Let be an holomorphic function with an essential singularity at . Then there is a number such that the image of any neighborhood of by contains . In other words, assumes every complex value, with the possible exception of , in any neighborhood of .
Remark. Little Picard theorem follows as a corollary: Given a nonconstant entire function , if it is a polynomial, it assumes every value in as a consequence of the fundamental theorem of algebra. If is not a polynomial, then has an essential singularity at ; Picard’s theorem implies that (and thus ) assumes every complex value, with one possible exception.
|Date of creation||2013-03-22 13:15:23|
|Last modified on||2013-03-22 13:15:23|
|Last modified by||Koro (127)|
|Synonym||great Picard theorem|