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.
@article{bwmeta1.element.doi-10_2478_v10062-012-0022-0,
author = {S. A. Plaksa and V. S. Shpakivskyi},
title = {On limiting values of Cauchy type integral in a harmonic algebra with two-dimensional radical},
journal = {Annales UMCS, Mathematica},
volume = {67},
year = {2013},
pages = {57-64},
zbl = {1284.30052},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_v10062-012-0022-0}
}
S. A. Plaksa; V. S. Shpakivskyi. On limiting values of Cauchy type integral in a harmonic algebra with two-dimensional radical. Annales UMCS, Mathematica, Tome 67 (2013) pp. 57-64. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_v10062-012-0022-0/
[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). | Zbl 0041.39707
[2] Salaev, V. V., Direct and inverse estimates for a singular Cauchy integral along aclosed curve, Mat. Zametki 19, no. 3 (1976), 365-380 (Russian). | Zbl 0345.44006
[3] Gerus, O. F., Finite-dimensional smoothness of Cauchy-type integrals, Ukrainian Math. J. 29, no. 5 (1977), 490-493. | Zbl 0429.30035
[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. | Zbl 0411.30026
[5] Ketchum, P. W., Analytic functions of hypercomplex variables, Trans. Amer. Math. Soc. 30 (1928), 641-667.[Crossref] | Zbl 55.0787.02
[6] Kunz, K. S., Application of an algebraic technique to the solution of Laplace’s equationin three dimensions, SIAM J. Appl. Math. 21, no. 3 (1971), 425-441.[Crossref] | Zbl 0235.31007
[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 threedimensionalLaplace 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 logarithmicorder on a spiral contour. I, Ukrainian Math. J. 42, no. 11 (1990), 1509-1517. | Zbl 0719.30029
[11] Shpakivskyi, V. S., Plaksa, S. A., Integral theorems and a Cauchy formula in a commutativethree-dimensional harmonic algebra, Bull. Soc. Sci. Lett. Łodź S´er. Rech. D´eform. 60 (2010), 47-54. | Zbl 1229.30026