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Proving an entire function is a polynomial under certain conditions

Bingk

New member
Jan 26, 2012
16
Hello,
This was an exam question which I wasn't sure how to solve:

Suppose [TEX]f[/TEX] is entire and [TEX]|f(z)| \leq C(1+ |z|)^n[/TEX] for all [TEX]z \in \mathbb{C}[/TEX] and for some [TEX]n \in \mathbb{N}[/TEX].
Prove that [TEX]f[/TEX] is a polynomial of degree less than or equal to [TEX]n[/TEX].

I know that f can be expressed as a power series, but I'm not sure how to show that the upper limit of the sum has to be less than or equal to n.

Thanks!
 

PaulRS

Member
Jan 26, 2012
37
Hints:

  • $f^{(n)}(z)$ is entire for all $n\in{\mathbb{N}}$.
  • $\frac{1}{2\pi\cdot i}\cdot \oint_\Gamma \frac{f(z)}{(z-w)^{n+1}}dz = \frac{f^{(n)}(w)}{n!}$ where $\Gamma$ is, say, a circle centered at $w$ of radius $R$.
  • What can you say, then, about $f^{(n)}(w)$ for some $n$ ? (Hint: try to find a uniform bound for $f^{(n)}(w)$ on the whole plane)