Introduction to Clifford Algebra

INTRODUCTION TO CLIFFORDANALYSISA NEW PERSPECTIVE
INTRODUCTION TO CLIFFORDANALYSISA NEW PERSPECTIVE
Contents
Preface
Chapter 1Complex Numbers
	1.1 Properties of Real Numbers
	1.2 Complex Numbers as Pairs of Real Numbers
	1.3 Solvability of z2 = −1
	1.4 Real and Imaginary Parts
	1.5 Geometrical Interpretation
	1.6 Absolute Value and Conjugate
	1.7 Trigonometric Form
	1.8 Exponential Form
	1.9 Geometrical Interpretation of the P
	1.10 Powers and Roots
Chapter 2Complex–Valued Functions:Cauchy–Riemann Equation andIntegral Theorem
	2.1 Complex–Valued Functions
	2.2 Limits of Complex–Valued Functions
	2.3 Continuity
	2.4 Two Special Cases of Complex–Valued Functions
		2.4.1. Complex–Valued Functions of a Complex Variable
		2.4.2. Complex–Valued Functions of a Real Variable
	2.5 Differentiation of Complex–Valued Functions withRespect to Real Variables
	2.6 Differentiation of Complex–Valued Functions withRespect to Complex Variables
	2.7 Rules for Complex Differentiation
	2.8 The Cauchy–Riemann System as NecessaryCondition
	2.9 The Cauchy–Riemann System as SufficientCondition
	2.10 Holomorphy
	2.11 The Cauchy Integral Theorem
		2.11.1. The Basic Statement
		2.11.2. Equality of Line Integrals Over Closed Curves
	2.12 The Cauchy Integral Formula
Chapter 3Clifford Algebras andCauchy–Riemann Operator
	3.1 Another Interesting Approach to ComplexNumbers
		3.1.1. The Usual Definition of Complex Numbers
		3.1.2. Complex Numbers as Linear Polynomials
	3.2 The Cauchy–Riemann Operator in HigherDimensions
	3.3 Clifford Algebras Defined by Equivalence Classes
	3.4 Dimension of Clifford Algebras
	3.5 Inversion of the Multiplication
Chapter 4Short Introduction to CliffordAnalysis
	4.1 Monogenic Functions and Cauchy–Riemann TypeOperators
		4.1.1. Clifford Number
		4.1.2. Clifford Valued Function
	4.2 Involution
	4.3 A Generalized Cauchy–Riemann Operator
	4.4 The Product Rule for Dq(v ·w) in An
Chapter 5Matrix Representation
	5.1 Recalling Some Notations
	5.2 Fundamental Matrices and CliffordMatrixRepresentation
		5.2.1. Construction of the FundamentalMatrices
	5.3 FundamentalMatricesUsingMATLAB Algorithms
		5.3.1. Algorithm to ConstructMatrices
		5.3.2. Algorithm to Construct Basis
Chapter 6Fundamental Solution for theCauchy–Riemann Operator
	6.1 Gauss Integral Formula for the Cauchy–RiemannOperator
	6.2 Green Integral Formula for the Cauchy-RiemannOperator
		6.2.1. Motivation or Differences
	6.3 Estimates for Integrals of An-Valued Functions
	6.4 Cauchy–Pompeiu Integral Formula
	6.5 Cauchy Integral Formula for Monogenic Function
Chapter 7Clifford Type Algebras
	7.1 Generalized Clifford Algebras
		7.1.1. From Classical to Generalized Clifford Algebras
		7.1.2. Clifford Type Algebra An(p | 2,aj(p), gi j(p))
		7.1.3. The Classical Cauchy-Riemann Operator in An(2,aj, gi j)
	7.2 Other Clifford Structure Types
	7.3 Higher Order Clifford Type Algebras
Chapter 8Fundamental Solution for D−l
	8.1 Fundamental Solution for D in An(2,aj, gi j)
	8.2 Cauchy–Pompeiu Type Formula for D inAn(2,aj, gi j)
	8.3 Meta–Monogenic and Anti–Meta–MonogenicFunctions of Order n
	8.4 Integral Representation Formulas Associated toDl, l 2 R
	8.5 A Cauchy–Pompeiu Formula for Meta–MonogenicFirst Order Functions in An,2
		8.5.1. IterativeWay to Obtain Higher Order Integral Formulas
Chapter 9Fundamental Solution for theSecond Order Operator
	9.1 Computation of the Fundamental Solution
	9.2 Integral Representation Formula
		9.2.1. Another Useful Cauchy–Pompeiu Formula
Chapter 10Distributional Solutions
	10.1 The Inhomogeneous Cauchy–Riemann Equation
	10.2 Distributional Solution of the InhomogeneousCauchy–Riemann Equation
	10.3 Weyl Lemma for Monogenic Functions
	10.4 Distributional Solution for Dl and D−l
	10.5 Discussion of the Special Solution for ˜
Chapter 11Applications Associated toOperators D and ˜D
	11.1 Applications Related to the Cauchy–PompeiuFormula
		11.1.1. Homogeneous Case
		11.1.2. Non Homogeneous Case
		11.1.3. Special Cases
		11.1.4. The n–Higher Order Integral Formulas
	11.2 Integral Representation for bi−˜D Fu
	11.3 Combined Operator D and ˜
Chapter 12Multi–Dimensional CliffordType Algebras
	12.1 Clifford Type Algebras
		12.1.1. Monogenic Functions in Amj ,2
		12.1.2. Harmonic Functions in Am j ,2
		12.1.3. Green’s Formula
	12.2 Multi–Dimensional Clifford Type Algebras
		12.2.1. The Am(s1)–Algebra
		12.2.2. The Am(s2)–Algebra
Chapter 13Clifford Fractional Operators
	13.1 Fractional Calculus
	13.2 Fractional Clifford Analysis
	13.3 Some Examples in R0,2
		13.3.1. Constant Functions in the Riemann–LiouvilleFractional Sense
		13.3.2. Plot Examples and Codes
Chapter 14Appendix: FundamentalSolutions in the Framework ofthe Theory of Distributions
References
About the Authors
Index
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