Article
Ufa Mathematical Journal
Volume 14, Number 4, pp. 56-68
Convergence of series of exponential monomials
Krivosheev A.S., Krivosheeva O.A.
DOI:10.13108/2022-14-4-56
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In the paper we study the convergence of series of exponential monomials, special cases of which are the series of exponentials, Dirichlet series and power series. We provide a description of the space of coefficients of series of exponential monomials converging in a given convex domain in the complex plane is described. Under a single natural restriction on the degrees of monomials, we provide a complete analogue of the Abel theorem for such series, which, in particular, implies results on the continued convergence of series of exponential monomials. We also obtain a complete analogue of the Cauchy-Hadamard theorem, in which we give a formula allowing to recover the convergence domain of these series by their coefficients. The obtained results
include, as special cases, all previously known results related
with the Abel and Cauchy-Hadamard theorems for exponential series, Dirichlet series and power series.