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Bicomplex Holomorphic Functions

The Algebra, Geometry and Analysis of Bicomplex Numbers

  • M. Elena Luna-Elizarrarás
  • Michael Shapiro
  • Daniele C. Struppa
  • Adrian Vajiac

Part of the Frontiers in Mathematics book series (FM)

Table of contents

  1. Front Matter
    Pages i-viii
  2. M. Elena Luna-Elizarrarás, Michael Shapiro, Daniele C. Struppa, Adrian Vajiac
    Pages 1-3
  3. M. Elena Luna-Elizarrarás, Michael Shapiro, Daniele C. Struppa, Adrian Vajiac
    Pages 5-28
  4. M. Elena Luna-Elizarrarás, Michael Shapiro, Daniele C. Struppa, Adrian Vajiac
    Pages 29-49
  5. M. Elena Luna-Elizarrarás, Michael Shapiro, Daniele C. Struppa, Adrian Vajiac
    Pages 51-72
  6. M. Elena Luna-Elizarrarás, Michael Shapiro, Daniele C. Struppa, Adrian Vajiac
    Pages 73-105
  7. M. Elena Luna-Elizarrarás, Michael Shapiro, Daniele C. Struppa, Adrian Vajiac
    Pages 107-111
  8. M. Elena Luna-Elizarrarás, Michael Shapiro, Daniele C. Struppa, Adrian Vajiac
    Pages 113-133
  9. M. Elena Luna-Elizarrarás, Michael Shapiro, Daniele C. Struppa, Adrian Vajiac
    Pages 135-177
  10. M. Elena Luna-Elizarrarás, Michael Shapiro, Daniele C. Struppa, Adrian Vajiac
    Pages 179-191
  11. M. Elena Luna-Elizarrarás, Michael Shapiro, Daniele C. Struppa, Adrian Vajiac
    Pages 193-199
  12. M. Elena Luna-Elizarrarás, Michael Shapiro, Daniele C. Struppa, Adrian Vajiac
    Pages 201-210
  13. M. Elena Luna-Elizarrarás, Michael Shapiro, Daniele C. Struppa, Adrian Vajiac
    Pages 211-217
  14. Back Matter
    Pages 219-231

About this book

Introduction

The purpose of this book is to develop the foundations of the theory of holomorphicity on the ring of bicomplex numbers. Accordingly, the main focus is on expressing the similarities with, and differences from, the classical theory of one complex variable. The result is an elementary yet comprehensive introduction to the algebra, geometry and analysis of bicomplex numbers.

Around the middle of the nineteenth century, several mathematicians (the best known being Sir William Hamilton and Arthur Cayley) became interested in studying number systems that extended the field of complex numbers. Hamilton famously introduced the quaternions, a skew field in real-dimension four, while almost simultaneously James Cockle introduced a commutative four-dimensional real algebra, which was rediscovered in 1892 by Corrado Segre, who referred to his elements as bicomplex numbers. The advantages of commutativity were accompanied by the introduction of zero divisors, something that for a while dampened interest in this subject. In recent years, due largely to the work of G.B. Price, there has been a resurgence of interest in the study of these numbers and, more importantly, in the study of functions defined on the ring of bicomplex numbers, which mimic the behavior of holomorphic functions of a complex variable.

While the algebra of bicomplex numbers is a four-dimensional real algebra, it is useful to think of it as a “complexification” of the field of complex 

numbers; from this perspective, the bicomplex algebra possesses the properties of a one-dimensional theory inside four real dimensions. Its rich analysis and innovative geometry provide new ideas and potential applications in relativity and quantum mechanics alike.

The book will appeal to researchers in the fields of complex, hypercomplex and functional analysis, as well as undergraduate and graduate students with an interest in one- or multidimensional complex analysis.

Keywords

bicomplex elementary functions bicomplex holomorphy bicomplex numbers complex holomorphic functions of 2 complex variables geometry in 4­‐dimensional space hyperbolic numbers hyperbolic-valued norm

Authors and affiliations

  • M. Elena Luna-Elizarrarás
    • 1
  • Michael Shapiro
    • 2
  • Daniele C. Struppa
    • 3
  • Adrian Vajiac
    • 4
  1. 1.Escuela Sup. de Física y MatemáticasInstituto Politécnico NacionalMexico CityMexico
  2. 2.Escuela Sup. de Física y MatemáticasInstituto Politécnico NacionalMexico CityMexico
  3. 3.Schmid College of Science and TechnologyChapman UniversityOrangeUSA
  4. 4.Schmid College of Science and TechnologyChapman University Schmid College of Science and TechnologyORANGEUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-24868-4
  • Copyright Information Springer International Publishing Switzerland 2015
  • Publisher Name Birkhäuser, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-24866-0
  • Online ISBN 978-3-319-24868-4
  • Series Print ISSN 1660-8046
  • Series Online ISSN 1660-8054
  • Buy this book on publisher's site
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