  We investigate a scalar characteristic exponential polynomial with complex
coefficients associated with a first order scalar differential-difference
equation. Our analysis provides necessary and sufficient conditions for
allocation of the roots in the complex open left half-plane what guarantees
asymptotic stability of the differential-difference equation. The conditions
are expressed explicitly in terms of complex coefficients of the characteristic
exponential polynomial, what makes them easy to use in applications. We show
examples including those for retarded PDEs in an abstract formulation.

Dynamical Systems