Why is the residue defined on a circle in complex analysis? I'm confused by the definition of the residue, which is as follows (from the book Complex Variables and Applications by James Ward Brown).

I don't know why the author restricted $z$ in $0<|z-z_0|<R_2$.
The Laurent series does not need this restriction, which are as follows.


In other words, $R_1$ in the definition of the Laurent series doesn't need to be zero, but in the definition of the residue it is zero.
I don't know why. I think the condition $R_1=0$ seems to be redundant and unnecessary in the definition of the residue. I think we could also get the definition of residue without this condition
So I'm very confused about this.
I'd appreciate it if you could help with my problem. Thank you!
 A: The thing about poles and residues is that a function can have more than one pole.
See this answer, in which there is a diagram representing a function with poles at $z_1,$ $z_2,$ $z_3,$ and $z_4.$
For convenience, here's a copy of that diagram:

The definition of the Laurent series, which only requires the function to be analytic on $R_1 < \lvert z - z_0 \rvert < R_2,$
allows us to apply that definition to the outer contour $C$ in this diagram.
If that definition required $R_1 = 0$ then it could not be applied to that contour, since there's no punctured disk containing $C$ on which the function is analytic.
(You can take care of one pole by centering the disk at one of the poles, say $z_1,$ but then there are still three other poles in the punctured disk that you can't get rid of without losing at least part of $C$.)
But if we don't set $R_1 = 0$ in the definition of the residue around a single point, then we could use either the contour $C$ in the diagram above or the contour $C_r(z_1)$ to define the residue of $z_1.$
And the problem with that is that these two contour integrals are not equal.
If you take a contour integral in an arbitrary annulus
$R_1 < \lvert z - z_0 \rvert < R_2$ around an isolated singular point $z_0,$ you're not necessarily getting the residue of $z_0$;
if $R_1 > 0$ you're actually getting the combined residues of $z_0$ and every other isolated singular point in the disk $\lvert z - z_0 \rvert \leq R_1.$
Setting $R_1 = 0$ in the definition of the residue of a point $z_0$ ensures that we're getting the reside of $z_0$ alone and not the combined residues of $z_0$ and any other nearby singular point.
