About analytical resuming multiple power series by using one-dimensional matrix methods of summation

In the theory of analytic functions of K. Weierstrass the concept of the analytical element (power series in C converging in a circle) and its analytic continuation are the main. The method of power series expansion at another, series proposed by Weierstrass, fundamentally solves the problem of analytic continuation, proved ineffective in a particular application. In the works of Hadamard, Mittag-Leffler, Le Roy, Lindelof the so-called summation methods that give good results for the analytic continuation of power series in the case of the star domains of the complex plane have been proposed. In the works of Arakelian a description of the areas, in which the restoration of the analytic continuation of the analytical element with a fixed center is possible by using the universal matrix methods of summation is received. This work is about the analytical continuation of multiple power series in the class offields of synthesis of spiral. Using one-dimensional matrix methods of summation of power series constructed multidimensional matrix methods of summation for multiple power series, which allows you to construct an analytic continuation of this number in the maximum spiral region called (m,α)-the star of the Mittag-Leffler function f defined by this row. This approbation built multidimensional matrix methods of summation of multiple power series is carried out using one-dimensional geometric progression. That is the domains of the complex plane, there is at least one infinite matrix "summarizing" all analytic elements with a given center. These domains were spiral relative to some point and were named Arakelian domains efficient summability.