What Is the Standard Name for This Theorem About Meromorphic Functions?

[nateHI]

Hi,

In my textbook the following theorem is designated "Proposition 3.4.2 part (vi)". There are 6 parts total in the overall theorem. I'll just type the part I'm interested in below. My question is, is there a more standard name for this theorem? I would like to find an additional introduction to it if possible.

Let $f$ be meromorphic on the open connected set $\Omega\subseteq \hat{\mathbb{C}}$ and let $A$ be the set of its poles in $\Omega$. Then:
(a) $A$ is a countable set.
(b) The accumulation points of $A$ are on the boundary of $\Omega$.
(c) The set $\Omega \setminus A$ is open.
(d) If $K$ is a compact subset of $\Omega$, then $A\cap K$ is a finite set.

 

[mathwonk]

all this follows just from the definition of a pole, since poles are isolated.

 

[nateHI]

all this follows just from the definition of a pole, since poles are isolated.

Agreed. For example, for (a), it's not difficult to show that there is a finite set around each pole and then use the fact that the union of finite sets are countable.

I'm looking for an additional introduction to meromorphic functions that includes this theorem. Any suggestions?

 

[micromass]

I am not sure what you are looking for. Could you tell us why you want another source for this theorem? We might be able to help you more then.

Anyway, the theorem looks a lot like the "identity theorem" for complex analysis: http://en.wikipedia.org/wiki/Identity_theorem This is not a coincidence, it can be explained by using Riemann surfaces.

 

[nateHI]

Could you tell us why you want another source for this theorem? We might be able to help you more then.

Sure. The books introduction to meromorphic functions is scattered throughout the text and relies on a separate section on the Riemann Sphere to build an intuition for $\hat{\mathbb{C}}$. The instructor doesn't like the use of the Riemann Sphere and told the class he would skip that section. I'd like to look for an introduction more in line with the professors teaching method but don't want to bother him with questions I can probably figure out on my own. He spent an entire class talking about that theorem so any book that covers it will probably be in line with his preferred teaching method.

 

[WWGD]

Just use the result that every uncountable subset of the plane has a limit point ( using Weirstrass' result that every bounded infinite subset has a limit point), and then a non-zero holomorphic function cannot have a limit point for its set of roots, and , like mathfunk said, poles cannot either.

 

[nateHI]

On second thought, this might be a good question for the professor after all. You can disregard unless you already found something.

Thanks anyway.

 

[mathwonk]

my point is this set of facts is so trivially derivable from the one fact that a pole is isolated, that it cannot be called a theorem. I.e. this is not a theorem these are "obvious consequences iof the definition". As such there is no guarantee they will appear explicitly in any other book. A true theorem is something like the residue theorem, or the argument principle.