Detailed proof that no essential singularity at infinity implies polynomial

Suppose $f(z)$ is holomorphic in the whole plane, and that $f(z)$ does not have an essential singularity at $\infty$. Prove that $f(z)$ is a polynomial.

I've tried following the hint given in this question. Since $f(z)$ has a nonessential singularity at $\infty$, so $g(z)=f(1/z)$ has a nonessential singularity at $0$. There are two cases:

1) $g(z)$ has a removable singularity at $0$. This means $\lim_{z\rightarrow 0}zg(z)=0$.

2) $g(z)$ has a pole at $0$. This means $g(z)=h(z)/z^k$ for some analytic function $h(z)$ such that $h(0)\neq 0$.

How can I finish each of these cases?

Solution 1:

1) Since $g$ is continuous, we can bound $g(z)$ inside some interval of $z$. Since $g(z)$ is defined as $f(1/z)$, it turns out that from the bound on $z$ we deduce that $f$ is a constant function:

There exists $M, \varepsilon >0$ such that $|g(z)|\leq M$ for all $|z| \in (0,\varepsilon)$. Hence $|f(z)|\leq M$ for all $|z| > 1/\varepsilon$. Letting $\varepsilon \to \infty$ $f$ is bounded and entire. It follows by Liouville's Theorem that $f$ is a constant function.

2) We use that $g$ has a Laurent expansion at $0$ and $f$ has a Taylor expansion since $f$ is entire. We can 'invert' the Laurent expansion so to say and by uniqueness of the Laurent expansion see that $f$ must be a polynomial.

Since the pole is of order $m$, the Laurent expansion of $g$ at $0$ is $$g(z) = \sum_{k=-m}^{\infty} a_k z^k$$ for $|z|\in (0,\varepsilon)$. We can invert this to get $f$: $$f(z) = \sum_{k = -\infty}^{m} a_{-k}z^k$$ for $|z|>1/\varepsilon$. The Taylor expansion of $f$ around $0$ given by $$f(z)=\sum_{k=0}^{\infty}b_kz^k$$ and the Laurent expansion must be equal by uniqueness, so $$f(z)=\sum_{k=0}^{m} b_k z^k$$ where $a_{-k}=b_k$ for all $k$. So $f$ is a polynomial.

See the following pdf: math.berkeley.edu/~mjv/Math185hw8.pdf