Complex Analysis - Derivation of Bode Integral I am trying to work through the steps deriving the Bode Integral in the following paper:
https://www.sciencedirect.com/science/article/pii/S2405896315025045
I am stuck on page 261, left column, on the section for the Gamma_pk contour integral, specifically the following unjustified statement:
$$ S(s)=S'(s)(p_{k}-s)$$
S(s) is the sensitivity transfer function.  Pk is a pole of the open-loop transfer function, and therefore a zero of S(s), which the contour was chosen to go around.  Please see figure 3 in the paper for a visual depiction.
Anyhow, I think it's just the complex plane throwing me off here, but I am unable to arrive at, or justify, that statement.  I come close, but can't quite get all the way there.
Thanks for any insights!
 A: It's helpful to consider one of the examples provided in the paper, namely that of an open-loop unstable system with sensitivity function $S(s)=\dfrac{s^2-1}{s^2+s+9}$. Then $$\frac{S'(s)}{S(s)}=\frac{d}{ds}\ln S(s)=\frac{2s}{s^2-1}-\frac{2s+1}{s^2+s+9}=\frac{s^2+20s+1}{(s^2-1)(s^2+s+9)}$$
or $$S(s)=S'(s)\cdot \frac{(s^2-1)(s^2+s+9)}{s^2+20s+1}.$$
It should be evident that this doesn't agree with their stated formula. That said, the reason $S(s)$ is open-loop unstable is due to the zero at $s=1$, and with this in mind we write $S(s)=S'(s)(s-1)\cdot f(s)$ where $$f(s)=\frac{(s+1)(s^2+s+9)}{s^2+20s+1}$$ is analytic in the right half-plane. This form seems entirely sufficient for the purposes of their argument.
Moreover, if we look to Lewis's notes on control theory (which your paper cites!) then the above is a case of their first equation on page 366: Given a zero of $S_L(s)$ at $s=p_j$ with positive real part, one may write
$$\ln\frac{S_L(s)}{S_L(\infty)}=\ln(s-p_j)+\ln f_j(s)$$
with $f_j(s)$ being analytic on and within the contour which runs around $p_j$. So I would discount the paper's arguments for the Bode integral formula in favor of those in Lewis's notes.
