Advances in Applied Clifford Algebras

, Volume 26, Issue 1, pp 417–434 | Cite as

Curvilinear Integral Theorems for Monogenic Functions in Commutative Associative Algebras



We consider an arbitrary finite-dimensional commutative associative algebra, \({\mathbb{A}_n^m}\), with unit, over the field of complex number with m idempotents. Let e 1 = 1,e 2,e 3 be elements of \({\mathbb{A}_n^m}\) 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 xe 1 + ye 2 + ze 3, where x,y,z are real. For mentioned monogenic function we prove curvilinear analogues of the Cauchy integral theorem, the Morera theorem and the Cauchy integral formula.

Mathematics Subject Classification

Primary 30G35 Secondary 30G12 


Commutative associative algebra Cauchy integral theorem Morera theorem Cauchy integral formula 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Lorch E.R.: The theory of analytic function in normed abelin vector rings. Trans. Am. Math. Soc. 54, 414–425 (1943)CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    Blum E.K.: A theory of analytic functions in Banach algebras. Trans. Am. Math. Soc. 78, 343–370 (1955)CrossRefMATHGoogle Scholar
  3. 3.
    Shpakivskyi V.S., Plaksa S.A.: Integral theorems and a Cauchy formula in a commutative three-dimensional harmonic algebra. Bull. Soc. Sci. Lett. Lódź 60, 47–54 (2010)MathSciNetMATHGoogle Scholar
  4. 4.
    Plaksa S.A., Shpakivskyi V.S.: Monogenic functions in a finite-dimensional algebra with unit and radical of maximal dimensionality. J. Algerian Math. Soc. 1, 1–13 (2014)Google Scholar
  5. 5.
    Plaksa S.A., Pukhtaievych R.P.: Constructive description of monogenic functions in n-dimensional semi-simple algebra. An. Şt. Univ. Ovidius Constanţa. 22(1), 221–235 (2014)MathSciNetGoogle Scholar
  6. 6.
    Gončarov V.: Sur l’intégrale de Cauchy dans le domaine hypercomplexe. Bull. Acad. Sci. URSS Cl. Sci. Math. 10, 1405–1424 (1932)Google Scholar
  7. 7.
    Ketchum P.W.: Analytic functions of hypercomplex variables. Trans. Am. Math. Soc. 30(4), 641–667 (1928)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Ketchum P.W.: A complete solution of Laplace’s equation by an infinite hypervariable. Am. J. Math. 51, 179–188 (1929)CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    Roşculeţ M.N.: O teorie a funcţiilor de o variabilă hipercomplexă în spaţiul cu trei dimensiuni. Stud. Cercet. Mat. 5(3–4), 361–401 (1954)MATHGoogle Scholar
  10. 10.
    Roşculeţ M.N.: Algebre liniare asociative şi comutative şi fincţii monogene ataşate lor. Stud. Cercet. Mat. 6(1–2), 135–173 (1955)MATHGoogle Scholar
  11. 11.
    Cartan E.: Les groupes bilinéares et les systèmes de nombres complexes. Ann. Fac. Sci. Toulouse 12(1), 1–64 (1989)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Burde D., de Graaf W.: Classification of Novicov algebras. Appl. Algebra Eng. Commun. Comput. 24(1), 1–15 (2013)CrossRefMATHGoogle Scholar
  13. 13.
    Burde D., Fialowski A.: Jacobi–Jordan algebras. Linear Algebra Appl. 459, 586–594 (2014)CrossRefMathSciNetMATHGoogle Scholar
  14. 14.
    Martin M.E.: Four-dimensional Jordan algebras. Int. J. Math. Game Theory Algebra 20(4), 41–59 (2013)Google Scholar
  15. 15.
    Shpakivskyi, V.S.: Constructive description of monogenic functions in a finite-dimensional commutative associative algebra. Adv. Pure Appl. Math. arXiv:1411.4643v1 (submitted)
  16. 16.
    Hille, E., Phillips, R.S.: Functional Analysis and Semi-Groups (Russian translation). Inostr. Lit., Moscow (1962)Google Scholar
  17. 17.
    Privalov, I.I.: Introduction to the Theory of Functions of a Complex Variable. GITTL, Moscow (1977) (Russian)Google Scholar
  18. 18.
    Shabat, B.V.: Introduction to Complex Analysis, Part 2. Nauka, Moskow (1976) (Russian)Google Scholar
  19. 19.
    Plaksa S.A., Pukhtaevich R.P.: Constructive description of monogenic functions in a three-dimensional harmonic algebra with one-dimensional radical. Ukr. Math. J. 65(5), 740–751 (2013)CrossRefMathSciNetMATHGoogle Scholar
  20. 20.
    Shpakivskyi, V.S., Kuzmenko, T.S.: Integral theorems for the quaternionic G-monogenic mappings. An. Şt. Univ. Ovidius Constanţa (accepted)Google Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Department of Complex Analysis and Potential TheoryInstitute of Mathematics of the National Academy of Sciences of UkraineKiev 4Ukraine

Personalised recommendations