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## Archive of Issues

Uzbekistan Tashkent
Year
2019
Volume
29
Issue
4
Pages
548-557
 Section Mathematics Title On volumes of matrix ball of third type and generalized Lie balls Author(-s) Rakhmonov U.S.a, Abdullayev J.Sh.b Affiliations Tashkent State Technical Universitya, National University of Uzbekistanb Abstract The third-type matrix ball and the generalized Lie ball that are connected with classical domains play a crucial role in the theory of several complex variable functions. In this paper the volumes of the third type matrix ball and the generalized Lie ball are calculated. The full volumes of these domains are necessary for finding kernels of integral formulas for these domains (kernels of Bergman, Cauchy-Szegö, Poisson etc.). In addition, it is used for the integral representation of a function holomorphic on these domains, in the mean value theorem and other important concepts. The results obtained in this article are the general case of results of Hua Lo-ken and his results in particular cases coincides with our results. Keywords classical domains, matrix ball of the first type, matrix balls of the second type, matrix balls of the third type, generalized Lie ball UDC 517.55 MSC 32A07, 32A50, 97I50, 97I60, 97I80 DOI 10.20537/vm190406 Received 26 July 2019 Language English Citation Rakhmonov U.S., Abdullayev J.Sh. On volumes of matrix ball of third type and generalized Lie balls, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2019, vol. 29, issue 4, pp. 548-557. References Hua L.-K. Garmonicheskii analiz funktsii mnogikh kompleksnykh peremennykh v klassicheskikh oblastyakh (Harmonic analysis of functions of several complex variables in classical domains), Moscow: Inostr. Lit., 1959. Pyatetskii-Shapiro I I. Geometriya klassicheskikh oblastei i teoriya avtomorfnykh funktsii (Geometry of classical domains and the theory of automorphic functions), Moscow: Gos. Izd. Fiz. Mat. Lit., 1961. Siegel C.L. Avtomorfnye funktsii neskol'kikh kompleksnykh peremennykh (Automorphic functions of several complex variables), Moscow: Inostr. Lit., 1954. Khudayberganov G., Hidirov B.B., Rakhmonov U.S. Automorphisms of matrix balls. Acta NUUz, 2010, no. 3, pp. 205-210 (in Russian). Khudayberganov G., Kytmanov A.M., Shaimkulov B.A. Analiz v matrichnykh oblastyakh (Analysis in matrix domains), Krasnoyarsk: Siberian Federal University, 2017. Khalknazarov A. The volume of the matrix ball in the space of matrices, Uzbek Mathematical Journal, 2012, no. 3, pp. 135-139 (in Russian). Khudayberganov G.Kh., Otemuratov B.P., Rakhmonov U.S. Boundary Morera theorem for the matrix ball of the third type, Journal of Siberian Federal University. Mathematics and Physics, 2018, vol. 11, issue 1, pp. 40-45. https://doi.org/10.17516/1997-1397-2018-11-1-40-45 Shaimkulov B.A. On holomorphic extendability of functions from part of the Lie sphere to the Lie ball, Siberian Mathematical Journal, 2003, vol. 44, issue 6, pp. 1105-1110. https://doi.org/10.1023/B:SIMJ.0000007486.06343.be Khudayberganov G., Rakhmonov U.S., Matyakubov Z.Q. Integral formulas for some matrix domains, Topics in Several Complex Variables, AMS, 2016, pp. 89-95. https://doi.org/10.1090/conm/662 Myslivets S.G. On the Szegö and Poisson kernels in the convex domains in ${\mathbb C}$${n}$ , Russian Mathematics, 2019, vol. 63, issue 1, pp. 35-41. https://doi.org/10.3103/s1066369x19010043 Rakhmonov U.S. Poisson Kernel for a matrix ball of the third type, Uzbek Mathematical Journal, 2012, no. 3, pp. 123-125. Khudayberganov G., Rakhmonov U.S. The Bergman and Cauchy-Szego kernels for matrix ball of the second type, Journal of Siberian Federal University. Mathematics and Physics, 2014, vol. 7, issue 3, pp. 305-310. http://mi.mathnet.ru/eng/jsfu375 Lankaster P. The theory of matrices, New York-London: Academic Press, 1969. Gantmakher F.R. The theory of matrices, Chelsea Publishing Company, 1977. Full text