Casorati–Weierstrass theorem

Statement

Let $c \in \mathbb{C}$, let $r > 0$, and let $f$ be holomorphic on the punctured disc $0 < |z - c| < r$. Assume that $c$ is an essential singularity of $f$. Then for each $\delta$ with $0 < \delta < r$, the image $f(\{z : 0 < |z - c| < \delta\})$ is dense in $\mathbb{C}$. Thus for each $w \in \mathbb{C}$ there is a sequence $z_n \to c$ with $z_n \neq c$ and $f(z_n) \to w$.

Remarks

Proof idea: if the image of a punctured disc misses a disc $|w - w_0| < \varepsilon$, then $g(z) = 1/(f(z) - w_0)$ is bounded near $c$. By Riemann's theorem, $g$ is holomorphic at $c$, and then $c$ is removable or a pole of $f$. Picard's great theorem is a much stronger statement. In the Russian literature, the theorem has the name of Sokhotski.

Explanations

L1 human

Near an essential singularity, the function comes as close as you like to each complex number. For example, $e^{1/z}$ comes close to $5$, to $-i$ and to $0$ on each small disc around $0$.