The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order (Dover Books on Mathematics)

The product of a collaboration between a mathematician and a chemist, this text is geared toward advanced undergraduates and graduate students. Not only does it explain the theory underlying the properties of the generalized operator, but it also illustrates the wide variety of fields to which these ideas may be applied. Rather than an exhaustive treatment, it represents an introduction that will appeal to a broad spectrum of students. Accordingly, the mathematics is kept as simple as possible.The first of the two-part treatment deals principally with the general properties of differintegral operators. The second half is mainly oriented toward the applications of these properties to mathematical and other problems. Topics include integer order, simple and complex functions, semiderivatives and semi-integrals, and transcendental functions. The text concludes with overviews of applications in the classical calculus and diffusion problems. **This is a corrected and cleaned scan of the original book** Preface ix Acknowledgments xiii Chapter 1 INTRODUCTION 1.1 Historical Survey 1 1.2 Notation 15 1.3 Properties of the Gamma Function 16 Chapter 2 DIFFERENTIATION AND INTEGRATION TO INTEGER ORDER 2.1 Symbolism 25 2.2 Conventional Definitions 27 2.3 Composition Rule for Mixed Integer Orders 30 2.4 Dependence of Multiple Integrals on Lower Limit 33 2.5 Product Rule for Multiple Integrals 34 2.6 The Chain Rule for Multiple Derivatives 36 2.7 Iterated Integrals 37 2.8 Differentiation and Integration of Series 38 2.9 Differentiation and Integration of Powers 39 2.10 Differentiation and Integration of Hypergeometrics 40 Chapter 3 FRACTIONAL DERIVATIVES AND INTEGRALS: DEFINITIONS AND EQUIVALENCES 3.1 Differintegrable Functions 46 3.2 Fundamental Definitions 47 3.3 Identity of Definitions 51 3.4 Other General Definitions 52 3.5 Other Formulas Applicable to Analytic Functions 57 3.6 Summary of Definitions 59 Chapter 4 DIFFER INTEGRATION OF SIMPLE FUNCTIONS 4.1 The Unit Function 61 4.2 The Zero Function 63 4.3 The Function $x-a$ 63 4.4 The Function $[x-a]$P 65 Chapter 5 GENERAL PROPERTIES 5.1 Linearity 69 5.2 Differintegration Term by Term 69 5.3 Homogeneity 75 5.4 Scale Change 75 5.5 Leibniz's Rule 76 5.6 Chain Rule 80 5.7 Composition Rule 82 5.8 Dependence on Lower Limit 87 5.9 Translation 89 5.10 Behavior Near Lower Limit 90 5.11 Behavior Far from Lower Limit 91 Chapter 6 DIFFERINTEGRATION OF MORE COMPLEX FUNCTIONS 6.1 The Binomial Function $[C-cx]P$ 93 6.2 The Exponential Function $exp(C-cx)$ 94 6.3 The Functions $x^{q}/[1-x]$ and $x^{p}/[1-x]$ and $[1-X]^{q-1}$ 95 6.4 The Hyperbolic and Trigonometric Functions $sinh(√x)$ and $sin(√x)$ 96 6.5 The Bessel Functions 97 6.6 Hypergeometric Functions 99 6.7 Logarithms 102 6.8 The Heaviside and Dirac Functions 105 6.9 The Sawtooth Function 107 6.10 Periodic Functions 108 6.11 Cyclodifferential Functions 110 6.12 The Function $x^{q-1}exp[-1/x]$ 112 Chapter 7 SEMI DERIVATIVES AND SEMIINTEGRALS 7.1 Definitions 115 7.2 General Properties 116 7.3 Constants and Powers 118 7.4 Binomials 120 7.5 Exponential and Related Functions 122 7.6 Trigonometric and Hyperbolic Functions 124 7.7 Bessel and Struve Functions 127 7.8 Generalized Hypergeometric Functions 129 7.9 Miscellaneous Functions 130 Chapter 8 TECHNIQUES IN THE FRACTIONAL CALCULUS 8.1 Laplace Transformation 133 8.2 Numerical Differintegration 136 8.3 Analog Differintegration 148 8.4 Extraordinary Differential Equations 154 8.5 Semidifferential Equations 157 8.6 Series Solutions 159 Chapter 9 REPRESENTATION OF TRANSCENDENTAL FUNCTIONS 9.1 Transcendental Functions as Hypergeometrics 162 9.2 Hypergeometrics with $K>L$ 165 9.3 Reduction of Complex Hypergeometrics 166 9.4 Basis Hypergeometrics 168 9.5 Synthesis of $K=L$ Transcendentals 172 9.6 Synthesis of $K=L-1$ Transcendentals 175 9.7 Synthesis of $K=L-2$ Transcendentals 177 Chapter 10 APPLICATIONS IN THE CLASSICAL CALCULUS 10.1 Evaluation of Definite Integrals and Infinite Sums 181 10.2 Abel's Integral Equation 183 10.3 Solution of Bessel's Equation 186 10.4 Candidate Solutions for Differential Equations 189 10.5 Function Families 192 Chapter 11 APPLICATIONS TO DIFFUSION PROBLEMS 11.1 Transport in a Semiinfinite Medium 198 11.2 Planar Geometry 201 11.3 Spherical Geometry 204 11.4 Incorporation of Sources and Sinks 207 11.5 Transport in Finite Media 210 11.6 Diffusion on a Curved Surface 216 References 219 Index 225 The product of a collaboration between a mathematician and a chemist, this text is geared toward advanced undergraduates and graduate students. Not only does it explain the theory underlying the properties of the generalized operator, but it also illustrates the wide variety of fields to which these ideas may be applied. Rather than an exhaustive treatment, it represents an introduction that will appeal to a broad spectrum of students. Accordingly, the mathematics is kept as simple as possible. The first of the two-part treatment deals principally with the general properties of differintegral operators. The second half is mainly oriented toward the applications of these properties to mathematical and other problems. Topics include integer order, simple and complex functions, semiderivatives and semi-integrals, and transcendental functions. The text concludes with overviews of applications in the classical calculus and diffusion problems. This is a corrected and cleaned scan of the original book

not Only Does This Text Explain The Theory Underlying The Properties Of The Generalized Operator, But It Also Illustrates The Wide Variety Of Fields To Which These Ideas May Be Applied. Topics Include Integer Order, Simple And Complex Functions, Semiderivatives And Semiintegrals, And Transcendental Functions. 1974 Edition.