Introduction to Clifford Algebra

185 Pages·5.764 MB·English

by  Johan Ceballos (editor)

MATHEMATICS RESEARCH DEVELOPMENTS

I C
NTRODUCTION TO LIFFORD
A
NALYSIS

A NEW PERSPECTIVE

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ATHEMATICS ESEARCH
D
EVELOPMENTS

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MATHEMATICS RESEARCH DEVELOPMENTS

I C
NTRODUCTION TO LIFFORD
A
NALYSIS

A NEW PERSPECTIVE

JOHAN CEBALLOS
NICOLÁS COLOMA
ANTONIO DI TEODORO
AND
FRANCISCO PONCE
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Library of Congress Cataloging-in-Publication Data
Names: Ceballos, Johan, Coloma, Nicolás, Di Teodoro, Antonio, Ponce, Francisco authors.
Title: Introduction to Clifford algebra / Johan Ceballos, Universidad de
Las Américas, Quito, Ecuador. Nicolás Coloma, University of Colorado Boulder, Co, USA, Antonio
Di Teodoro, Universidad San Francisco de Quito, Quito, Ecuador, and Francisco Ponce, Universidad
San Francisco de Quito, Quito, Ecuador, University of Colorado Boulder, Co,USA, Antonio Di
Teodoro, Universidad San Francisco de Quito, Quito, Ecuador and Francisco Ponce, Universidad San
Francisco de Quito, Quito, Ecuador.
Identifiers: LCCN 2020040956 (print) | LCCN 2020040957 (ebook) | ISBN 9781536185331
(paperback) | ISBN 9781536186642 (adobe pdf)
Subjects: LCSH: Clifford algebras.
Classification: LCC QA199 .C43 2020 (print) | LCC QA199 (ebook) | DDC 512/.57--dc23
LC record available at https://lccn.loc.gov/2020040956
LC ebook record available at https://lccn.loc.gov/2020040957
Published by Nova Science Publishers, Inc. † New York
Contents
Preface ix
1 ComplexNumbers 1
1.1 PropertiesofRealNumbers . . . . . . . . . . . . . . . . . . . . 1
1.2 ComplexNumbersasPairsofRealNumbers . . . . . . . . . . 2
1.3 Solvabilityofz2= 1 . . . . . . . . . . . . . . . . . . . . . . 3
−
1.4 RealandImaginaryParts . . . . . . . . . . . . . . . . . . . . . 4
1.5 GeometricalInterpretation . . . . . . . . . . . . . . . . . . . . 4
1.6 AbsoluteValueandConjugate . . . . . . . . . . . . . . . . . . 5
1.7 TrigonometricForm . . . . . . . . . . . . . . . . . . . . . . . . 6
1.8 ExponentialForm . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.9 GeometricalInterpretationoftheProduct . . . . . . . . . . . . 7
1.10 PowersandRoots . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Complex–ValuedFunctions 13
2.1 Complex–ValuedFunctions . . . . . . . . . . . . . . . . . . . . 13
2.2 LimitsofComplex–ValuedFunctions . . . . . . . . . . . . . . 14
2.3 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4 TwoSpecialCasesofComplex–ValuedFunctions . . . . . . . . 15
2.4.1. Complex–ValuedFunctionsofaComplexVariable . . . 15
2.4.2. Complex–ValuedFunctionsofaRealVariable . . . . . 16
2.5 RealDifferentiation . . . . . . . . . . . . . . . . . . . . . . . . 16
2.6 Differentiation of Complex–Valued Functions with Respect to
ComplexVariables . . . . . . . . . . . . . . . . . . . . . . . . 17
2.7 RulesforComplexDifferentiation . . . . . . . . . . . . . . . . 18
2.8 Necessity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.9 Sufficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
vi Contents
2.10 Holomorphy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.11 TheCauchyIntegralTheorem . . . . . . . . . . . . . . . . . . 22
2.11.1. TheBasicStatement . . . . . . . . . . . . . . . . . . . 22
2.11.2. EqualityofLineIntegralsOverClosedCurves . . . . . 23
2.12 TheCauchyIntegralFormula . . . . . . . . . . . . . . . . . . . 24
3 CliffordAlgebrasandCauchy–RiemannOperator 27
3.1 AnotherInterestingApproachtoComplex
Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.1.1. TheUsualDefinitionofComplexNumbers . . . . . . . 28
3.1.2. ComplexNumbersasLinearPolynomials . . . . . . . . 28
3.2 Cauchy–RiemannOperator . . . . . . . . . . . . . . . . . . . . 30
3.3 DefinitionbyEquivalenceClasses . . . . . . . . . . . . . . . . 32
3.4 DimensionofCliffordAlgebras . . . . . . . . . . . . . . . . . 36
3.5 InversionoftheMultiplication . . . . . . . . . . . . . . . . . . 38
4 ShortIntroductiontoCliffordAnalysis 41
4.1 MonogenicFunctionsandCauchy–RiemannType
Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.1.1. CliffordNumber . . . . . . . . . . . . . . . . . . . . . 42
4.1.2. CliffordValuedFunction . . . . . . . . . . . . . . . . . 42
