# Multidimensional Analogs of the Cauchy–Riemann System and Representations of Their Solutions via Harmonic Functions

## Abstract

At present, there are numerous multidimensional generalizations of holomorphic vectors. The most general of these is the four-dimensional generalization of the Cauchy–Riemann system. In the present work, by introducing two quaternion functions and the notion of quaternion differentiation, we obtain, for the first time, a five-dimensional generalization of holomorphic vectors. By using the representation of holomorphic vectors via the quaternion harmonic function and its derivatives, we consider the Riemann–Hilbert problem and one problem in a layer. A new solution of the Riemann–Hilbert problem in the five-dimensional half space is obtained.

- Authors:

- Al-Farabi Kazakh National University (Kazakhstan)

- Publication Date:

- OSTI Identifier:
- 22771553

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Mathematical Sciences

- Additional Journal Information:
- Journal Volume: 229; Journal Issue: 2; Other Information: Copyright (c) 2018 Springer Science+Business Media, LLC, part of Springer Nature; http://www.springer-ny.com; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1072-3374

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICAL METHODS AND COMPUTING; CAUCHY PROBLEM; FOUR-DIMENSIONAL CALCULATIONS; HARMONICS; MANY-DIMENSIONAL CALCULATIONS; MATHEMATICAL SOLUTIONS; RIEMANN FUNCTION

### Citation Formats

```
Tokibetov, J. A., Abduakhitova, G. E., and Sarsekeeva, A. S.
```*Multidimensional Analogs of the Cauchy–Riemann System and Representations of Their Solutions via Harmonic Functions*. United States: N. p., 2018.
Web. doi:10.1007/S10958-018-3671-X.

```
Tokibetov, J. A., Abduakhitova, G. E., & Sarsekeeva, A. S.
```*Multidimensional Analogs of the Cauchy–Riemann System and Representations of Their Solutions via Harmonic Functions*. United States. doi:10.1007/S10958-018-3671-X.

```
Tokibetov, J. A., Abduakhitova, G. E., and Sarsekeeva, A. S. Thu .
"Multidimensional Analogs of the Cauchy–Riemann System and Representations of Their Solutions via Harmonic Functions". United States. doi:10.1007/S10958-018-3671-X.
```

```
@article{osti_22771553,
```

title = {Multidimensional Analogs of the Cauchy–Riemann System and Representations of Their Solutions via Harmonic Functions},

author = {Tokibetov, J. A. and Abduakhitova, G. E. and Sarsekeeva, A. S.},

abstractNote = {At present, there are numerous multidimensional generalizations of holomorphic vectors. The most general of these is the four-dimensional generalization of the Cauchy–Riemann system. In the present work, by introducing two quaternion functions and the notion of quaternion differentiation, we obtain, for the first time, a five-dimensional generalization of holomorphic vectors. By using the representation of holomorphic vectors via the quaternion harmonic function and its derivatives, we consider the Riemann–Hilbert problem and one problem in a layer. A new solution of the Riemann–Hilbert problem in the five-dimensional half space is obtained.},

doi = {10.1007/S10958-018-3671-X},

journal = {Journal of Mathematical Sciences},

issn = {1072-3374},

number = 2,

volume = 229,

place = {United States},

year = {2018},

month = {2}

}

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