# Clarifications regarding definitions of Taylor, Laurent etc. series

• Avichal
In summary, Laurent series is a series that is convergent on a disk in the complex plane while Taylor series is a series that is convergent on a annulus. Every holomorphic function that is below the annulus has a convergent Laurent series, but the exact values of the coefficients depend on the annulus. Practically, a Laurent series will contain terms with negative powers, while a Taylor series will not.
Avichal
I want to know the difference between various kinds of series like Taylor, Laurent and Asymptotic.

I have some understanding but I want some clarifications. Here is what I understand:-
1) Taylor series is just f(0) + x.f'(0) + x2.f''(0)/2! + ...
2) Laurent series is applied when taylor series cannot be applied like when f(0) is not defined.
3) Asymptotic series is around infinty point.

Am I correct? Correct me if I'm wrong

Here are some crucial differences:
1. A Laurent series is convergent on an annulus in the complex plane, whereas the Taylor series is on a disk.

2. EVERY function that is holomorphic on r < | z - a | < R has a unique, convergent Laurent series (defined on this disk. The exact values of the coefficients depend on the annulus). This is a very strong claim, this is far from being true for a Taylor series. You might have R = infinity and r=0.

3. From a practical point of view: a Laurent series contains in general terms with negative powers, e.g. (z-a)^(-n) (when r > 0).

Here are some crucial differences:
1. A Laurent series is convergent on an annulus in the complex plane, whereas the Taylor series is on a disk.

2. EVERY function that is holomorphic on r < | z - a | < R has a unique, convergent Laurent series (defined on this disk. The exact values of the coefficients depend on the annulus). This is a very strong claim, this is far from being true for a Taylor series. You might have R = infinity and r=0.

3. From a practical point of view: a Laurent series contains in general terms with negative powers, e.g. (z-a)^(-n) (when r > 0).

Before that I have one more confusion. What does around a point mean when finding series of some function.
When finding taylor series, that means we use derivative at that point to approximate.

In case of laurent series, what does 'around some point' mean?
Also suppose I want a series expansion of e1/x what would I apply? Laurent or asymptotic?

Also am I write about asymptotic series i.e. it is series around point = infinity?

## 1. What is the difference between Taylor and Laurent series?

The main difference between Taylor and Laurent series is that Taylor series is centered around a single point, while Laurent series includes both positive and negative powers of the variable centered at a point.

## 2. How do you determine the convergence of a Taylor or Laurent series?

The convergence of a Taylor series can be determined by using the ratio test or the root test. For Laurent series, the convergence depends on the annular region within which the series is centered and the behavior of the function at the boundary of this region.

## 3. Can a Taylor series be used to approximate a function at any point?

No, a Taylor series can only be used to approximate a function at points within its radius of convergence. If the point is outside this radius, the series may not converge to the function's value at that point.

## 4. How can you find the coefficients of a Taylor or Laurent series?

The coefficients of a Taylor series can be found by taking derivatives of the function at the center point. For Laurent series, the coefficients can be found by using the formula for the general term of a power series and solving for the coefficients.

## 5. What are some applications of Taylor and Laurent series in real-world problems?

Taylor and Laurent series are used in fields such as physics, engineering, and economics to approximate and analyze functions. They are also used in numerical analysis to solve differential equations and in computer graphics to create smooth curves and surfaces.

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