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Riemann mapping theorem (Theorem)

Let $ U$ be a simply connected open proper subset of $ \mathbb{C}$, and let $ a\in U$. There is a unique analytic function $ f:U\rightarrow\mathbb{C}$ such that

  1. $ f(a)=0$, and $ f'(a)$ is real and positive;
  2. $ f$ is injective;
  3. $ f(U)=\{z\in \mathbb{C}:\vert z\vert<1\}$.

Remark. As a consequence of this theorem, any two simply connected regions, none of which is the whole plane, are conformally equivalent.



"Riemann mapping theorem" is owned by Koro.
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See Also: conformal radius


Attachments:
proof of Riemann mapping theorem (Proof) by rspuzio
Green functions and conformal mapping (Topic) by rspuzio
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Cross-references: conformally equivalent, plane, regions, consequence, injective, positive, real, analytic function, proper subset, open, simply connected
There are 7 references to this entry.

This is version 2 of Riemann mapping theorem, born on 2002-12-11, modified 2002-12-11.
Object id is 3728, canonical name is RiemannMappingTheorem.
Accessed 4522 times total.

Classification:
AMS MSC: 30A99 (Functions of a complex variable :: General properties :: Miscellaneous)
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