On limiting values of Cauchy type integral in a harmonic algebra with two-dimensional radical

S. A. Plaksa, V. S. Shpakivskyi

Abstract


We consider a certain analog of Cauchy type integral taking values in a three-dimensional harmonic algebra with two-dimensional radical. We establish sufficient conditions for an existence of limiting values of this integral on the curve of integration.

Keywords


Three-dimensional harmonic algebra; Cauchy type integral; limiting values; closed Jordan rectifiable curve.

Full Text:

PDF

References


Davydov, N. A., The continuity of an integral of Cauchy type in a closed domain, Dokl. Akad. Nauk SSSR 64, no. 6 (1949), 759–762 (Russian).

Salaev, V. V., Direct and inverse estimates for a singular Cauchy integral along a closed curve, Mat. Zametki 19, no. 3 (1976), 365–380 (Russian).

Gerus, O. F., Finite-dimensional smoothness of Cauchy-type integrals, Ukrainian Math. J. 29, no. 5 (1977), 490–493.

Gerus, O. F., Some estimates of moduli of smoothness of integrals of the Cauchy type, Ukrainian Math. J. 30, no. 5 (1978), 594–601.

Ketchum, P. W., Analytic functions of hypercomplex variables, Trans. Amer. Math. Soc. 30 (1928), 641–667.

Kunz, K. S., Application of an algebraic technique to the solution of Laplace’s equation in three dimensions, SIAM J. Appl. Math. 21, no. 3 (1971), 425–441.

Mel’nichenko, I. P., The representation of harmonic mappings by monogenic functions, Ukrainian Math. J. 27, no. 5 (1975), 499–505.

Mel’nichenko, I. P., Algebras of functionally invariant solutions of the threedimensional Laplace equation, Ukrainian Math. J. 55, no. 9 (2003), 1551–1559.

Mel’nichenko, I. P., Plaksa, S. A., Commutative algebras and spatial potential fields, Inst. Math. NAS Ukraine, Kiev, 2008 (Russian).

Plaksa, S. A., Riemann boundary-value problem with infinite index of logarithmic order on a spiral contour. I, Ukrainian Math. J. 42, no. 11 (1990), 1509–1517.

Shpakivskyi, V. S., Plaksa, S. A., Integral theorems and a Cauchy formula in a commutative three-dimensional harmonic algebra, Bull. Soc. Sci. Lett. Łódz Ser. Rech. Deform. 60 (2010), 47–54.




DOI: http://dx.doi.org/10.2478/v10062-012-0022-0
Date of publication: 2015-07-15 00:00:00
Date of submission: 2016-01-11 19:12:10


Statistics


Total abstract view - 673
Downloads (from 2020-06-17) - PDF - 434

Indicators



Refbacks

  • There are currently no refbacks.


Copyright (c) 2016 S. A. Plaksa, V. S. Shpakivskyi