I have something confusing from a lecture that I'm watching:
Consider this integral:
$$\int_c \frac{dz}{z(z-2)^4}$$
The contour is $C: |z-2| = 1$
So this circle, graphically, is centered at 2 with a radius of 1.
I am told this function is analytic everywhere except for two places: 0 and 2. The 0 is outside the circle and the 2 is inside the circle/domain since the circle is centered at 2. So the circle is a punctured disk.
In my lecture I am told that the interval $0 \lt \left | (z-2) \right | \lt 2$. I don't see how this follows. Shouldn't it be $1 \lt | (z-2)| \lt 3$ because if z = 4.9999... the resulting value still lies within the circle. Aren't we looking for values of z such that (z-2) lie within the circle?