ON THE REPRESENTATION OF ANALYTIC FUNCTIONS BY DIRICHLET SERIES

Abstract

We have earlier proved (Dokl. Akad. Nauk SSSR 164 (1965), 40-42; Mat. Sb. 70 (112) (1966), 132-144) a theorem on the representation of an arbitrary function analytic in a closed convex region by a Dirichlet series in the open region . In this paper we prove that any function analytic in an open convex finite region and continuous in can be represented by a Dirichlet series with coefficients which can be computed by means of specific already-known formulas.We also prove that if the convex region is bounded by a regular analytic curve, then any function analytic in can be expanded in a Dirichlet series in . These two theorems are based on the following theorem from the theory of entire functions:Let be a finite open region, the support function of , , and a function satisfying the conditions Then there exists an entire function of exponential type with growth indicator and completely regular growth, which satisfies the following conditions:1) All the zeros of are simple, and 0$ SRC=http://ej.iop.org/images/0025-5734/9/1/A05/tex_sm_2048_img10.gif/>.2) We have the estimate r_0.$ SRC=http://ej.iop.org/images/0025-5734/9/1/A05/tex_sm_2048_img11.gif/>3) The sequence is part of a sequence , , which depends on the region but not on the function .In this paper we prove an analogous theorem for entire functions of arbitrary finite order .