## Injective Entire Functions

Prove that all entire functions that are also injective, take the form with and .

**Solution:**

We take , which is holomorphic everywhere except the origin. Now, we try to find out what type of singularity is the origin for .

If the origin is a removable singularity for , then is bounded on a closed disk centred at the origin, which implies that is bounded outside a closed circle containinf the origin. But is bounded on this closed circle, because is continuous, therefore, is bounded. Because is entire and bounded, by Liouiville’s Theorem, is constant. This contradicts the injectivity of . So the origin is not a removable singularity for .

Suppose now that is an essential singularity for . Then, by Cassorati-Weierstrass Theorem, if we chose a punctured disk centred at the origin, then is dense in . This implies is dense in . But is open because any holomorphic mapping is an open mapping. Then , which is again a contradiction with the injectivity of .

Therefore is a pole for . Since the Laurent expansion is unique, and the principal part of is the same as the analytic part of , it follows that the analytic part of has finitely many terms, which implies that is a polynomial. Because is injective, the polynomial can have at most one root. Because is not constant, we conclude that the only expression of can be of the form , where and .