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Cousin’s problems I and II: the bicomplex case

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Abstract

We present the bicomplex versions of Cousin’s problems I and II as well as their relationship with the bicomplex versions of Weierstrass’ and Mittag–Leffler’s theorems. We establish relations between these theorems and Cousin’s problems, which reveal peculiarities of bicomplex meromorphic function theory.

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References

  1. Alpay, D., Luna-Elizarrarás, M.E., Shapiro, M., Struppa, D.C.: Basics of Functional Analysis with Bicomplex Scalars, and Bicomplex Schur Analysis. Springer Briefs in Mathematics. Springer, Berlin (2014)

    Book  Google Scholar 

  2. Bamdad, H., Vahidi, A.: Extension functors of Cousin cohomology modules. Bull. Iran. Math. Soc. 44, 253–267 (2018). https://doi.org/10.1007/s41980-018-0009-x

    Article  MathSciNet  Google Scholar 

  3. Bers, L.: Introduction to Several Complex Variables. Courant Institute of Mathematical Sciences. New York University, New York (1962–1963)

  4. Charak, K.S., Rochon, D.: On Factorization of Bicomplex Meromorphic Functions. Trends in Mathematics, pp. 55–68. Birkhäuser, Basel (2008)

    Google Scholar 

  5. Charak, K.S., Rochon, D., Sharma, N.: Normal families of bicomplex meromorphic functios. Ann. Polinici Math. (2011). https://doi.org/10.4064/ap103-3-6

    Article  Google Scholar 

  6. Colombo, F., Sabadini, F., Struppa, D.C., Vajiac, A., Vajiac, M.B.: Singularities of functions of one and several bicomplex variables. Ark. Mat. (2011). https://doi.org/10.1007/s11512-010-0126-0

    Article  MathSciNet  Google Scholar 

  7. Conway, J.B.: Functions of One Complex Variable. Springer, New York (2000)

    Google Scholar 

  8. Luna-Elizarrarás, M.E., Pérez-Regalado, C.O., Shapiro, M.: On the Laurent series for bicomplex holomorphic functions. Complex Var. Elliptic Equ. 62(9), 1266–1286 (2017). https://doi.org/10.1080/17476933.2016.1250404

    Article  MathSciNet  Google Scholar 

  9. Luna-Elizarrarás, M.E., Pérez-Regalado, C.O., Shapiro, M.: Singularities of bicomplex holomorphic functions. Math. Methods Appl. Sci. (2021). https://doi.org/10.1002/mma.7522

    Article  Google Scholar 

  10. Luna-Elizarrarás, M.E., Shapiro, M., Struppa, D.C., Vajiac, A.: Bicomplex Holomorphic Functions: The Algebra, Geometry and Analysis. Frontiers in Mathematics. Birkhäuser, Basel (2015)

    Book  Google Scholar 

  11. Reséndis, O.L.F., Téllez, G., Tovar, L.M.: Directional derivatives and Stolz condition for bicomplex holomorphic functions. In: Cerejeiras, P., Reissig, M., Sabadini, I., Toft, J. (eds.) Current Trends in Analysis, Its Applications and Computation. Trends in Mathematics Research Perspectives, pp. 201–208. Birkhauser, Basel (2021)

    Google Scholar 

  12. Rönn, S.: Bicomplex Algebra and Function Theory (2018)

  13. Silverman, R.A.: Introductory Complex Analysis. Dover Publications, New York (1972)

    Google Scholar 

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Authors and Affiliations

  1. Instituto Politécnico Nacional, Escuela Superior de Física y Matemáticas, Edif. 9, U.P. Adolfo López Mateos, C.P. 07338, Mexico City, Mexico

    Yesenia Bravo Ortega & Luis Manuel Tovar Sánchez

  2. Universidad Autónoma Metropolitana, Unidad Azcapotzalco, Área de Análisis Matemático y sus Aplicaciones, C.P. 02128, Mexico City, Mexico

    Lino Feliciano Reséndis Ocampo

Authors
  1. Yesenia Bravo Ortega
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  2. Lino Feliciano Reséndis Ocampo
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  3. Luis Manuel Tovar Sánchez
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Correspondence to Luis Manuel Tovar Sánchez.

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For Enrique Ramírez de Arellano on the 52 anniversary (a Mayan Calendar Round) of the Henkin-Ramirez kernel.

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Bravo Ortega, Y., Reséndis Ocampo, L.F. & Tovar Sánchez, L.M. Cousin’s problems I and II: the bicomplex case. Bol. Soc. Mat. Mex. 29 (Suppl 1), 99 (2023). https://doi.org/10.1007/s40590-023-00547-7

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  • DOI: https://doi.org/10.1007/s40590-023-00547-7

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