# Isolated Singularities Foundation Questions

• Apr 26th 2009, 07:52 AM
Isolated Singularities Foundation Questions
In my class we are learning about isolated singularities, here is my understanding:

1. Removable Singularity: a function f that is analytic in an open disc except for at the origin.
Property:
a) f is bounded in the neighorhood of that point.
b) All positive terms.

2. Pole of Order m: the coefficient of the -1 term of the power series.
Property:
a) $\displaystyle |f(z)| \rightarrow \infty$
b) Finite many negative terms.

3. Essential Singularity: A function f with Laurent series that has infinitely many negative terms.
Property:
a) The image f(U) is dense in complex plane.

Are my properties listed correctly?

Here are my questions:

Why is property 2a true?

If a function f is completely holomorphic in a disc D, including its center, then it doesn't have a singularity, right?

This question is more in general, but how does a meromorphic function tie to all this? A meromorphic can only be a pole?

Thank you.
• Apr 26th 2009, 08:21 AM
manjohn12
A meromorphic function is analytic everywhere except at poles. If a function has a pole of order $\displaystyle m$, then we can write it as: $\displaystyle f(z) = \frac{g(z)}{(z-z_0)^{m}}$. Now as $\displaystyle z \to \infty$, $\displaystyle |f(z)| \to \infty$.