دانلود کتاب Introduction to Clifford Algebra

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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 Blank Page Blank Page