Bicomplex Holomorphic Functions by Michael Shapiro
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.
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Choquet Integral and Monotone Sublinear Operators
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Exploring Information Geometry
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Lie Models for Spaces
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Hoop Algebras
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Mild Differentiability Conditions for Newton's Method in Banach Spaces
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Heat Kernel on Lie Groups and Maximally Symmetric Spaces
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Direct and Inverse Sturm-Liouville Problems
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Algebraic Approximation: A Guide to Past and Current Solutions
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Advances in the Theory of Varieties of Semigroups
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Developments of Harmonic Maps, Wave Maps and Yang-Mills Fields into Biharmonic Maps, Biwave Maps and Bi-Yang-Mills Fields
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Poncelet Porisms and Beyond
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Stability of Vector Differential Delay Equations
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Applications of Holomorphic Functions in Geometry
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A Primer for a Secret Shortcut to PDEs of Mathematical Physics
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Ulam’s Conjecture on Invariance of Measure in the Hilbert Cube
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Functional Analysis in Asymmetric Normed Spaces
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Topics in Uniform Approximation of Continuous Functions
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Infinite Matrices and their Finite Sections
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Oblique Derivative Problems for Elliptic Equations in Conical Domains
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Approximation of Additive Convolution-Like Operators
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Flag-transitive Steiner Designs
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Functions of Omega-Bounded Type
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q-Clan Geometries in Characteristic 2
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Lifting Modules
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Applied Pseudoanalytic Function Theory
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Homotopy Theory of C*-Algebras
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The Geometry of Filtering
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Functional Identities
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Module Theory, Extending Modules and Generalizations
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Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws
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Positive Solutions to Indefinite Problems
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Nonlocal Perimeter, Curvature and Minimal Surfaces for Measurable Sets
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Geometric Analysis of Quasilinear Inequalities on Complete Manifolds
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Special Functions and Generalized Sturm-Liouville Problems
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Variational Source Conditions, Quadratic Inverse Problems, Sparsity Promoting Regularization
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Global Well-posedness of Nonlinear Parabolic-Hyperbolic Coupled Systems
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Shafarevich-Tate Groups
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Variational and Monotonicity Methods in Nonsmooth Analysis
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Quaternionic Approximation
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Newton's Method: an Updated Approach of Kantorovich's Theory
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Introduction to Complex Theory of Differential Equations
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Extremum Problems for Eigenvalues of Elliptic Operators
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Artinian Modules over Group Rings
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Exponentially Convergent Algorithms for Abstract Differential Equations
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Geometric Qp Functions
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Schwarz-Pick Type Inequalities
“This text is one of the very few books entirely dedicated to bicomplex numbersThe 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)
Michael Shapiro, the editor, has published widely in the fields of linguistics, poetics, and semiotics. In the application of Peirce's theory of signs to language, his two Sense volumes - The Sense of Grammar and The Sense of Change - have been dubbed instant classics.
| SKU | Unavailable |
| ISBN 13 | 9783319248660 |
| ISBN 10 | 3319248669 |
| Title | Bicomplex Holomorphic Functions |
| Author | Daniele Struppa |
| Series | Frontiers In Mathematics |
| Condition | Unavailable |
| Binding Type | Paperback |
| Publisher | Birkhauser Verlag AG |
| Year published | 2015-12-18 |
| Number of pages | 231 |
| Cover note | Book picture is for illustrative purposes only, actual binding, cover or edition may vary. |
| Note | Unavailable |