Geodesic
On limiting values of Cauchy type integral in a harmonic algebra with two-dimensional radical
Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 67 (2013) no. 1.

Voir la notice de l'article provenant de la source Library of Science

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. 
@article{AUM_2013_67_1_a5,
     author = {Plaksa, S. A. and Shpakivskyi, V. S.},
     title = {On limiting values of {Cauchy} type integral in a harmonic algebra with two-dimensional radical},
     journal = {Annales Universitatis Mariae Curie-Sk{\l}odowska. Mathematica },
     publisher = {mathdoc},
     volume = {67},
     number = {1},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/AUM_2013_67_1_a5/}
}
TY  - JOUR
AU  - Plaksa, S. A.
AU  - Shpakivskyi, V. S.
TI  - On limiting values of Cauchy type integral in a harmonic algebra with two-dimensional radical
JO  - Annales Universitatis Mariae Curie-Skłodowska. Mathematica 
PY  - 2013
VL  - 67
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AUM_2013_67_1_a5/
LA  - en
ID  - AUM_2013_67_1_a5
ER  - 
%0 Journal Article
%A Plaksa, S. A.
%A Shpakivskyi, V. S.
%T On limiting values of Cauchy type integral in a harmonic algebra with two-dimensional radical
%J Annales Universitatis Mariae Curie-Skłodowska. Mathematica 
%D 2013
%V 67
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AUM_2013_67_1_a5/
%G en
%F AUM_2013_67_1_a5
Plaksa, S. A.; Shpakivskyi, V. S. On limiting values of Cauchy type integral in a harmonic algebra with two-dimensional radical. Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 67 (2013) no. 1. http://geodesic.mathdoc.fr/item/AUM_2013_67_1_a5/

[1] 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).

[2] 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).

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

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

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

[6] 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.

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

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

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

[10] 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.

[11] 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.