Expansion of real valued meromorphic functions into Fourier trigonometric series

Abstract

In the main part of the paper, on the basis of contour integration of complex meromorphic functions whose singularities lie onto an integration contour, in the first step, a concept of improper integrals absolute existence of meromorphic functions, as more general one with respect to the concept of improper integrals convergence (existence), is introduced into analysis. In the second step, in the case when a modulus of complex parameter tends to infinity, an interval of improper integrals convergence of parametric meromorphic functions is defined. In accordance with this, it is shown that the class of real valued meromorphic functions, whose finitely many isolated singularities lie onto a real axis segment, may be expanded into Fourier trigonometric series, separately. At all points of the segment, at which the meromorphic functions are continuous ones, the Fourier trigonometric series is summable and its sum is equal to the function values at those points. Finally, that all is illustrated by two representative examples.