The theorem I was trying to understand is taken in "Introduction to Numerical Analysis, Stoer,Bulirsch", and it's the following :
$\textbf{Theorem :}$ Let $\lambda$ be a simple zero of the characteristic polynomial of the $n\times n$ matrix $A$, and $x, y^{H}$ corresponding right and left eigenvectors of $A$ respectively and let $C$ be an arbitrary $n\times n$ matrix. Then there exists a function $\lambda(\epsilon)$ which is analytic for $\epsilon$ sufficiently small, $\vert \epsilon \vert < \epsilon_{0}, \epsilon_{0} > 0$ such that $\lambda(0) = \lambda, \lambda'(0) = \frac{y^{H}C x}{y^{H}x}$ and $\lambda(\epsilon)$ is a simple zero of the characteristic polynomial of $A + \epsilon C$. One has, in first approximation, $\lambda(\epsilon) = \lambda + \epsilon \frac{y^{H}C x}{y^{H}x}$ (excluding terms of order greater of equal $\epsilon^{2}$)
At a certain point the proof justify the characterization of a simple zero using an integral formula "according to another theorem in complex analysis". Since this theorem doesn't belong to my references I was hoping to find some here.
Any help would be appreciated.
$$\frac{1}{2i \pi}\int_{\gamma}z\dfrac{f'(z)}{f(z)}dz$$
is the sum of zeros of $f$ inside the closed loop $\gamma$ (proof with the residue formula:see here by taking $g(z):=f$). If this loop is small enough as to enclose only a (simple) root, you get the representation given in the text.
Besides, for the other part of your question, there is a natural connection between the so-called Rayleigh quotients $X^TCX/X^TX$ and eigenvalues as you can see in this text.
See as well this reference.