Prove that the image of a nonconstant entire function \(E : \mathbb{C} \to \mathbb{C}\) is dense in \(\mathbb{C}\).
Solution
Suppose the image were not dense. Then some open disk \[ D(a,r) \] would be disjoint from \(E(\mathbb C)\). In particular \[ |E(z)-a|\ge r \qquad(z\in\mathbb C). \] Thus \[ F(z)=\frac1{E(z)-a} \] is an entire bounded function. Liouville’s theorem makes \(F\) constant, and therefore \(E\) constant, contrary to hypothesis. Hence the image of every nonconstant entire function is dense in \(\mathbb C\).