# analytic isomorphism

#### dwsmith

##### Well-known member
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.

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#### dwsmith

##### Well-known member
Re: analytic ismoprhism

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.
How can I do this without invoking this theorem?

#### dwsmith

##### Well-known member
Re: analytic ismoprhism

Let $f:\mathbb{D}\to A_{1,2}$.
Let $U$ be an open neighborhood of 0 of radius $\epsilon > 0$.
Then $f(U)\subset A_{1,2}$.
Letting $\epsilon$ be small enough we would have that $f(U)$ is not in the annulus.