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Research and Exposition in Mathematics -- Volume 28

   Enlarged Picture

Vladislav V. Kravchenko

Applied Quaternionic Analysis

vi + 136 pages, soft cover, ISBN 3-88538-228-8, EUR 24.00, 2003

Quaternionic analysis is the most natural and close generalization of complex analysis that preserves many of its important features. The present book is meant as an introduction and invitation to this theory and its applications (in fact it was inspired by a course given by the author to graduate engineering students). Restricting ourselves to Maxwell's equations and the Dirac equation only we show the progress achieved in applied quaternionic analysis during the last five years, emphasising results which can not so easily be obtained by other methods. Thus, the main objective of this work is to introduce the reader to some topics of quaternionic analysis whose selection is motivated by particular models from the theory of electromagnetic and spinor fields, and to show the usefulness and necessity of applying the tools of quaternionic analysis to these kinds of problems.

List of Contents:

  Chapter 1: Introduction 1
  Chapter 2: Elements of quaternionic analysis  
2.1 Quaternions 5
2.2 Complex quaternions 10
2.3 Complex quaternionic functions 14
2.4 The Moisil-Theodoresco differential operator 14
2.5 The operator D+αI 18
2.6 The operator D+Mα 39
Chapter 3: Physical models reducing to the operator Dα  
3.1 Maxwell's equations 51
3.2 Time-harmonic Maxwell's equations 57
3.3 Boundary value problems for electromagnetic fields 65
3.4 Chiral media 72
3.5 The Dirac equation for a free particle 76
Chapter 4: Fields in inhomogeneus media  
4.1 Vektorräume 87
4.2 Spinor fields 109
Bibliography 121