Bicomplex Holomorphic Functions

The Algebra, Geometry and Analysis of Bicomplex Numbers
Buch | Softcover
VIII, 231 Seiten
2015 | 1st ed. 2015
Springer International Publishing (Verlag)
978-3-319-24866-0 (ISBN)

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Bicomplex Holomorphic Functions - M. Elena Luna-Elizarrarás, Michael Shapiro, Daniele C. Struppa, Adrian Vajiac
58,84 inkl. MwSt

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 thatfor 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.

Introduction.- 1.The Bicomplex Numbers.- 2.Algebraic Structures of the Set of Bicomplex Numbers.- 3.Geometry and Trigonometric Representations of Bicomplex.- 4.Lines and curves in BC.- 5.Limits and Continuity.- 6.Elementary Bicomplex Functions.- 7.Bicomplex Derivability and Differentiability.- 8.Some properties of bicomplex holomorphic functions.- 9.Second order complex and hyperbolic differential operators.- 10.Sequences and series of bicomplex functions.- 11.Integral formulas and theorems.- Bibliography.

"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 self-contained. 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)

Erscheinungsdatum
Reihe/Serie Frontiers in Mathematics
Zusatzinfo VIII, 231 p. 23 illus.
Verlagsort Cham
Sprache englisch
Maße 168 x 240 mm
Themenwelt Mathematik / Informatik Mathematik Analysis
Mathematik / Informatik Mathematik Angewandte Mathematik
Naturwissenschaften Physik / Astronomie
Schlagworte bicomplex elementary functions • bicomplex holomorphy • bicomplex numbers • complex holomorphic functions of 2 complex variabl • complex holomorphic functions of 2 complex variables • Functions of a Complex Variable • geometry in 4-dimensional space • geometry in 4‐dimensional space • hyperbolic numbers • hyperbolic-valued norm • Mathematical Applications in the Physical Sciences • mathematics and statistics • Several Complex Variables and Analytic Spaces
ISBN-10 3-319-24866-9 / 3319248669
ISBN-13 978-3-319-24866-0 / 9783319248660
Zustand Neuware
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