Abstract
We give a representation of the class of Cullen-regular functions in split-quaternions. We consider each Cullen’s form of split-quaternions, which provides corresponding Cauchy-Riemann equations for split-quaternionic variables. Using Cullen’s form, we research hyperholomorphy and the properties of functions of split-quaternionic variables which are expressed by hyperbolic coordinates.
Highlights
The skew field of real quaternions, denoted by H, has the formH = q | q = x + ix + jx + kx, xr ∈ R (r =, ), where R is the set of real numbers and i, j, and k are imaginary units with i = j = k =, ij = –ji = k, jk = –kj = i, ki = –ik = j.Theories and applications of functions of a quaternionic variable have been led by holomorphic functions of one complex variable
Split-quaternions are elements of four-dimensional algebra introduced by Cockle [ ]
We investigate the structure of a Cullen-regular function and corresponding split-Cauchy-Riemann systems for Cullenvariables and research properties of hyperholomorphic functions, represented by Cullenregular functions
Summary
Kim et al [ , ] obtained the regularity of functions on the form of reduced quaternions in Clifford analysis. Kim and Shon [ , ] researched corresponding Cauchy-Riemann systems and the properties of hyperholomorphic functions with values in modified split-quaternions.
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