Bicomplex Holomorphic Functions
The Algebra, Geometry and Analysis of Bicomplex Numbers
Authors: LunaElizarrarás, M.E., Shapiro, M., Struppa, D.C., Vajiac, A.
 Presents a comprehensive study of the analysis and geometry of bicomplex numbers
 Offers a fundamental reference work for the field of bicomplex analysis
 Develops a solid foundation for potential new applications in relativity, dynamical systems and quantum mechanics
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 About this book

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 realdimension four, while almost simultaneously James Cockle introduced a commutative fourdimensional 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 fourdimensional 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 onedimensional 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.
 Reviews

“This text is one of the very few books entirely dedicated to bicomplex numbers. The purpose of the book is to give an extensive description of algebraic, geometric and analytic aspects of bicomplex numbers. … The text is well written and selfcontained. It can be used as a comprehensive introduction to the algebra, the geometry and the analysis of bicomplex numbers.” (Alessandro Perotti, Mathematical Reviews, January, 2017)
“The authors present a very interesting contribution to the field of hypercomplex analysis. This work bundles all the individual results known from the literature and forms a rich theory of the algebra and geometry of bicomplex numbers and bicomplex functions. It is well written with many details and examples. … The book is recommended as a text book for supplementary courses in complex analysis for undergraduate and graduate students and also for self studies.” (Wolfgang Sprößig, zbMATH 1345.30002, 2016)
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Bibliographic Information
 Bibliographic Information

 Book Title
 Bicomplex Holomorphic Functions
 Book Subtitle
 The Algebra, Geometry and Analysis of Bicomplex Numbers
 Authors

 M. Elena LunaElizarrarás
 Michael Shapiro
 Daniele C. Struppa
 Adrian Vajiac
 Series Title
 Frontiers in Mathematics
 Copyright
 2015
 Publisher
 Birkhäuser Basel
 Copyright Holder
 Springer International Publishing Switzerland
 eBook ISBN
 9783319248684
 DOI
 10.1007/9783319248684
 Softcover ISBN
 9783319248660
 Series ISSN
 16608046
 Edition Number
 1
 Number of Pages
 VIII, 231
 Number of Illustrations and Tables
 23 b/w illustrations
 Topics