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Table of contents

  1. Front Matter
    Pages i-xv
  2. Klaus Gürlebeck, Klaus Habetha, Wolfgang Sprößig
    Pages 1-42
  3. Klaus Gürlebeck, Klaus Habetha, Wolfgang Sprößig
    Pages 43-74
  4. Klaus Gürlebeck, Klaus Habetha, Wolfgang Sprößig
    Pages 75-93
  5. Klaus Gürlebeck, Klaus Habetha, Wolfgang Sprößig
    Pages 95-150
  6. Klaus Gürlebeck, Klaus Habetha, Wolfgang Sprößig
    Pages 151-167
  7. Klaus Gürlebeck, Klaus Habetha, Wolfgang Sprößig
    Pages 169-202
  8. Klaus Gürlebeck, Klaus Habetha, Wolfgang Sprößig
    Pages 203-264
  9. Klaus Gürlebeck, Klaus Habetha, Wolfgang Sprößig
    Pages 265-301
  10. Klaus Gürlebeck, Klaus Habetha, Wolfgang Sprößig
    Pages 303-318
  11. Klaus Gürlebeck, Klaus Habetha, Wolfgang Sprößig
    Pages 319-328
  12. Klaus Gürlebeck, Klaus Habetha, Wolfgang Sprößig
    Pages 329-359
  13. Back Matter
    Pages 361-390

About this book

Introduction

This book presents applications of hypercomplex analysis to boundary value and initial-boundary value problems from various areas of mathematical physics. Given that quaternion and Clifford analysis offer natural and intelligent ways to enter into higher dimensions, it starts with quaternion and Clifford versions of complex function theory including series expansions with Appell polynomials, as well as Taylor and Laurent series. Several necessary function spaces are introduced, and an operator calculus based on modifications of the Dirac, Cauchy-Fueter, and Teodorescu operators and different decompositions of quaternion Hilbert spaces are proved. Finally, hypercomplex Fourier transforms are studied in detail.

All this is then applied to first-order partial differential equations such as the Maxwell equations, the Carleman-Bers-Vekua system, the Schrödinger equation, and the Beltrami equation. The higher-order equations start with Riccati-type equations. Further topics include spatial fluid flow problems, image and multi-channel processing, image diffusion, linear scale invariant filtering, and others. One of the highlights is the derivation of the three-dimensional Kolosov-Mushkelishvili formulas in linear elasticity. 

Throughout the book the authors endeavor to present historical references and important personalities. The book is intended for a wide audience in the mathematical and engineering sciences and is accessible to readers with a basic grasp of real, complex, and functional analysis.

Keywords

hypercomplex numbers hypercomplex versions of Fourier transforms initial-boundary value problems quaternionic anaylsis quaternionic operator calculus

Authors and affiliations

  • Klaus Gürlebeck
    • 1
  • Klaus Habetha
    • 2
  • Wolfgang Sprößig
    • 3
  1. 1.Bauhaus-Universität WeimarWeimarGermany
  2. 2.RWTH AachenAachenGermany
  3. 3.TU Bergakademie FreibergFreibergGermany

Bibliographic information