2

I have been studying this proof for a bit and I was wondering if anyone is aware of a more succinct and elegant proof of the uniqueness theorem. Maybe this isn't possible, but I figured it couldn't hurt to ask.

Also, if we assume connectedness of the domain does that make for a simpler argument perhaps? enter image description here

MathIsHard
  • 2,733
  • 16
  • 47
  • By Rouchè's theorem every nonzero holomorphic function has isolated zeros – Bob Aug 02 '18 at 21:28
  • @Bob: and how are you proposing to prove Rouchè's theorem more sucinctly and elegantly than the above? – Rob Arthan Aug 02 '18 at 21:50
  • 1
    @MathIsHard: The result is false if $G$ is not connected. There may be more succinct arguments or more "elegant" arguments, but I doubt it. – Rob Arthan Aug 02 '18 at 22:00
  • @Bob How could I use that in the argument of the uniqueness theorem? Perhaps with that (and not needing to prove Rouche's theorem) could you prove the uniqueness theorem? – MathIsHard Aug 02 '18 at 22:53
  • 1
    @mathishard $f-g$ would have a non-isolated zero, so it is null in a neighbourhood of that point and then using connectedness you get the conclusion – Bob Aug 03 '18 at 03:01
  • oh I see. Thank you. I appreciate your time. This is exactly the type of thing I was hoping to find :) – MathIsHard Aug 03 '18 at 03:26
  • @MathIsHard see also here: https://math.stackexchange.com/questions/2101297/possible-alternate-proof-of-uniqueness-of-power-series/2835314#2835314 – Bob Aug 03 '18 at 04:20
  • However, as pointed out by Rob Arthan, it is a bit unfair to use a bazooka in order to get rid of a fly... – Bob Aug 03 '18 at 04:27
  • I know what you mean... I’m studying for three big tests at the end of the month and this is kind of just a last thing I want to try and get quick though in this case. Using other machinery seems to do that. This is also similar to how my teacher did it in class. I have a partial proof in my notes but the end uses an example and doesn’t really finish the proof. It also didn’t mention the Rouche theorem but uses the idea I believe. – MathIsHard Aug 03 '18 at 06:28

0 Answers0