Asymptotic solutions of characteristic equations

We give rigorous asymptotic results about solutions λ∈C of the equationaλneλb-c=0with a, b>0, n∈N and c∈C,|c|=1. These results are extended to rather general perturbations of this characteristic equation as they arise in the study of eigenvalues of linear operators. Applications are given to eigenvalue problems of differential-difference equations, elongational flow and population dynamics.