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Fractional Elementary Bicomplex Functions in the Riemann–Liouville Sense

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Abstract

In this paper, we present the development of fractional bicomplex calculus in the Riemann–Liouville sense, based on the modification of the Cauchy–Riemann operator using the one-dimensional Riemann–Liouville derivative in each direction of the bicomplex basis. We introduce elementary functions such as analytic polynomials, exponential, trigonometric, and some properties of these functions. Furthermore, we present the fractional bicomplex Laplace operator connected with the fractional Cauchy–Riemann operator.

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Notes

  1. A function h is \({\mathbb {R}}\)–analytic in an open interval I if for every \(x\in I\), it admits a Taylor expansion with center x and converges in a neighborhood \(I_x\subset I\) of x.

  2. In fact, it converges uniformly over the entire complex plane.

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Acknowledgements

The authors would like to thank the reviewers for the many useful comments which lead to improvements in the paper.

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Correspondence to Antonio Di Teodoro.

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This article is part of the Topical Collection on Proceedings ICCA 12, Hefei, 2020, edited by Guangbin Ren, Uwe Kähler, Rafal Ablamowicz, Fabrizio Colombo, Pierre Dechant, Jacques Helmstetter, G. Stacey Staples, Wei Wang.

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Coloma, N., Di Teodoro, A., Ochoa-Tocachi, D. et al. Fractional Elementary Bicomplex Functions in the Riemann–Liouville Sense. Adv. Appl. Clifford Algebras 31, 63 (2021). https://doi.org/10.1007/s00006-021-01165-0

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