  The inverse Mills ratio is $R:=\varphi/\Psi$, where $\varphi$ and $\Psi$ are,
respectively, the probability density function and the tail function of the
standard normal distribution. Exact bounds on $R(z)$ for complex $z$ with $\Re
z\ge0$ are obtained, which then yield logarithmically exact bounds on
high-order derivatives of $R$. The main idea of the proof is a non-asymptotic
version of the so-called stationary-phase method.
