Monogenic Functions in Commutative Algebras

  • Vitalii Shpakivskyi
Conference paper
Part of the Trends in Mathematics book series (TM)


Let An be an arbitrary n-dimensional commutative associative algebra over the field of complex numbers. Let e1 = 1, e2, e3 be elements of An which are linearly independent over the field of real numbers. We consider monogenic (i.e., continuous and differentiable in the sense of Gateaux) functions of the variable xe1 + ye2 + ze3, where x, y, z are real, and obtain a constructive description of all mentioned monogenic functions by means of holomorphic functions of complex variables. It follows from this description that monogenic functions have Gateaux derivatives of all orders. The relations between monogenic functions and partial differential equations are investigated.



This research is partially supported by Grant of Ministry of Education and Science of Ukraine (Project No. 0116U001528).


  1. 1.
    P.W. Ketchum, Analytic functions of hypercomplex variables. Trans. Am. Math. Soc. 30(4), 641–667 (1928)MathSciNetCrossRefGoogle Scholar
  2. 2.
    M.N. Roşculeţ, Algebre infinite asociate la ecuaţii cu derivate parţiale, omogene, cu coeficienţi constanţi de ordin oarecare. Studii şi Cercetǎri Matematice 6(3–4), 567–643 (1955)Google Scholar
  3. 3.
    M.N. Roşculeţ, Algebre infinite, comutative, asociate la sisteme de ecuaţii cu derivate parţiale. Studii şi Cercetǎri Matematice 7(3–4), 321–371 (1956)Google Scholar
  4. 4.
    V.S. Shpakivskyi, T.S. Kuzmenko, Monogenic functions of double variable. Zb. Pr. Inst. Mat. NAN Ukr. 10(4–5), 372–378 (2013)zbMATHGoogle Scholar
  5. 5.
    J.C. Vignaux, Y. Durañona, A. Vedia, Sobre la teoria de las funciones de una variable compleja hiperbolica. Univ. Nac. La Plata. Publ. Fac. Ci. fis. mat. 104, 139–183 (1935)zbMATHGoogle Scholar
  6. 6.
    A.E. Motter, M.A.F. Rosa, Hyperbolic calculus. Adv. Appl. Clifford Alg. 8(1), 109–128 (1998)MathSciNetCrossRefGoogle Scholar
  7. 7.
    F. Messelmi, Analysis of dual functions. Annu. Rev. Chaos Theory Bifurcations Dyn. Syst. 4, 37–54 (2013)Google Scholar
  8. 8.
    M.E. Luna-Elizarrarás, M. Shapiro, D.C. Struppa, A. Vajiac, Bicomplex Holomorphic Functions: The Algebra, Geometry and Analysis of Bicomplex Numbers (Birkhäuser, Basel, 2015)Google Scholar
  9. 9.
    S.A. Plaksa, R.P. Pukhtaievych, Monogenic functions in a finite-dimensional semi-simple commutative algebra. An. Şt. Univ. Ovidius Constanţa. 22(1), 221–235 (2014)MathSciNetzbMATHGoogle Scholar
  10. 10.
    S.V. Grishchuk, S.A. Plaksa, Monogenic functions in a biharmonic algebra. Ukr. Math. J. 61(12), 1865–1876 (2009)MathSciNetCrossRefGoogle Scholar
  11. 11.
    S.A. Plaksa, V.S. Shpakovskii, Constructive description of monogenic functions in a harmonic algebra of the third rank. Ukr. Math. J. 62(8), 1251–1266 (2011)MathSciNetCrossRefGoogle Scholar
  12. 12.
    S.A. Plaksa, R.P. Pukhtaevich, Constructive description of monogenic functions in a three-dimensional harmonic algebra with one-dimensional radical. Ukr. Math. J. 65(5), 740–751 (2013)MathSciNetCrossRefGoogle Scholar
  13. 13.
    I.P. Mel’nichenko, The representation of harmonic mappings by monogenic functions. Ukr. Math. J. 27(5), 499–505 (1975)MathSciNetCrossRefGoogle Scholar
  14. 14.
    E. Cartan, Les groupes bilinéares et les systèmes de nombres complexes. Annales de la faculté des sciences de Toulouse 12(1), 1–64 (1898)MathSciNetCrossRefGoogle Scholar
  15. 15.
    V.S. Shpakivskyi, Constructive description of monogenic functions in a finite-dimensional commutative associative algebra. Adv. Pure Appl. Math. 7(1), 63–75 (2016)MathSciNetzbMATHGoogle Scholar
  16. 16.
    V.S. Shpakivskyi, Monogenic functions in finite-dimensional commutative associative algebras. Zb. Pr. Inst. Mat. NAN Ukr. 12(3), 251–268 (2015)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Vitalii Shpakivskyi
    • 1
  1. 1.Institute of Mathematics of the National Academy of Sciences of UkraineKyivUkraine

Personalised recommendations