# Picard’s 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.

Title Picard’s theorem PicardsTheorem 2013-03-22 13:15:23 2013-03-22 13:15:23 Koro (127) Koro (127) 9 Koro (127) Theorem msc 32H25 great Picard theorem EssentialSingularity CasoratiWeierstrassTheorem ProofOfCasoratiWeierstrassTheorem