Great Picard theorem

Statement

Let $a$ be an essential isolated singularity of a holomorphic function $f$ . In every punctured neighborhood of $a$ , the function $f$ assumes every value in $\mathbb C$ , with at most one exception, infinitely many times.

Comparison with Casorati–Weierstrass

The Casorati–Weierstrass theorem says that the image is dense. Great Picard says much more: apart from at most one omitted value, each value is attained, and is attained infinitely often arbitrarily close to the singularity.

Entire-function consequence

Applying the theorem at infinity gives the little Picard theorem for nonconstant entire functions.

Sharpness

The exceptional value can occur. The function $e^{1/z}$ has an essential singularity at $0$ and never takes the value $0$ , while taking every nonzero complex value infinitely often near $0$ .

References

John B. Conway, Functions of One Complex Variable II, Springer, 1995. Publisher record. Relevant: Chapter XII.