Relations between roots and coefficients of the transcendental equations

Abstract

Abstract. It is proved that there exist the relations between coefficients of the transcendental equations and the infinite number of their roots, similar to Vieta's formulae.These relations may be obtained for the entire analytic functions using theorems of residues and argument principle. In particular the meromorphic functions will be considered.

A Lyapunov functional for a system with both lumped and distributed delay

“…for t 0, where z(t, ϕ) is a solution of system (11), with the initial function ϕ ∈ PC([−h, 0], R 2n ), given by (12).…”

“…Górecki and Białas published two articles [2,12] whose concern relations between roots of the transcendental equations and their coefficients. These results are helpful in the stability analysis of the time delay systems.…”

Design of the oscillatory systems with the extremal dynamic properties

“…It is also possible to enlarge the formula (22) on the system with time-delay using the method described in [7][8][9][10]. Let us consider a differential equation with time delay h > 0.…”

“…In consequence the necessary condition is fulfilled. We apply the Theorem 3 proved in [7]. The relation between coefficients and the roots of the quasipolynomial equations of the type (R1) is given by the following formula:…”

A Lyapunov functional for a neutral system with a time-varying delay

“…Górecki and Białas published two articles [23,24] whose concern relations between roots of the transcendental equations and their coefficients. These results are helpful in the stability analysis of the time delay systems.…”