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.
MSC2010 numbers : 32A05, 30B30
For multiple power series centered at the origin we consider the problem of its analytic continuability into a sectorial domain. The condition for continuability is formulated in terms of a holomorphic function that interpolates the series coefficients. For series in one variable this problem has been studied in the works of E. Lindelöf, N. Arakelian, and others.
We study the problem of analytic continuation of a power series across an open arc on the boundary of the circle of convergence. The answer is given in terms of a meromorphic function of a special form that interpolates the coefficients of the series. We find the conditions for the sum of the series to extend analytically to a neigbourhood of the arc, to a sector defined by the arc, or to the whole complex plane except some arc on the convergence disk.
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