Differentiable functions in a three-dimensional associative noncommutative algebra

We consider a three-dimensional associative noncommutative algebra Ã2 over the field C, which contains the algebra of bicomplex numbers B(C) as a subalgebra. In this paper we consider functions of the form Φ(ζ)=f1(ξ1, ξ2,ξ3)I1+ f2(ξ1, ξ2,ξ3)I2+ f3(ξ1, ξ2,ξ3)ρ of the variable ζ= ξ1I1+ ξ2I2+ ξ3ρ, where ξ1, ξ2, ξ3 are independent complex variables and f1, f2, f3 are holomorphic functions of three complex variables. We construct in an explicit form all functions defined by equalities dΦ =dζ·Φ´(ζ) or dΦ = Φ´(ζ) ·dζ. The obtained descriptions we apply to representation of the mentioned class of functions by series. Also we established integral representations of these functions.

Keywords:

noncommutative algebra, differentiable function, Cauchy-Riemann conditions, constructive description power series, integral representation,

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N. M. Krylov, On Rowan Hamilton's quaternions and the notion of monogenicity. Dokl. Akad. Nauk SSSR. 55(9) (1947) 799-800 (in Russian).
A. S. Meilikhzon, On the monogenicity of quaternions. Dokl. Akad. Nauk SSSR. 59(3) (1948) 431-434 (in Russian).
M. E. Luna-Elizarraras, M. Shapiro, A Survey on the (Hyper-) Derivatives in Complex, Quaternionic and Clifford Analysis. Milan J. Math. 79(2) (2001) 521-542.
V. V. Kravchenko, M. V. Shapiro, Integral representations for spatial models of mathematical physics. Pitman Research Notes in Mathematics, Addison Wesley Longman Inc. (1996).
F. Brackx, R. Delanghe, F. Sommen, Clifford Analysis. Pitman, London. (1982).
R. A. El-Nabulsi, Fractional Dirac operators and deformed field theory on Clifford algebra, 42 (2009) 2614-2622.
D. Baleanu, J. Restrepo, D. Suragan, A class of time-fractional Dirac type operators, Chaos, Solitons and Fractals. 143 (2021).
E. Study, Uber Systeme von complexen Zahlen. Gott. Nachr. (1889).
D. Alpay, M. E. Luna-Elizarraras, M. Shapiro, D. C. Struppa, Basics of functional analysis with bicomplex scalars, and bicomplex schur analysis. SpringerBriefs in Mathematics, Springer. (2014).
M. E. Luna-Elizarraras, M. Shapiro, D. C. Struppa, A. Vajiac, Bicomplex holomorphic functions: the algebra, geometry and analysis of bicomplex numbers. Springer. (2015).
T. Kuzmenko, V. Shpakivskyi, G-monogenic mappings in a three-dimensional noncommutative algebra. Complex Variable and Elliptic Equations (submitted).
S. N. Volovel'skaya, The experience of construction of elements of the theory of functions in a commutative associative system with three units. Zapiski Nauchno-Issledovatel'skogo Instituta Matematiki i Mehaniki i Har'kovskogo Matematicheskogo Obshchestva. 16 (1939) 143-157 (in Russian).
S. V. Gryshchuk, S. A. Plaksa, Monogenic functions in the biharmonic boundary value problem. Math. Meth. Appl. Sci. 39(11) (2016) 2939-2952.
S. V. Gryshchuk, S. A. Plaksa, Reduction of a Schwartz-type boundary value problem for biharmonic monogenic functions to Fredholm integral equations. Open Mathematics. 15(1) (2017) 374-381.
I. P. Mel'nichenko, S. A. Plaksa, Commutative algebras and spatial potential fields. Kiev: Inst. Math. NAS Ukraine. (2008) (in Russian).
V. S. Shpakivskyi, Hypercomplex method for solving linear differential levels with partial derivatives. Proc. of the IAMM NAS of Ukraine. 32 (2018) 147-168.
V. S. Vladimirov, Methods of the theory of functions of several complex variables. Moscow: Nauka. (1964) 414. (in Russian).
S. Bock and K. Gurlebeck, On a generalized Appell system and monogenic power series. Math. Meth. Appl. Sci. 33 (2010) 394-411.