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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 function of this ebook is to improve the rules of the idea of holomorphicity at the ring of bicomplex numbers. for this reason, the main target is on expressing the similarities with, and alterations from, the classical conception of 1 advanced variable. the result's an ordinary but entire creation to the algebra, geometry and research of bicomplex numbers.

Around the center of the 19th century, a number of mathematicians (the top identified being Sir William Hamilton and Arthur Cayley) took an interest in learning quantity structures that prolonged the sector of complicated numbers. Hamilton famously brought the quaternions, a skew box in real-dimension 4, whereas nearly concurrently James Cockle brought a commutative 4-dimensional genuine algebra, which was once rediscovered in 1892 by way of Corrado Segre, who noted his parts as bicomplex numbers. the benefits of commutativity have been followed via the advent of 0 divisors, whatever that for your time dampened curiosity during this topic. lately, due mostly to the paintings of G.B. expense, there was a resurgence of curiosity within the examine of those numbers and, extra importantly, within the learn of capabilities outlined at the ring of bicomplex numbers, which mimic the habit of holomorphic services of a fancy variable.

While the algebra of bicomplex numbers is a 4-dimensional genuine algebra, it really is valuable to think about it as a “complexification” of the sector of complex

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

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

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

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56 Chapter 3. 7: The cubes I0 and Ii and their vertices. 8: The cubes J0 and Jj and their vertices. 2. 9: The cubes K0 and Kk and their vertices. 1 shows: • the vertices of the 4-dimensional cube (there are sixteen of them). • The vertices of each of its 3-dimensional sides. • At which vertices the 3-dimensional sides intersect. Observe that the first column gives all the vertices and the other columns show which cube corresponds to each vertex; besides, each row shows the cubes with a fixed vertex.

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.

This isomorphism sends the bicomplex numbers 1 and i into the canonical basis in C2 ( j) and it induces the following isomorphism (of real linear spaces) between R4 and C2 ( j): R4 (x1 , y1 , x2 , y2 ) −→ (x1 + j x2 , y1 + j y2 ) ∈ C2 ( j) . 6) are different: this shows once again that inside BC the “complex sets” C2 (i) and C2 ( j) play distinct roles. 2. Linear spaces and modules in BC 31 One more difference that one notes considering BC as a C(i)- or a C( j)-linear space is, for example, that the set {1, i} is linearly independent when BC is seen as a C( j)-linear space, but the same set is linearly dependent in the C(i)-linear space BC.

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