How can I find the first few terms of the Laurent series for

In summary, for the given function, 1/(z*(z-1)(z-2)^2), the partial fraction expansion yields 1/(z-1)- 1/(4z) - 3/(4(z-2)) + 1/(2(z-2)^2). The term -1/(4z) is already in the desired form, while the other terms can be expanded using Taylor series to obtain the Laurent series for each annular ring between singular points. The residue of the function at the origin can be found by evaluating the coefficient of 1/z in the Laurent series.
  • #1
Jonobro
7
0

Homework Statement


For each of the following functions find the first few terms of each of the Laurent series about the origin, that is, one series for each annular ring between singular points. Find the residue of each function at the origin.

The function is...

1/(z*(z-1)(z-2)^2)

Homework Equations


N/A

The Attempt at a Solution


I did partial fraction expansion and got 1/(z-1)- 1/(4z) - 3/(4(z-2)) + 1/(2(z-2)^2) but am not sure where to go from here... Any help would be appreciated.
 
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  • #2
The ## -1/(4z) ## term already has the form you want. So how about Taylor expanding the other terms about ## z = 0 ## to make them into power series in ## z ##?
 

FAQ: How can I find the first few terms of the Laurent series for

1. How can I find the first few terms of the Laurent series for a given function?

To find the first few terms of the Laurent series for a given function, you can use the general formula for Laurent series expansion, which involves finding the coefficients of the series. This can be done by using techniques such as the Cauchy integral formula or the method of partial fractions. Alternatively, you can also use software programs like Mathematica or Wolfram Alpha to calculate the Laurent series for you.

2. What is the difference between a Laurent series and a Taylor series?

A Taylor series is a type of power series that represents a function as an infinite sum of powers of its variable, while a Laurent series is a type of power series that includes both positive and negative powers of the variable. This means that a Taylor series is only valid for functions that are analytic (smooth) at a given point, while a Laurent series can be used for functions that have poles or singularities at that point.

3. Can I find the Laurent series for any function?

No, not all functions have a Laurent series representation. A function must be analytic (smooth) at a given point for its Laurent series to exist at that point. Functions that have poles or essential singularities at a given point do not have a Laurent series representation at that point.

4. How do I know when to use a Laurent series instead of a Taylor series?

You should use a Laurent series when the function you are working with has poles or essential singularities at a given point. In these cases, a Taylor series would not be able to accurately represent the function, so a Laurent series is needed to include both positive and negative powers of the variable.

5. Are there any applications of Laurent series in real-world problems?

Yes, Laurent series have many applications in various fields such as physics, engineering, and economics. They can be used to approximate functions and solve differential equations, and they are also used in signal processing and control systems. Additionally, Laurent series are used in the study of complex analysis and in the development of numerical methods for solving complex problems.

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