Little Picard theorem

Statement

If $f:\mathbb C\to\mathbb C$ is a nonconstant entire function, then the complement

\mathbb C\setminus f(\mathbb C)

contains at most one point. Equivalently, a nonconstant entire function cannot omit two distinct complex values.

Derivation from Great Picard

If $f$ is a transcendental entire function, then $f(1/w)$ has an essential singularity at $w=0$ . The great Picard theorem implies that it assumes every complex value, with at most one exception, infinitely often near $0$ . If $f$ is a nonconstant polynomial, it assumes every complex value by applying the fundamental theorem of algebra to $f(z)-a$ .

Sharpness

One omitted value is possible: the exponential function $e^z$ is entire and never vanishes. It assumes every value in $\mathbb C^\times$ .

References

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