I have understood up to the Cauchy Integral Formula, including it's derivation. I would like to understand the jump from the formula to the general expression, which utilizes the formula for derivatives. Book: Fundamentals of Complex Analysis with Applications to Engineering and Science by E.B. Saff and A.D. Snider.
Thusfar, $\zeta$ has represented the poles of a function. $\Gamma$ represents contours, $z$ inside of $\Gamma$.
The text gives Cauchy's Integral Formula:
$$f(z_0) = \frac{1}{2\pi i}\int\frac{f(z)}{z-z_0} dz$$
The text then says: "If in Cauchy's formula we replace $z$ by $\zeta$ and $z_0$ by $z$, then we obtain
$$f(z) = \frac{1}{2\pi i} \int \frac{f(\zeta)}{\zeta-z} d\zeta$$
The advantage of this representation is that it suggests a formula for the derivative $f'(z)$,obtained by formally differentiating with respect to z under the integral sign. thus we are led to suspect that
$$f'(z) = \frac{1}{2\pi i} \int\frac{f(\zeta)}{(\zeta-z)^2} d\zeta$$
In verifying this equation we shall actually establish a more general theorem..."
What I would like to understand is why these replacements happen, how this thought process came about, and how this necessarily differs from the original representation.
In the first equation, $$f(z_0)=\frac 1{2\pi i}\int_\Gamma\frac{f(z)}{z-z_0}\,dz$$ we see that the value of the given integral is equal the value of the function $f$ at $z_0$. So, the value of $f$ at a single point $z_0$ is represented by the integral. Note here that $z_0$ is a constant.
Now, since $z$ represents complex values in the interior of $\Gamma$ we can view the Cauchy integral as representing the values for all points in the interior of $\Gamma$. Of course to do this, we must change the variable of integration. Hence $$f(z)=\frac1{2\pi i}\int_{\Gamma}\frac{f(\zeta)}{\zeta-z}\,d\zeta.$$