I'm studying the book An introduction ot infinite dimensional linear system Theory by Curtain and Zawart.
In page 199, exercise 4.15, the authors claim that the poles, $p_{n}$, of
$G(s) = \frac{e^{\frac{-s}{2}}}{s(s+\frac{\pi}{2}e^{-s})}$
assymptotically $p_{n}\simeq -\log (4n-3)\pm \frac{\pi}{2}(4n-3)i$, for $n\in \mathbb{N}$.
I know that $G$ has a pole in $s=0$ and infinitely many poles in $s=W_{k}(-\frac{\pi}{2})$, where $W_{k}$, for $k\in \mathbb{Z}$, is the $k$-th branch of the Lambert W function. However, I have no idea how they found that asymptotic approximation. Anyone can help me?
Additionally, I am wondering if this expression could be generalized for
$G(s) = \frac{e^{\frac{-s}{2}}}{s(s+a e^{-s})}$
where $a$ is a nonzero constant.
Thanks in advance.
Gustavo