In Complex Var.& Apps. The conditions of the proof of residues at infinity is given as subsequent.
Suppose that a function $f$ is analytic throughout the finite plane except for a finite number of singular points interior to a positively oriented simple closed contour $C$. Next, let $R_1$ denote a positive number which is large enough that C lies inside the circle $|z|=R_1...$ The function f is evidently analytic throughout the domain $R_1<|z|<\infty$ and the point at infinity is then said to be an isolated singular point of $f$.
Now let $C_0$ denote a circle $|z|=R_0$, oriented in the clockwise direction, where $R_0>R_1$. The residue of $f$ at infinity is defined by means of the equation $$\int_{C_0}f(z)dz=2\pi i \underset{z=\infty}{\operatorname{Res}}f(z)$$
The book goes on to establish the Laurent expansion coefficients for $f(z)$ as $$c_n=\frac{1}{2\pi i}\int_{-C_0}\frac{f(z)dz}{z^{n+1}}.$$ Manipulating for the series expansion of $\frac{1}{z^2}f(\frac{1}{z})$ yields $\sum_{n=-\infty}^{\infty}\frac{c_{n-2}}{z^n}$; $\quad$ $0<|z|<\frac{1}{R_1}$.
I understand that the $\frac{1}{z^2}f(\frac{1}{z})$ is the result of a change-of-variable in the contour integral.
The interesting question is: how does this change-of-variable correspond with the manipulations in the series expansion? And is there a direct verification of this WITHOUT using the equivalence of the residues?