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| Preface | |
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| Historical Survey | |
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| The Origin of the Fractional Calculus | |
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| The Contributions of Abel and Liouville | |
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| A Longstanding Controversy | |
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| Riemann's Contribution, Errors by Noted Mathematicians | |
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| The Mid-Nineteenth Century | |
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| The Origin of the Riemann-Liouville Definition | |
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| The Last Decade of the Nineteenth Century | |
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| The Twentieth Century | |
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| The Modern Approach | |
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| The Iterated Integral Approach | |
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| The Differential Equation Approach | |
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| The Complex Variable Approach | |
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| The Weyl Transform | |
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| The Fractional Derivative | |
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| The Definitions of Grunwald and Marchaud | |
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| The Riemann-Liouville Fractional Integral | |
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| Definition of the Fractional Integral | |
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| Some Examples of Fractional Integrals | |
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| Dirichlet's Formula | |
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| Derivatives of the Fractional Integral and the Fractional Integral of Derivatives | |
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| Laplace Transform of the Fractional Integral | |
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| Leibniz's Formula for Fractional Integrals | |
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| The Riemann-Liouville Fractional Calculus | |
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| The Fractional Derivative | |
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| A Class of Functions | |
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| Leibniz's Formula for Fractional Derivatives | |
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| The Law of Exponents | |
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| Integral Representations | |
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| Representations of Functions | |
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| Integral Relations | |
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| Laplace Transform of the Fractional Derivative | |
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| Fractional Differential Equations | |
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| Motivation: Direct Approach | |
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| Motivation: Laplace Transform | |
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| Motivation: Linearly Independent Solutions | |
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| Solution of the Homogeneous Equation | |
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| Explicit Representation of Solution | |
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| Relation to the Green's Function | |
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| Solution of the Nonhomogeneous Fractional Differential Equation | |
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| Convolution of Fractional Green's Functions | |
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| Reduction of Fractional Differential Equations to Ordinary Differential Equations | |
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| Semidifferential Equations | |
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| Further Results Associated with Fractional Differential Equations | |
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| Fractional Integral Equations | |
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| Fractional Differential Equations with Nonconstant Coefficients | |
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| Sequential Fractional Differential Equations | |
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| Vector Fractional Differential Equations | |
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| Some Comparisons with Ordinary Differential Equations | |
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| The Weyl Fractional Calculus | |
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| Good Functions | |
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| A Law of Exponents for Fractional Integrals | |
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| The Weyl Fractional Derivative | |
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| The Algebra of the Weyl Transform | |
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| A Leibniz Formula | |
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| An Application to Ordinary Differential Equations | |
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| Some Historical Arguments | |
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| Abel's Integral Equation and the Tautochrone Problem | |
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| Heaviside Operational Calculus and the Fractional Calculus | |
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| Potential Theory and Liouville's Problem | |
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| Fluid Flow and the Design of a Weir Notch | |
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| Appendix A. Some Algebraic Results | |
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| Some Identities Associated with Partial Fraction Expansions | |
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| Zeros of Multiplicity Greater than One | |
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| Complementary Polynomials | |
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| Some Reduction Formulas | |
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| Some Algebraic Identities | |
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| Appendix B. Higher Transcendental Functions | |
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| The Gamma Function and Related Functions | |
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| Bessel Functions | |
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| Hypergeometric Functions | |
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| Legendre and Laguerre Functions | |
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| Appendix C. The Incomplete Gamma Function and Related Functions | |
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| The Incomplete Gamma Function | |
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| Some Functions Related to the Incomplete Gamma Function | |
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| Laplace Transforms | |
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| Numerical Results | |
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| Appendix D. A Brief Table of Fractional Integrals and Derivatives | |
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| References | |
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| Index of Symbols | |
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| Index | |