In the Wigner-Weisskopf theory of spontaneous decay, the Laplace transform for the amplitude is found to be $$L(s)=\frac{1}{s+i\omega+G(s)}$$ where $G(s)$ is a complex function. $L(s)$ has a pole at $s=-i\omega-G(s)$ and so the approximation is made that $G(s)$ can be replaced with $G(s=-i\omega)=\frac{1}{2}\Gamma+iS$ provided that $G(s)$ is slowly varying around $s=-i\omega$ and that $\Gamma\ll\omega$.
I don't understand this second condition; why must it not be the case that $S\ll\omega$ as well? Otherwise, at $s=-i\omega$, the denominator would still be large, and there may exist some other $s$ where it is much smaller. I must be wrong because it turns out that $S$ diverges and yet the theory still works, but I can't see why.