Abstract
In this paper, we give an extended quaternion as a matrix form involving complex components. We introduce a semicommutative subalgebra ℂ ℂ 2 of the complex matrix algebra M 4 , ℂ . We exhibit regular functions defined on a domain in ℂ 4 but taking values in ℂ ℂ 2 . By using the characteristics of these regular functions, we propose the corresponding Cauchy–Riemann equations. In addition, we demonstrate several properties of these regular functions using these novel Cauchy–Riemann equations. Mathematical Subject Classification is 32G35, 32W50, 32A99, and 11E88.
Highlights
Introduced by Hamilton in 1894, quaternions form an algebra generated as a noncommutative division algebra
From the notion of regularity over C(C2), we propose corresponding Cauchy–Riemann equations and several properties of regular functions in C(C2)
Let us define the regular functions to be applied in C(C2) by using the differential operator D k(Dk)(k 1, 2, 3, 4) for functions defined in C(C2)
Summary
Introduced by Hamilton in 1894, quaternions form an algebra generated as a noncommutative division (associative) algebra. Fueter [4] defined regular functions over the quaternion field identified with R4 Based on this definition, Fueter investigated a generalization of the Cauchy–Riemann equations in the complex holomorphic function theory. Using a generalization of the Cauchy–Riemann equation, Ryan [6] introduced a regularity of quaternion-valued functions and developed a regular function theory of complex Clifford algebra. Sommen [13] constructed kernels and monogenic and holomorphic functions, leading to connections between the theory of holomorphic functions of several variables and the theory of monogenic functions Based on these studies, in this paper, we propose a form distinct from the previously attempted expansion of quaternions.
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