EXPANSION OF THE RECIPROCAL OF AN ENTIRE FUNCTION WITH ZEROS IN THE HALF-PLANE IN A SERIES OF PARTIAL FRACTIONS

Abstract

We study the problem of expanding the reciprocal of an entire function of exponential type with simple zeros located at a half-plane in a series of partial fractions. Our main result is the development of a theorem of M. G. Krein on the properties of an entire function with simple real zeros, the inverse of which can be decomposed into simple fractions. These results demonstrate the possibility of using specific examples. The first of these is the Laplace transform, taken in a finite interval with decreasing it original. Second - regularly encountered in the analysis and probability errors integral.