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

by Oldham, Keith B.; Spanier, Jerome

• ISBN13: 9780486450018
• ISBN10: 0486450015
• Format: Paperback
• Copyright: 2006-04-28
• Publisher: Dover Publications

Summary

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.

Author Biography

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.

Table of Contents

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