  We give an explicit formula for the logarithmic potential of the asymptotic
zero-counting measure of the sequence
$\left\{\frac{\mathrm{d}^n}{\mathrm{d}z^n}\left(R(z)\exp{T(z)}\right)\right\}$.
Here, $R(z)$ is a rational function with at least two poles, all of which are
distinct, and $T(z)$ is a polynomial. This is an extension of a recent
measure-theoretic refinement of P\'olya's Shire theorem for rational functions.
