The Fractional Calculus Theory and

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. Topics include integer order, simple and complex functions, semiderivatives and semiintegrals, and transcendental functions. The text concludes with overviews of applications in the classical calculus and diffusion problems. 1974 ed.

Keith B. Oldham is Professor of Chemistry at Trent University in Ontario, and Jerome Spanier is a research mathematician at the University of California at Irvine.

Preface

ix

Acknowledgments

xiii

Errata

xv

Chapter 1 INTRODUCTION

1.1 Historical Survey

1

(14)

1.2 Notation

15

(1)

1.3 Properties of the Gamma Function

16

(9)

Chapter 2 DIFFERENTIATION AND INTEGRATION TO INTEGER ORDER

2.1 Symbolism

25

(2)

2.2 Conventional Definitions

27

(3)

2.3 Composition Rule for Mixed Integer Orders

30

(3)

2.4 Dependence of Multiple Integrals on Lower Limit

33

(1)

2.5 Product Rule for Multiple Integrals

34

(2)

2.6 The Chain Rule for Multiple Derivatives

36

(1)

2.7 Iterated Integrals

37

(1)

2.8 Differentiation and Integration of Series

38

(1)

2.9 Differentiation and Integration of Powers

39

(1)

2.10 Differentiation and Integration of Hypergeometrics

40

(6)

Chapter 3 FRACTIONAL DERIVATIVES AND INTEGRALS: DEFINITIONS AND EQUIVALENCES

3.1 Differintegrable Functions

46

(1)

3.2 Fundamental Definitions

47

(4)

3.3 Identity of Definitions

51

(1)

3.4 Other General Definitions

52

(5)

3.5 Other Formulas Applicable to Analytic Functions

57

(2)

3.6 Summary of Definitions

59

(2)

Chapter 4 DIFFERINTEGRATION OF SIMPLE FUNCTIONS

4.1 The Unit Function

61

(2)

4.2 The Zero Function

63

(1)

4.3 The Function x - a

63

(2)

4.4 The Function [x - a]p

65

(4)

Chapter 5 GENERAL PROPERTIES

5.1 Linearity

69

(1)

5.2 Differintegration Term by Term

69

(6)

5.3 Homogeneity

75

(1)

5.4 Scale Change

75

(1)

5.5 Leibniz's Rule

76

(4)

5.6 Chain Rule

80

(2)

5.7 Composition Rule

82

(5)

5.8 Dependence on Lower Limit

87

(2)

5.9 Translation

89

(1)

5.10 Behavior Near Lower Limit

90

(1)

5.11 Behavior Far from Lower Limit

91

(2)

Chapter 6 DIFFERINTEGRATION OF MORE COMPLEX FUNCTIONS

6.1 The Binomial Function [C - cx]p

93

(1)

6.2 The Exponential Function exp(C - cx)

94

(1)

6.3 The Functions xq/[1 - x] and xp/[1 - x] and [1 - x]q-1

95

(1)

6.4 The Hyperbolic and Trigonometric Functions sinh(square root of x) and sin(square root of x)

96

(1)

6.5 The Bessel Functions

97

(2)

6.6 Hypergeometric Functions

99

(3)

6.7 Logarithms

102

(3)

6.8 The Heaviside and Dirac Functions

105

(2)

6.9 The Sawtooth Function

107

(1)

6.10 Periodic Functions

108

(2)

6.11 Cyclodifferential Functions

110

(2)

6.12 The Function xq-1exp[- 1/x]

112

(3)

Chapter 7 SEMIDERIVATIVES AND SEMIINTEGRALS

7.1 Definitions

115

(1)

7.2 General Properties

116

(2)

7.3 Constants and Powers

118

(2)

7.4 Binomials

120

(2)

7.5 Exponential and Related Functions

122

(2)

7.6 Trigonometric and Hyperbolic Functions

124

(3)

7.7 Bessel and Struve Functions

127

(2)

7.8 Generalized Hypergeometric Functions

129

(1)

7.9 Miscellaneous Functions

130

(3)

Chapter 8 TECHNIQUES IN THE FRACTIONAL CALCULUS

8.1 Laplace Transformation

133

(3)

8.2 Numerical Differintegration

136

(12)

8.3 Analog Differintegration

148

(6)

8.4 Extraordinary Differential Equations

154

(3)

8.5 Semidifferential Equations

157

(2)

8.6 Series Solutions

159

(3)

Chapter 9 REPRESENTATION OF TRANSCENDENTAL FUNCTIONS

9.1 Transcendental Functions as Hypergeometrics

162

(3)

9.2 Hypergeometrics with K > L

165

(1)

9.3 Reduction of Complex Hypergeometrics

166

(2)

9.4 Basis Hypergeometrics

168

(4)

9.5 Synthesis of K= L Transcendentals

172

(3)

9.6 Synthesis of K= L — 1 Transcendentals

175

(2)

9.7 Synthesis of K = L — 2 Transcendentals

177

(4)

Chapter 10 APPLICATIONS IN THE CLASSICAL CALCULUS

10.1 Evaluation of Definite Integrals and Infinite Sums

181

(2)

10.2 Abel's Integral Equation

183

(3)

10.3 Solution of Bessel's Equation

186

(3)

10.4 Candidate Solutions for Differential Equations

189

(3)

10.5 Function Families

192

(6)

Chapter 11 APPLICATIONS TO DIFFUSION PROBLEMS

11.1 Transport in a Semiinfinite Medium

198

(3)

11.2 Planar Geometry

201

(3)

11.3 Spherical Geometry

204

(3)

11.4 Incorporation of Sources and Sinks

207

(3)

11.5 Transport in Finite Media

210

(6)

11.6 Diffusion on a Curved Surface

216

(3)

References

219

(6)

Index

225

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The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order

Summary

Author Biography

Table of Contents

Supplemental Materials