The Fractional Calculus

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