## Cute Complex Analysis Problem

**Problem:** Let be entire with

Then, if either or for some we have that is constant.

**Proof:** Evidently we must find some way to employ Liouville’s theorem, but how? It may seem hard at first to do this, but in fact the reason the problem is “cute” is because it has a “cute” solution (surprisingly simple). To do this we note that if is entire then so is . But, note that

But, remembering that

and for finite

And thus, by limiting process it follows that

And thus,

And thus it follows by Liouville’s theorem that is constant, but this is only possible if is constant.

Now, suppose that , then we merely note that

and applying the exact same logic as in the previous part we may conclude that for some . Thus, .

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