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Why cant an annulus be analytically isomorphic to the punctured unit disc?

$A_{r,R}$ is an annulus

Theorem: $A_{r,R}$ is analytically isomorphic to $A_{s,S}$ iff $R/r = S/s$.

If our annulus $A_{1,2}$, then $R/r = 2$ and the punctured disc would be $\lim\limits_{s\to 0}1/s = \infty$.

So here is a counter example.

$A_{r,R}$ is an annulus

Theorem: $A_{r,R}$ is analytically isomorphic to $A_{s,S}$ iff $R/r = S/s$.

If our annulus $A_{1,2}$, then $R/r = 2$ and the punctured disc would be $\lim\limits_{s\to 0}1/s = \infty$.

So here is a counter example.

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