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'Picard's theorem'
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| Title of object: |
Picard's theorem |
| Canonical Name: |
PicardsTheorem |
| Type: |
Definition |
| Created on: |
2002-12-11 10:24:50.897315-05 |
| Modified on: |
2002-12-12 20:38:45.256997-05 |
| Classification: |
msc:32H25 |
| Synonyms: |
Picard's theorem=great Picard theorem |
Revision comment (for changes between this and next version):
| Changes for correction #1352 ('connections between Picard's theorems'). |
Preamble:
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Content:
Let $f$ be an holomorphic function with an essential singularity at $w\in \mathbb{C}$. Then there is a number $z_0\in \mathbb{C}$ such that the image of any neighborhood of $w$ by $f$ contains $\mathbb{C}-\{z_0\}$. In other words, $f$ assumes every complex value, with the possible exception of $z_0$, in any neighborhood of $w$.
\emph{Remark.} Little Picard theorem follows as a corollary:
Given an entire function $f$, if it is a polynomial, it assumes every value in $\mathbb{C}$ as a consequence of the fundamental theorem of algebra. If $f$ is not a polynomial, then $g(z)=f(1/z)$ has an essential singularity at $0$; Picard's theorem implies that $g$ (and thus $f$) assumes every complex value, with one possble exception. |
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