Mathematics & Physics 2015, 8(2), 173–183 УДК 517.55 Analytic Continuation of Power Series by Means of Interpolating the Coefficients by Meromorphic Functions Aleksandr J.Mkrtchyan∗ Institute of Mathematics and Computer Science Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041 Russia Received 07.04.2015, received in revised form 20.04.2015, accepted 25.04.2015 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. <...> Introduction The problem of analytic continuation and finding singular points of a function has a rich and long history. <...> It has been studied by many prominent mathematicians such as Carlson, Polya, Hadamard, and others (see, for example, [1]). <...> The Cauchy-Hadamard theorem yields that nlim →∞ n|fn| = 1. <...> All rights reserved c – 173 – (2) (3) Recall (see, e.g. [2]) that the indicator function hϕ(θ) for an entire function ϕ is defined as the upper limit (1) Aleksandr J.Mkrtchyan Analytic Continuation of Power Series by Means of Interpolating the Coefficients . <...> The first one asks about the conditions for continuation to the whole complex plane except ∂D1 \∆σ. <...> The answer is given by Polya’s theorem. <...> The series (1) extends analytically to C, possibly except the arc ∂D1 \γσ, if and only if there exists an entire function of exponential type ϕ(ζ) interpolating the coefficients fn such that hϕ(θ) σ| sin θ| for |θ| π. <...> Two other questions concern continuation to the sector C \ ∆σ defined by the arc γσ = ∂D1\∆σ, or to a neighbourhood of this arc. <...> Both of them are answered by Arakelian’s theorems. <...> The sum of the series (1) extends analytically to the sector C\∆σ if and only if there is an entire function ϕ of exponential type interpolating the coefficients of the series fn whose indicator function hϕ(θ) satisfies the condition hϕ(θ) σ| sinθ| for |θ| < π 2 . (4) The continuation property of f(z) to a neighbourhood of the arc γσ was studied in [6] (see also [7]). <...> The open arc γσ = C\∆σ is an arc of regularity of the series (1) if and only if there <...>