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- Title
Hyperholomorphicity by Proposing the Corresponding Cauchy–Riemann Equation in the Extended Quaternion Field.
- Authors
Kim, Ji-Eun
- Abstract
In algebra, the sedenions, an extension of the octonion system, form a 16-dimensional noncommutative and nonassociative algebra over the real numbers. It can be expressed as two octonions, and a function and differential operator can be defined to treat the sedenion, expressed as two octonions, as a variable. By configuring elements using the structure of complex numbers, the characteristics of octonions, the stage before expansion, can be utilized. The basis of a sedenion can be simplified and used for calculations. We propose a corresponding Cauchy–Riemann equation by defining a regular function for two octonions with a complex structure. Based on this, the integration theorem of regular functions with a sedenion of the complex structure is given. The relationship between regular functions and holomorphy is presented, presenting the basis of function theory for a sedenion of the complex structure.
- Subjects
NONASSOCIATIVE algebras; CAUCHY-Riemann equations; QUATERNIONS; COMPLEX numbers; DIFFERENTIAL operators; CAYLEY numbers (Algebra); NONCOMMUTATIVE algebras
- Publication
Axioms (2075-1680), 2024, Vol 13, Issue 5, p291
- ISSN
2075-1680
- Publication type
Article
- DOI
10.3390/axioms13050291