On the notion of a quaternionic holomorphic function
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866917588939833344 |
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| author | Parfenov, Michael |
| author_facet | Parfenov, Michael |
| contents | A physically more adequate definition of a quaternionic holomorphic (H-holomorphic) function of one quaternionic variable compared to known ones and a quaternionic generalization of Cauchy-Riemann's equations are presented. At that a class of introduced H-holomorphic functions consists of those quaternionic functions whose left and right derivatives become equal after the transition to 3D space. The presented theory demonstrates a complete similarity of the algebraic properties and differentiation rules between the classes of H-holomorphic and ordinary complex holomorphic functions, including the fact that quaternionic multiplication of the H-holomorphic functions behaves as commutative and the fact that each H-holomorphic function can be created from its complex holomorphic analogue by replacing a complex variable by a quaternion one. A fairly large number of detailed examples are given to illustrate the presented theory efficiency. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_08487 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the notion of a quaternionic holomorphic function Parfenov, Michael Complex Variables 30G35 A physically more adequate definition of a quaternionic holomorphic (H-holomorphic) function of one quaternionic variable compared to known ones and a quaternionic generalization of Cauchy-Riemann's equations are presented. At that a class of introduced H-holomorphic functions consists of those quaternionic functions whose left and right derivatives become equal after the transition to 3D space. The presented theory demonstrates a complete similarity of the algebraic properties and differentiation rules between the classes of H-holomorphic and ordinary complex holomorphic functions, including the fact that quaternionic multiplication of the H-holomorphic functions behaves as commutative and the fact that each H-holomorphic function can be created from its complex holomorphic analogue by replacing a complex variable by a quaternion one. A fairly large number of detailed examples are given to illustrate the presented theory efficiency. |
| title | On the notion of a quaternionic holomorphic function |
| topic | Complex Variables 30G35 |
| url | https://arxiv.org/abs/2402.08487 |