The Fractional Calculus

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Cover Title page Preface Acknowledgments Chapter 1 INTRODUCTION 1.1 Historical Survey 1.2 Notation 1.3 Properties of the Gamma Function Chapter 2 DIFFERENTIATION AND INTEGRATION TO INTEGER ORDER 2.1 Symbolism 2.2 Conventional Definitions 2.3 Composition Rule for Mixed Integer Orders 2.4 Dependence of Multiple IntegraIs on Lower Limit 2.5 Product Rule for Multiple IntegraIs 2.6 The Chain Rule for Multiple Derivatives 2.7 Iterated Integrais 2.8 Differentiation and Integration of Series 2.9 Differentiation and Integration of Powers 2.10 Differentiation and Integration of Hypergeometrics Chapter 3 FRACTIONAL DERIVATIVES AND INTEGRALS: DEFINITIONS AND EQUIVALENCES 3.1 Differintegrable Functions 3.2 Fundamental Definitions 3.3 Identity of Definitions 3.4 Other General Definitions 3.5 Other Formulas Applicable to Analytic Functions 3.6 Summary of Definitions Chapter 4 DIFFERINTEGRATION OF SIMPLE FUNCTIONS 4.1 The Unit Function 4.2 The Zero Function 4.3 The Function x-a 4.4 The Function [x-a]^p Chapter 5 GENERAL PROPERTIES 5.1 Linearity 5.2 Differintegration Term by Term 5.3 Homogeneity 5.4 Scale Change 5.5 Leibniz's Rule 5.6 Chain Rule 5.7 Composition Rule 5.8 Dependence on Lower Limit 5.9 Translation 5.10 Behavior Near Lower Limit 5.11 Behavior Far from Lower Limit Chapter 6 DIFFERINTEGRATION OF MORE COMPLEX FUNCTIONS 6.1 The Binomial Function [C-cx]^p 6.2 The Exponential Function exp(C-cx) 6.3 The Functions x^q /[1-x] and x^p/[l-x] and [1-x]^{q-l} 6.4 The Hyperbolic and Trigonometric Functions sinh(√x) and sin(√x) 6.5 The Bessel Functions 6.6 Hypergeometric Functions 6.7 Logarithms 6.8 The Heaviside and Dirac Functions 6.9 The Sawtooth Function 6.10 Periodic Functions 6.11 Cyclodifferential Functions 6.12 The Function x^{q-1} exp[-1/x] Chapter 7 SEMIDERIVATIVES AND SEMIINTEGRALS 7.1 Definitions 7.2 General Properties 7.3 Constants and Powers 7.4 Binomials 7.5 Exponential and Related Functions 7.6 Trigonometric and Hyperbolic Functions 7.7 Bessel and Struve Functions 7.8 Generalized Hypergeometric Functions 7.9 Miscellaneous Functions Chapter 8 TECHNIQUES lN THE FRACTIONAL CALCULUS 8.1 Laplace Transformation 8.2 Numerical Differintegration 8.3 Analog Differintegration 8.4 Extraordinary Differential Equations 8.5 Semidifferential Equations 8.6 Series Solutions Chapter 9 REPRESENTATION OF TRANSCENDENTAL FUNCTIONS 9.1 Transcendental Functions as Hypergeometrics 9.2 Hypergeometrics with K> L 9.3 Reduction of Complex Hypergeometrics 9.4 Basis Hypergeometrics 9.5 Synthesis of K = L Transcendentals 9.6 Synthesis of K = L-1 Transcendentals 9.7 Synthesis of K = L-2 Transcendentals Chapter 10 APPLICATIONS lN THE CLASSICAL CALCULUS 10.1 Evaluation of Definite IntegraIs and Infinite Sums 10.2 Abel's Integral Equation 10.3 Solution of Bessel's Equation 10.4 Candidate Solutions for DifferentiaI Equations 10.5 Function Families Chapter 11 APPLICATIONS TO DIFFUSION PROBLEMS II.1 Transport in a Semiinfinite Medium 11.2 Planar Geometry 11.3 Spherical Geometry 11.4 Incorporation of Sources and Sinks 11.5 Transport in Finite Media 11.6 Diffusion on a Curved Surface References Index