# Introduction to Clifford Analysis: A New Perspective

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Series: Mathematics Research Developments
BISAC: MAT002000
DOI: 10.52305/EYQO1733

This book pursues to exhibit how we can construct a Clifford type algebra from the classical one. The basic idea of these lecture notes is to show how to calculate fundamental solutions to either first–order differential operators of the form D=∑_(i=0)^n▒〖e_i δ_i〗or second–order elliptic differential operators¯D D, both with constant coefficients or combinations of this kind of operators. After considering in detail how to find the fundamental solution we study the problem of integral representations in a classical Clifford algebra and in a dependent–parameter Clifford algebra which generalizes the classical one. We also propose a basic method to extend the order of the operator, for instance D^n,n∈N and how to produce integral representations for higher order operators and mixtures of them. Although the Clifford algebras have produced many applications concerning boundary value problems, initial value problems, mathematical physics, quantum chemistry, among others; in this book we do not discuss these topics as they are better discussed in other courses. Researchers and practitioners will find this book very useful as a source book.

The reader is expected to have basic knowledge of partial differential equations and complex analysis. When planning and writing these lecture notes, we had in mind that they would be used as a resource by mathematics students interested in understanding how we can combine partial differential equations and Clifford analysis to find integral representations. This in turn would allow them to solve boundary value problems and initial value problems. To this end, proofs have been described in rigorous detail and we have included numerous worked examples. On the other hand, exercises have not been included.

ISBN: N/A

Preface

Chapter 1. Complex Numbers

Chapter 2. Complex–Valued Functions

Chapter 3. Clifford Algebras and Cauchy–Riemann Operator

Chapter 4. Short Introduction to Clifford Analysis

Chapter 5. Matrix Representation

Chapter 6. Fundamental Solution for the Cauchy–Riemann operator

Chapter 7. Clifford Type Algebras

Chapter 8. Fundamental Solution D_λ

Chapter 9. Fundamental Solution for the Second Order Operator

Chapter 10. Distributional Solutions

Chapter 11. Applications Associated to Operators Dand ∆ Þ

Chapter 12. Multi–Dimensional Clifford Type Algebras

Chapter 13. Clifford Fractional Operators 117

Chapter 14. Appendix