Abstract
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.
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Lorch E.R.: The theory of analytic function in normed abelin vector rings. Trans. Am. Math. Soc. 54, 414–425 (1943)
Blum E.K.: A theory of analytic functions in Banach algebras. Trans. Am. Math. Soc. 78, 343–370 (1955)
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)
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)
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)
Gončarov V.: Sur l’intégrale de Cauchy dans le domaine hypercomplexe. Bull. Acad. Sci. URSS Cl. Sci. Math. 10, 1405–1424 (1932)
Ketchum P.W.: Analytic functions of hypercomplex variables. Trans. Am. Math. Soc. 30(4), 641–667 (1928)
Ketchum P.W.: A complete solution of Laplace’s equation by an infinite hypervariable. Am. J. Math. 51, 179–188 (1929)
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)
Roşculeţ M.N.: Algebre liniare asociative şi comutative şi fincţii monogene ataşate lor. Stud. Cercet. Mat. 6(1–2), 135–173 (1955)
Cartan E.: Les groupes bilinéares et les systèmes de nombres complexes. Ann. Fac. Sci. Toulouse 12(1), 1–64 (1989)
Burde D., de Graaf W.: Classification of Novicov algebras. Appl. Algebra Eng. Commun. Comput. 24(1), 1–15 (2013)
Burde D., Fialowski A.: Jacobi–Jordan algebras. Linear Algebra Appl. 459, 586–594 (2014)
Martin M.E.: Four-dimensional Jordan algebras. Int. J. Math. Game Theory Algebra 20(4), 41–59 (2013)
Shpakivskyi, V.S.: Constructive description of monogenic functions in a finite-dimensional commutative associative algebra. Adv. Pure Appl. Math. arXiv:1411.4643v1 (submitted)
Hille, E., Phillips, R.S.: Functional Analysis and Semi-Groups (Russian translation). Inostr. Lit., Moscow (1962)
Privalov, I.I.: Introduction to the Theory of Functions of a Complex Variable. GITTL, Moscow (1977) (Russian)
Shabat, B.V.: Introduction to Complex Analysis, Part 2. Nauka, Moskow (1976) (Russian)
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)
Shpakivskyi, V.S., Kuzmenko, T.S.: Integral theorems for the quaternionic G-monogenic mappings. An. Şt. Univ. Ovidius Constanţa (accepted)
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Shpakivskyi, V.S. Curvilinear Integral Theorems for Monogenic Functions in Commutative Associative Algebras. Adv. Appl. Clifford Algebras 26, 417–434 (2016). https://doi.org/10.1007/s00006-015-0561-x
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DOI: https://doi.org/10.1007/s00006-015-0561-x