In systems theory one often deals with the frequency domain representation of systems instead of their time-domain representations: If a LTI system $S$ is given by the convolution with $h(t) = Ce^{At}B$ for some $A \in \mathbb{R}^{n \times n}$, $B \in \mathbb{R}^{n \times 1}$ and $C \in \mathbb{R}^{1 \times n}$, then the Laplace transform of $h$ is given by the known formula $\hat{h}(s) = C(sI-A)^{-1}B$ and defined only on $$D = \{s \in \mathbb{C} \mid \operatorname{Re}(s) > \max \operatorname{Re} \sigma(A)\},$$ where $\sigma(A)$ is the set of eigenvalues of $A$. However, in many theorems in system theory one uses the transfer function $G(s) = C(sI-A)^{-1}B$ and assumes it to be defined on $E = \mathbb{C} \setminus \sigma(A)$, i.e. $G$ is the analytical continuation of $\hat{h}$ from $D$ to $E$.
Since in the usual proof of the Nyquist stability theorem one uses the argument principle on the meromorphic function $G$ together with the contour $\Gamma = [-R,R]i \cup R e^{i[-\pi/2,\pi/2]}$ (for some $R > 0$) which includes parts of the imaginary axis, which are not part of the original domain $D$ if $h$ is unstable (i.e. $A$ has eigenvalues in the closed right hand plane), it is not clear how the absence of poles of the sensitivity $$\frac{1}{1+G}$$ implies the stability of the original feedback with the system $S$
Question: What is a rigorous proof of this theorem that incorporates the domain $D$ of the original Laplace transform $\hat{h}$?
What is important here is to characterize the location of the the poles of the closed-loop system in terms of the behavior of the open-loop transfer function. While it is true that the Laplace transform $\hat h(s)$ of the open-loop impulse response has a region of convergence, this region of convergence is only useful for recovering the impulse response $h(t)$ from its Laplace transform. Different regions of convergence for $\hat h(s)$ yield different functions $h(t)$ obtained from the inverse Laplace transform.
Another point is that if one consider the region of convergence as the domain of the Laplace transform $\hat h(s)$ of the impulse response $h(t)$, then $\hat h(s)$ has no poles. However, those poles contain crucial information about the behavior of the open-loop transfer and that of the closed-loop transfer. In fact the region of convergence just contains some information about the rightmost pole of the open-loop, which is not enough.
So, we need to extend $\hat h(s)$ to the whole complex plane except at the eigenvalues of $A$, which yields the transfer function $G(s)$.
The proof of the Nyquist using the argument principle is then routine from this point and can be found in many monographs.
I am open to comments and remarks. I will update this response based on them. This is just a starting point.