First Hopf boundary of scalar delayed negative feedback ($q^{\prime }=-\beta q-\gamma q(T-1)$)

For nonnegative instantaneous damping $\beta$ and positive delayed gain $\gamma$, characteristic roots obey $\lambda +\beta +\gamma e^{-\lambda }=0$. There is no nonnegative real root. The first imaginary crossing has $\omega \in [\pi /2,\pi )$ and $\beta =-\omega \cot \omega$, $\gamma =\omega /\sin \omega$. In particular stability is retained below this first boundary and $\gamma =\pi /2$ is exact only at $\beta =0$. For small positive $\beta$, the critical gain is $\pi /2+2\beta /\pi +O({\beta }^2)$ and the scaled period tends to four. If the time delay is dimensional $\tau$, that period is multiplied by $\tau$.