SINGULAR POINTS FOR THE SUM OF A SERIES OF EXPONENTIAL MONOMIALS

A problem of distribution of singular points for sums of series of exponential monomials on the boundary of its convergence domain is studied. The influence of a multiple sequence Λ = {λk, nk}k=1 of the series in the presence of singular points on the arc of the boundary, the ends of which are located at a certain distance R from each other, is investigated. In this regard, the condensation indices of the sequence and the relative multiplicity of its points are considered. It is proved that the finiteness of the condensation index and the zero relative multiplicity are necessary for the existence of singular points of the series sum on the R-arc. It is also proved that for one of the sequence classes Λ, these conditions give a criterion. Special cases of this result are the well-known results for the singular points of the sums of the Taylor and Dirichlet series, obtained by J. Hadamard, E. Fabry, G. Pólya, W.H.J. Fuchs, P. Malliavin, V. Bernstein and A. F. Leont’ev, etc.