Article
Ufa Mathematical Journal
Volume 6, Number 1, pp. 71-79
Cauchy--Hadamard theorem for exponential series
Merzlyakov S. G.
DOI:10.13108/2014-6-1-71
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In this paper we study the connection between the growth of coefficients of an exponentials series with its convergence domain in finite-dimensional real and complex spaces. Among the first results of the subject is the well-known Cauchy--Hadamard formula. We obtain exact conditions on the exponentials and a convex region in which there is a generalization of the Cauchy--Hadamard theorem. To the sequence of coefficients of exponential series we associate a space of sequences forming a commutative ring with unit. A study of the properties of this ring allows us to obtain the results on solvability of non-homogeneous systems of convolution equations.