Question about Laurent series and analytic functions. I don't think I understand when a Laurent series vs Taylor series should be used. More importantly, I don't think I understand how a Taylor series can be analytic with a singularity. A few questions:

*

*Can an analytic function (as one where the derivative, defined as a limit of ratios, existed) exist even if there's a singularity at a point?

*So only analytic functions can be expressed as power series? Is that correct?

*So in the example below, it has a singularity yes but it can be expressed as a Taylor series and not a Laurent series. Why is this?

*If the radius R = 0, what does that mean? Does it converge at the singularity? What does that mean?

*I guess my main question is: Can a function be analytic in a domain even if it has a singularity?

For example, I know this can be expressed as a Taylor series: $\frac{1}{3-z}$ around the point z = 1
$$\frac{1}{2 - (z - 1)} = \frac{1}{2} \cdot \frac{1}{1 - \frac{z - 1}{2}}$$
Note that the leading 1 in the denominator is in the right form so we can use:
$$\frac{1}{1-z} = 1 + z + z^2 + ...\sum_{n=0}^{\infty} z^n$$ and this converges if: \left | z \right | < 1
So even though this series has a singularity at z = 3, it can be expressed as a Taylor series. But is this Taylor series analytic everywhere? It's not at z=3 right?
The series above: $$\frac{1}{2} \cdot \frac{1}{1 - \frac{z - 1}{2}} = \frac{1}{2}( 1 + (\frac{z-1}{2}) + (\frac{z - 1}{2})^2 + ...$$ converges if $$\left | \frac{z-1}{2} \right | < 1$$
Here's a quote about Laurent series:

A “Laurent series” for a function is a generalization of the Taylor series, and like a Taylor series, it is essentially a power series. With a Laurent series, however, the powers can be negative. A major advantage of the Laurent series over the Taylor seriesis that Laurent series can be expanded around singular points for a given function, and can then be used to analyze the function in the neighborhoods of its singularities. This analysis will be needed when we develop “residue theory” for computing all sorts of weird integrals that can arise in applications.
By the way, a singular point for a function f is a point on the complex plane where f
might not be analytic

What does it mean to expand around singular points for a given function?
 A: There is a lot to address in this question, maybe more than ought to be addressed in one question in a forum like this. First of all, I will suggest that you read about Laurent series, analyticity and singularities in an introductory text on complex analysis. Some good alternatives are Theodore Gamelin's Complex Analysis and Saff and Snider's Fundamentals of Complex Analysis. That said, I will briefly address some of the points here:
A analytic function can have a limit at a singular point. This is what we call a removable singularity, and it means that the function can be redefined such that it is analytic in that point. In this case the Laurent series will equal the Taylor series. In addition to removable singularities, we distinguish between poles and essential singularities. These are defined through properties of the Laurent series around the point in question.
The Laurent series is, as you mention, a generalized Taylor series. It is generalized in the sense that we allow for terms with negative exponents. A Laurent series is therefore usually written as
$$\sum_{n = -\infty}^{\infty} a_n (z-z_0)^n,$$
which is a notation for the formal series
$$\sum_{n = 1}^{\infty} \frac{a_{-n}}{(z-z_0)^n} + \sum_{n=0}^{\infty} a_n (z-z_0)^n.$$
Obsere that the last parts looks exactly like a Taylor series. Whereas a Taylor series converges in a disc, a Laurent series converges in an annulus. An annulus is a region in the complex plane of the form $r < |z-z_0| < R$ (You should be able to see this from the expression above if you are familiar with power series). Note that we allow for $r=0$, where the annulus is a punctured disc, and $R= \infty$. This is why a Laurent series is defined around a singularity. $z_0$ would typically be a singularity and the Laurent series is a representation of the function around that singularity. Intuitively the terms with negative exponents "make up for" the singularity, and how many such terms are included in the series tells us what kind of singularity we are near. If there are none, it is a removable singulary. If there are finitely many but at least one, we call it a pole, whereas if there are infinitely many, we are dealing with an essential singularity.
Yes, analytic functions can be expressed locally as power series. In some textbooks this is in fact the definition of analyticity. Be aware that there is not just one Laurent series representation of an analytic function. As mentioned, the representation is local, and will depend on the domain you are studying.
