of Operational Calculus
    2.6.1 Ordinary Differential Equations
    2.6.2 Partial Differential Equations

  • 3. Generalized Functions
    3.1 Introduction
    3.2 Generalized Functions — Functional Approach
    3.2.1 Introduction
    3.2.2 Distributions of One Variable
    3.2.3 Distributional Convergence
    3.2.4 Algebraic Operations on Distributions
    3.3 Generalized Functions — Sequential Approach
    3.3.1 The Identification Principle
    3.3.2 Fundamental Sequences
    3.3.3 Definition of Distributions
    3.3.4 Operations with Distributions
    3.3.5 Regular Operations
    3.4 Delta Sequences
    3.4.1 Definition and Properties
    3.4.2 Distributions as a Generalization of Continuous Functions
    3.4.3 Distributions as a Generalization of Locally Integrable Functions
    3.4.4 Remarks about Distributional Derivatives
    3.4.5 Functions with Poles
    3.4.6 Applications
    3.5 Convergent Sequences
    3.5.1 Sequences of Distributions
    3.5.2 Convergence and Regular Operations
    3.5.3 Distributionally Convergent Sequences of Smooth Functions
    3.5.4 Convolution of Distribution with a Smooth Function of Bounded Support
    3.5.5 Applications
    3.6 Local Properties
    3.6.1 Inner Product of Two Functions
    3.6.2 Distributions of Finite Order
    3.6.3 The Value of a Distribution at a Point
    3.6.4 The Value of a Distribution at Infinity
    3.6.5 Support of a Distribution
    3.7 Irregular Operations
    3.7.1 Definition
    3.7.2 The Integral of Distributions
    3.7.3 Convolution of Distributions
    3.7.4 Multiplication of Distributions
    3.7.5 Applications
    3.8 Hilbert Transform and Multiplication Forms
    3.8.1 Definition of the Hilbert Transform
    3.8.2 Applications and Examples

  • References
    Index

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