Bicomplex Holomorphic Functions: The Algebra, Geometry and by M. Elena Luna-Elizarrarás, Visit Amazon's Michael Shapiro

By M. Elena Luna-Elizarrarás, Visit Amazon's Michael Shapiro Page, search results, Learn about Author Central, Michael Shapiro, , Daniele C. Struppa, Adrian Vajiac

The objective of this booklet is to increase the rules of the speculation of holomorphicity at the ring of bicomplex numbers. consequently, the main target is on expressing the similarities with, and variations from, the classical idea of 1 advanced variable. the result's an easy but entire creation to the algebra, geometry and research of bicomplex numbers.

Around the center of the 19th century, a number of mathematicians (the most sensible recognized being Sir William Hamilton and Arthur Cayley) got interested in learning quantity structures that prolonged the sphere of advanced numbers. Hamilton famously brought the quaternions, a skew box in real-dimension 4, whereas nearly at the same time James Cockle brought a commutative 4-dimensional genuine algebra, which was once rediscovered in 1892 by means of Corrado Segre, who talked about his components as bicomplex numbers. some great benefits of commutativity have been observed via the advent of 0 divisors, whatever that for some time dampened curiosity during this topic. in recent times, due principally to the paintings of G.B. expense, there was a resurgence of curiosity within the learn of those numbers and, extra importantly, within the learn of capabilities outlined at the ring of bicomplex numbers, which mimic the habit of holomorphic features of a posh variable.

While the algebra of bicomplex numbers is a 4-dimensional actual algebra, it's helpful to consider it as a “complexification” of the sphere of complex

numbers; from this attitude, the bicomplex algebra possesses the houses of a one-dimensional concept inside of 4 genuine dimensions. Its wealthy research and cutting edge geometry offer new rules and capability purposes in relativity and quantum mechanics alike.

The ebook will attract researchers within the fields of complicated, hypercomplex and useful research, in addition to undergraduate and graduate scholars with an curiosity in a single- or multidimensional complicated analysis.

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

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1: The vertices of the four-dimensional cube. 6) has at least a solution Θ ∈ C(i).

Because of properties (i)–(iii) we will say that | · |k is the hyperbolic-valued (D-valued) norm on the BC-module BC. 13) where the norm of the product and the product of the norms are related with an inequality. We believe that one could say that the hyperbolic norm of bicomplex numbers is better suited to the algebraic structure of the latter although, of course, one has to allow hyperbolic values for the norm. 1. 21) 48 Chapter 2. Algebraic Structures of the Set of Bicomplex Numbers with |Z| the Euclidean norm of Z, then one has: √ 2 · |Z| · |W |k .

The (topological) boundary of the 4-dimensional cube is the union of the 3-dimensional cubes C0 , C1 , I0 , Ii , J0 , Jj , K0 and Kk . Proof. , are the “extreme cubes” when the described process applies to each of the following segments: [0, 1], [0, i], [0, j], [0, k]. In order to see more precisely how the vertices of the cubes are glued together, we will write down the vertices explicitly. 6: The cubes C0 and C1 and their vertices. The vertices of C0 are: 0, i, j, k, i + j, i + k, j + k, i + j + k.

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