Problem 8

Exercise 8.8 (Entire functions, singularities, and injectivity)   Let $f : \mathbb{C} \to \mathbb{C}$ be entire with

$$f(z) = \sum_{n=0}^\infty a_nz^n$$ (8.35)

1. Show that $f$ has an essential singularity at infinity if $a_n\neq 0$ for infinitely many $n$.
2. Show that if $f$ is injective, then $f(z)=a_0+a_1z$.

Proof. For (a), suppose $f$ has no essential singularity at infinity. Then one of the limits

$$\lim_{z\to\infty}f(z) \quad \lim_{z\to\infty}1/f(z)$$ (8.36)

exists. If the first one exists, then $f$ is bounded, indicating $f$ is constant, so suppose the first one does not exist and the second one does. Suppose $1/f(z)\to a\neq 0$. Then the limit $f(z)\to 1/a$ exists, a contradiction. Therefore, $1/f(z)\to 0$, and $\vert f(z)\vert\to\infty$, which indicates $f$ is a polynomial.

For (b), if $f(z)$ is not a polymomial, then $f$ has an essential singularity at infinity, which means $f$ is not injective. Therefore suppose $f(z)=a_0+\cdots+a_nz^n$. Injectivity means that $f(z)=a_n(z-r)^n$ for some unique root $r$. Substitute

$$f(e^{2\pi i/n}+r) = a_n$$ (8.37)

$$f(1+r) = a_n$$ (8.38)

and apply the injectivity of $f$ to find $1=e^{2\pi i/n}$, which means $n=1$. Therefore, $f(z)=a_0+a_1z$. $\qedsymbol$