A property of exponential polynomials of importance in dynamical systems theory

Abstract

It is the object of this note to establish a simple theorem of relevance to the study of linear oscillatory systems involving a time lag. The result, concerns the connection between the zeros of an exponential polynomial P τ(λ) and those of the related constant coefficient polynomial P 0 (λ). The exponential polynomial P τ(λ) appears as the characteristic function associated with the differential-difference equation describing the system response, with τ signifying the time lag. In particular, it is shown that as the exponential polynomial P τ(λ) tends uniformly to the corresponding ordinary constant coefficient polynomial P 0(λ) as λ→0, so each zero of P ν(λ) is approached by a number of zeros of P τ(λ) equal in number to its multiplicity ; the remaining zeros of P τ(λ) are shown to lie outside a circle of arbitrary large radius centred on the origin.