Roots of a characteristic equation with complex coefficients associated with differential-difference equations

The paper proves a necessary-and-sufficient stability region for the transcendental equation s − λ − γ e^{−sτ} = 0 by reducing to s − a − η e^{−sτ} = 0 (with a = Re λ, η = e^{−i Im λ τ}γ) and giving an explicit three-case description Λ_{τ,a} (equations (8)–(10)), valid when a ≤ 1/τ; it also shows that for a > 1/τ no η yields all roots in C−. The model independently derives the same boundary conditions from the imaginary-axis root locus η = (iω − a)e^{iωτ} and then uses a Nyquist/winding-number argument for sufficiency and a transversality check for necessity/orientation. The geometric content and final criteria agree with the paper’s Λ_{τ,a} description, though the proofs use different tools (the paper: first-crossing analysis and monotonicity of argument increments; the model: argument principle and Nyquist). Minor technical slips in the model (an overly strong large-arc bound) do not affect correctness of the main argument. Overall, both are correct and consistent with one another. Key matches include the shift reduction (Lemma 2), the three explicit cases for Λ_{τ,a}, the modulus/argument description of the imaginary-axis locus, the special role of the radius cutoff |η| < |η_π|, and the restriction a ≤ 1/τ (paper: Theorem 6 and points 12–16; model: conclusion). Citations: main result and Λ_{τ,a} in (8)–(10) and Theorem 6, the reduction lemma and first-crossing/derivative analysis, and the radius-cutoff/argument-increment calculus all appear explicitly in the paper .