Bicomplex Holomorphic Functions: The Algebra, Geometry and Analysis of Bicomplex Numbers

ISBN-13: 9783319248660 / Angielski / Miękka / 2015 / 231 str.

M. Elena Luna-Elizarraras; Michael Shapiro; Daniele C. Struppa

Opis

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.

Recenzje

"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)

Spis treści

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.