4.2 Involution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.3 AGeneralizedCauchy–RiemannOperator . . . . . . . . . . . . 48
4.4 TheProductRuleforD (v w)inA . . . . . . . . . . . . . . . 51
q n
·
5 MatrixRepresentation 55
5.1 RecallingSomeNotations . . . . . . . . . . . . . . . . . . . . 55
5.2 FundamentalMatricesandCliffordMatrix
Representation . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.2.1. ConstructionoftheFundamentalMatrices . . . . . . . . 59
5.3 FundamentalMatricesUsingMATLABAlgorithms . . . . . . . 61
5.3.1. AlgorithmtoConstructMatrices . . . . . . . . . . . . . 61
5.3.2. AlgorithmtoConstructBasis . . . . . . . . . . . . . . 63
6 FundamentalSolutionfortheCauchy–RiemannOperator 67
6.1 GaussIntegralFormula . . . . . . . . . . . . . . . . . . . . . . 69
6.2 GreenIntegralFormula . . . . . . . . . . . . . . . . . . . . . . 70
6.2.1. MotivationorDifferences . . . . . . . . . . . . . . . . 70
Contents vii
6.3 EstimatesofIntegrals . . . . . . . . . . . . . . . . . . . . . . . 72
6.4 Cauchy–PompeiuIntegralFormula . . . . . . . . . . . . . . . . 74
6.5 CauchyIntegralFormula . . . . . . . . . . . . . . . . . . . . . 76
7 CliffordTypeAlgebras 79
7.1 GeneralizedCliffordAlgebras . . . . . . . . . . . . . . . . . . 79
7.1.1. FromClassicaltoGeneralizedCliffordAlgebras . . . . 79
7.1.2. CliffordTypeAlgebraA (p 2,α (p),γ (p)) . . . . . . 82
n j ij
|
7.1.3. TheClassicalCauchy-RiemannOperatorinA (2,α ,γ ) 83
n j ij
7.2 OtherCliffordStructureTypes . . . . . . . . . . . . . . . . . . 85
7.3 HigherOrderCliffordTypeAlgebras . . . . . . . . . . . . . . . 88
8 FundamentalSolutionD 91
λ
8.1 FundamentalSolutionforDinA (2,α ,γ ) . . . . . . . . . . . 91
n j ij
8.2 Cauchy–PompeiuTypeFormulaforDin
A (2,α ,γ ). . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
n j ij
8.3 Meta–MonogenicandAnti–Meta–Monogenic
FunctionsofOrdern . . . . . . . . . . . . . . . . . . . . . . . 96
8.4 IntegralRepresentationFormulasAssociatedtoD ,λ R . . . 97
λ
∈
8.5 ACauchy–PompeiuFormula for Meta–MonogenicFirstOrder
FunctionsinA . . . . . . . . . . . . . . . . . . . . . . . . . 98
n∗,2
8.5.1. IterativeWaytoObtainHigherOrderIntegralFormulas 100
9 FundamentalSolutionfortheSecondOrder Operator 105
9.1 ComputationoftheFundamentalSolution . . . . . . . . . . . . 105
9.2 IntegralRepresentationFormula . . . . . . . . . . . . . . . . . 110
9.2.1. AnotherUsefulCauchy–PompeiuFormula . . . . . . . 111
10 DistributionalSolutions 113
10.1 ClassicalSolutions . . . . . . . . . . . . . . . . . . . . . . . . 114
10.2 InhomogeneousEquation . . . . . . . . . . . . . . . . . . . . . 116
10.3 WeylLemmaforMonogenicFunctions . . . . . . . . . . . . . 117
10.4 DistributionalSolutionforD andD . . . . . . . . . . . . . . 117
λ λ
10.5 DiscussionoftheSpecialSolutionfo−r∆˜ . . . . . . . . . . . . . 119
11 ApplicationsAssociatedtoOperatorsDand∆˜ 123
11.1 ApplicationsRelatedtotheCauchy–Pompeiu
Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
viii Contents
11.1.1. HomogeneousCase . . . . . . . . . . . . . . . . . . . . 123
11.1.2. NonHomogeneousCase . . . . . . . . . . . . . . . . . 124
11.1.3. SpecialCases . . . . . . . . . . . . . . . . . . . . . . . 125
11.1.4. Then–HigherOrderIntegralFormulas . . . . . . . . . 127
11.2 IntegralRepresentationforbi ∆˜ Functions . . . . . . . . . . . 128
11.3 CombinedOperatorDand∆˜ −. . . . . . . . . . . . . . . . . . . 129
12 Multi–DimensionalCliffordTypeAlgebras 131
12.1 CliffordTypeAlgebras . . . . . . . . . . . . . . . . . . . . . . 132
12.1.1. MonogenicFunctionsinA . . . . . . . . . . . . . . 132
m∗j,2
12.1.2. HarmonicFunctionsinA . . . . . . . . . . . . . . . 133
m∗j,2
12.1.3. Green’sFormula . . . . . . . . . . . . . . . . . . . . . 133
12.2 Multi–DimensionalCliffordTypeAlgebras . . . . . . . . . . . 134
12.2.1. TheA (σ )–Algebra . . . . . . . . . . . . . . . . . . . 135
m 1
12.2.2. TheA (σ )–Algebra . . . . . . . . . . . . . . . . . . . 137
m 2
13 CliffordFractionalOperators 141
13.1 FractionalCalculus . . . . . . . . . . . . . . . . . . . . . . . . 141
13.2 FractionalCliffordAnalysis . . . . . . . . . . . . . . . . . . . 143
13.3 SomeExamplesinR . . . . . . . . . . . . . . . . . . . . . . 145
0,2
13.3.1. ConstantFunctionsintheRiemann–Liouville
FractionalSense . . . . . . . . . . . . . . . . . . . . . 145
13.3.2. PlotExamplesandCodes . . . . . . . . . . . . . . . . 147
14 Appendix 151
References 155
AbouttheAuthors 165
Index 167