PlanetMath (more info)
 Math for the people, by the people. Sponsor PlanetMath
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
Owner confidence rating: High Entry average rating: No information on entry rating
Picard's theorem (Theorem)

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 $w$

Remark. Little Picard theorem follows as a corollary: Given a nonconstant 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 possible exception.




"Picard's theorem" is owned by Koro.
(view preamble | get metadata)

View style:

See Also: essential singularity, Casorati-Weierstrass theorem, proof of Casorati-Weierstrass theorem

Other names:  great Picard theorem
Log in to rate this entry.
(view current ratings)

Cross-references: implies, fundamental theorem of algebra, consequence, polynomial, entire function, little Picard theorem, complex, contains, neighborhood, image, number, essential singularity, holomorphic function
There are 6 references to this entry.

This is version 6 of Picard's theorem, born on 2002-12-11, modified 2008-05-01.
Object id is 3735, canonical name is PicardsTheorem.
Accessed 9490 times total.

Classification:
AMS MSC32H25 (Several complex variables and analytic spaces :: Holomorphic mappings and correspondences :: Picard-type theorems and generalizations)

Pending Errata and Addenda
None.
[ View all 4 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | prove | add result | add corollary | add example | add (any)