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Poincaré–Bertrand and Hilbert formulas for the Cauchy–Cimmino singular integrals

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Abstract

The Cimmino system offers a natural and elegant generalization to four-dimensional case of the Cauchy–Riemann system of first order complex partial differential equations. Recently, it has been proved that many facts from the holomorphic function theory have their extensions onto the Cimmino system theory. In the present work a Poincaré–Bertrand formula related to the Cauchy–Cimmino singular integrals over piecewise Lyapunov surfaces in \(\mathbb {R}^4\) is derived with recourse to arguments involving quaternionic analysis. Furthermore, this paper obtains some analogues of the Hilbert formulas on the unit 3-sphere and on the 3-dimensional space for the theory of Cimmino system.

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Change history

  • 19 February 2019

    Unfortunately, the name of the communicating editor has been published incorrectly. The correct name is Rafał Abłamowicz

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Authors and Affiliations

  1. SEPI-ESIME-Zacatenco, Instituto Politécnico Nacional, 07738, CDMX, Mexico

    Juan Bory Reyes

  2. Facultad de Informática y Matemática, Universidad de Holguín, Holguín, 80100, Cuba

    Ricardo Abreu Blaya

  3. Departamento de Matemáticas, Instituto Tecnológico Autónomo de México, 01080, CDMX, Mexico

    Marco Antonio Pérez-de la Rosa

  4. Department of Mathematics, Faculty of Sciences and Literature, Izmir University of Economics, 35330, Izmir, Turkey

    Baruch Schneider

Authors
  1. Juan Bory Reyes
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  2. Ricardo Abreu Blaya
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  3. Marco Antonio Pérez-de la Rosa
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  4. Baruch Schneider
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Corresponding author

Correspondence to Marco Antonio Pérez-de la Rosa.

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Communicated by Rafa Abamowicz

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Bory Reyes, J., Abreu Blaya, R., Pérez-de la Rosa, M.A. et al. Poincaré–Bertrand and Hilbert formulas for the Cauchy–Cimmino singular integrals. Adv. Appl. Clifford Algebras 27, 2933–2960 (2017). https://doi.org/10.1007/s00006-017-0809-8

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  • DOI: https://doi.org/10.1007/s00006-017-0809-8

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