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# 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.

## Mathematics Subject Classification

32H25*no label found*

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