# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
On zeroes of some transcendental equations. (English) Zbl 0603.34069

The equations treated in this paper are of the form (*) $P\left(z\right)+Q\left(z\right){e}^{-Tz}=0$ and arise from the consideration of delay-differential equations with a single delay T, $T\ge 0$. The equation is called stable if all zeroes lie in $Re\left(z\right)<0$ and unstable if at least one zero lies in $Re\left(z\right)>0$. 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 $Re\left(z\right)>\delta$ $\left(\delta >0\right)$ 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) $P\left(0\right)+Q\left(0\right)\ne 0;$ (iv) there are at most a finite number of zeroes of P(z) $+Q\left(z\right)$ in the right half-plane; (v) $F\left(y\right)\equiv {|P\left(iy\right)|}^{2}-{|Q\left(iy\right)|}^{2}$ for real y, has at most a finite number of real zeroes.

Under these conditions, the following statements are true: (a) If the equation $F\left(y\right)=0$ has no positive roots, then if (*) is stable [unstable] at $T=0$ it remains stable [unstable] for all $T\ge 0·$

(b) Suppose that $F\left(y\right)=0$ has at least one positive root and each positive root is simple. As T increases, stability switches may occur. There exists a positive ${T}^{*}$ such that (*) is unstable for all $T>{T}^{*}$. As T varies from 0 to ${T}^{*}$, at most a finite number of stability switches may occur.

##### MSC:
 34K20 Stability theory of functional-differential equations 30C15 Zeros of polynomials, etc. (one complex variable)