Abstract
In this paper we study the sets of regularity on the boundary of the domain of convergence for a given multiple power series. As such sets we consider collections of polyarcs on the frames of polydisk of convergence of the series. In terms of an entire function, interpolating the coefficients of series, we find the sizes of polyarcs, constituting the regular set. To compute the sizes of polyarcs we essentially use the set of linear majorants for the logarithm of the module of interpolating entire function.
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References
L. Bieberbach, Analytic Continuation (Nauka, Moscow, 1967).
N. U. Arakelyan and V. A. Martirosyan, “Localization of singularities on the boundary of the circle of convergence”, Soviet J. Contemp. Math. Anal., 22(1), 3–21, 1987.
N. U. Arakelian and V. A. Martirosian, “The location of singularities of power series on the circle of convergence II”, Soviet J. Contemp. Math. Anal., 23(3), 123–137, 1988.
N. U. Arakelian, “Approximation by entire functions and analytic continuation”, Computational Mathematical Series, 19, 295–313, 1992.
N. Arakelian, W. Luh, J. Muller, “On the localization of singularities of lacunar power series”, Complex Variables and Elliptic Equations, 52(7), 561–573, 2007.
V. V. Shabat, Introduction to Complex Analysis (Nauka, Moscow, 1987).
V. K. Ivanov, “A characterization of the growth of an entire function of two variables and its application to the summation of double power series”, Mat. sbornik, 47(89)(1), 3–16, 1959.
L. I. Ronkin, Introduction to the Theory of Entire Functions of Several Variables (Nauka, Moscow, 1971).
I. A. Antipova, “Inversion of many-dimensional Mellin transforms and solutions of algebraic equations”, Mat. sbornik, 198(4), 3–20, 2007.
T. M. Sadykov, A. K. Tsikh, Hypergeometric and Algebraic Functions of Several Variables (Nauka, Moscow, 2014).
Yu. V. Sidorov, M. V. Fedoryuk, M. I. Shubin, Lectures on Theory of Complex Variable (Nauka, Moscow, 1989).
A. K. Tsikh, Multidimensional Residues and Their Applications (AMS. 103, Providence, 1992).
O.N. Zhdanov, A. K. Tsikh, “Investigation of multiple Mellin-Barnes integrals by means of multidimensional residues” Sib. Mat. J., 39(2), 281–298, 1998.
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Original Russian Text © A. J. Mkrtchyan, 2015, published in Izvestiya NAN Armenii. Matematika, 2015, No. 1, pp. 51–64.
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Mkrtchyan, A.J. On analytic continuation of multiple power series beyond the domain of convergence. J. Contemp. Mathemat. Anal. 50, 22–31 (2015). https://doi.org/10.3103/S1068362315010033
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DOI: https://doi.org/10.3103/S1068362315010033