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Bicomplex holomorphic functions. The algebra, geometry and analysis of bicomplex numbers. (English) Zbl 1345.30002

Frontiers in Mathematics. Cham: Birkhäuser/Springer (ISBN 978-3-319-24866-0/pbk; 978-3-319-24868-4/ebook). viii, 231 p. (2015).
Alternatively to W. R. Hamilton’s theory [“On quaternions, or on a new system of imaginaries in algebra”, Philos. Mag., III. Ser. 25, 489–495 (1844)] of noncommutative quaternions, J. Cockle [“On certain functions resembling quaternions and on a new imaginary in algebra”, Philos. Mag., III. Ser. 33, 43–59 (1848)] considered the ring of commutative quaternions which he called tessarines. Only in [Math. Ann. 40, 413–467 (1892; JFM 24.0640.01)] C. Segre considered the same objects and called them bicomplex numbers. The appearance of zero-divisors was the price paid for the commutativity. Otherwise it is easy to give two idempotents: \( 1/2(1+ij)\) and \(1/2(1-ij)\), which is very useful for the development of some kind of function theory.
The aim of the authors is to give a comprehensive description and to realize a detailed consideration of bicomplex numbers from the point of view of algebra, geometry and even analysis (in the sense of some kind of function theory). For the first time a comprehensive treatment of the bicomplex numbers, their geometric and algebraic properties up to analysis and function theoretic aspects, is presented.
In Chapter 1 one can find the definition of bicomplex numbers \(\mathbb{BH}\), \((ij=ji=k, i^2=j^2=-1, k^2=1)\). This definition differs from the definition usually given for instance in “https://en.wikipedia.org/wiki/Bicomplex number” (\(ij=ji=k, i^2=k^2=-1, j^2=1\)). The definition is equal to the so-called cotangerines. But it is also proved that both classes are isomorphic. The class of bicomplex numbers also contains the subclass of hyperbolic numbers (\(=\) bireal numbers or duplex numbers). Conjugation, moduli, Euclidean norm, invertibility, zero-divisors and the representation by idempotents of bicomplex numbers are discussed.
In Chapter 2 the ring of bicomplex numbers \(\mathbb{BC}\) is defined. Linear spaces and modules in \(\mathbb{BC}\) are considered. Corresponding real and complex algebras are introduced. Matrix representations can be found analogously to the complex numbers. Several bilinear forms with factorizations are presented. Even a partial ordering on the set of hyperbolic numbers is studied. With the help of the pair of idempotents for any bicomplex number \(Z=\alpha{\mathbf e}+\beta{\mathbf e}^\dagger\) the “hyperbolic norm” \(|Z|_k=|\alpha|{\mathbf e}+|\beta|{\mathbf e}^\dagger\) is introduced.
Chapter 3 is devoted to geometric and trigonometric representations of bicomplex numbers. The authors give the reader a nice visual impression of “seeing” the 4-dimensional world. Bicomplex numbers are represented in complex and alternatively also in hyperbolic terms. For all the results the representation of bicomplex numbers with a pair of idempotents plays a crucial role.
In Chapter 4 lines and curves in \(\mathbb{BC}\) are discussed. Straight lines in \(\mathbb{BC}\) are discussed very detailed. For instance there is an answer to the question: Can any two-dimensional plane in \(\mathbb{BC} \) be seen as a complex line in \(\mathbb{BC}\)? Six conditions define hyperbolic lines in \(\mathbb{BC}\). Parametric representations of hyperbolic lines are described completely. Consequently, it follows a subsection on hyperbolic and complex curves in \(\mathbb{BC}\). It is very interesting (also from the topological point of view) to learn more on bicomplex spheres and balls with hyperbolic radius. The multiplicative group structure of bicomplex spheres is considered. Some remarks on limits and continuity are needed for a bicomplex analysis (short Chapter 5). One has to consider special features.
In Chapter 6 elementary bicomplex functions (polynomials, exponential functions, logarithm functions, trigonometric functions) are introduced, and their properties are dicussed. In particular an analogue of the Fundamental Theorem of Algebra is formulated. Again the representation by a pair of (orthogonal) idempotents is essentially used. Many properties are similar to the complex case, but there are also particularities.
Bicomplex derivability and differentiability form the main content of Chapter 7. The proofs in this chapter show lots of difficulties. One has to be very careful in the details. For instance in any theorem on this topic, one has to consider the role of zero-divisors. It is interesting to read the interplay of real differentiability and the derivability of bicomplex functions. The authors also discuss the relations between bicomplex holomorphy and the holomorphy in two variables. Translations of differential expressions written in cartesian coordinates to the corresponding expressions in the idempotent coordinates are described.
Chapter 8 is devoted to the study of properties of bicomplex holomorphic functions. In particular the zeroes of such functions are considered. Again zero-divisors play an important role. Relations between bicomplex, complex and hyperbolic holomorphies and anti-holomorphies are discussed. A geometric interpretation of the derivative is given and even a “bicomplex Riemann Mapping Theorem” is formulated.
Any theory of holomorphic functions is connected with harmonic functions – the null-solutions of the Laplacian. In the bicomplex approach (Chapter 9), there are three candidates for Laplacians: the \(\mathbb{C}({\mathbf i})\)-Laplacian, the \(\mathbb{C}({\mathbf j})\)-Laplacian (complex Laplacians) and the hyperbolic Laplacian. Relations between null solutions of these Laplacians and \(\mathbb{BC}\)-holomorphic functions are studied. Hyperbolic and complex analogues to the wave operator are presented. Corresponding conjugate harmonic functions are derived.
In Chapter 10 the theory of bicomplex Taylor series is discussed. Using the representation of bicomplex numbers in pairs of idempotents, the results are consequences of the classical complex theory in connection with results presented before. Nevertheless, one has to be careful in the transition of classical contents.
In the last chapter a Stokes’ formula related to the bicomplex Cauchy-Riemann opertors is derived. In particular also a bicomplex Borel-Pompeiu formula is presented. The integration theory is to understood in the context of complexified Clifford analysis.
In summary, 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. In a monolithic way similarities and differences with the classical theory of one complex variable are expressed. Possible applications in physics (relativity, quantum mechanics) are mentioned. The book is recommended as a text book for supplementary courses in complex analysis for undergraduate and graduate students and also for self studies.

MSC:

30-02 Research exposition (monographs, survey articles) pertaining to functions of a complex variable
30G35 Functions of hypercomplex variables and generalized variables

Citations:

JFM 24.0640.01
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