The equations treated in this paper are of the form (*) and arise from the consideration of delay-differential equations with a single delay T, . The equation is called stable if all zeroes lie in and unstable if at least one zero lies in . A stability switch is said to occur if the equation changes from stable to unstable, or vice-versa, as T varies through particular values. The main result is the following theorem.
Assume that: (i) P(z), Q(z) are analytic functions in which have no common imaginary zero; (ii) the conjugates of P(-iy) and Q(-iy) are P(iy) and Q(iy) for real y; (iii) (iv) there are at most a finite number of zeroes of P(z) in the right half-plane; (v) for real y, has at most a finite number of real zeroes.
Under these conditions, the following statements are true: (a) If the equation has no positive roots, then if (*) is stable [unstable] at it remains stable [unstable] for all
(b) Suppose that has at least one positive root and each positive root is simple. As T increases, stability switches may occur. There exists a positive such that (*) is unstable for all . As T varies from 0 to , at most a finite number of stability switches may occur.