Laurent series expansion help

The laurent series for 1/(z-1)^2 is -1/z^2 + 2/z - 1 + 2z + ...In summary, in order to find the laurent series for 1/(z^2-1)^2 valid in 0 < |z-1| < 2 and |z + 1| > 2, we need to expand the other factor in the equation as well. The laurent series for 1/(z-1)^2 is -1/z^2 + 2/z - 1 + 2z + ..., and we can use this to find the desired series for the given regions.
  • #1
wakko101
68
0
The problem:

find the laurent series for 1/(z^2-1)^2 valid in 0 < |z-1| < 2 and |z + 1| > 2

we know that f(z) has poles of order 2 at 1 and -1...

In the first region, there are no poles (since z=-1 isn't a part of it). We can write the equation as a product of 1/(z-1)^2 and 1/(z+1)^2. The laurent series for the latter is simply itself, so is the whole equation's expansion simply itself as well? That doesn't seem right to me, because my reasoning for the second region is that we find the expansion for 0 < |z+1| < 1 and subtract it from the first. However, if it too is its own laurent series, then we will get 0.

Any help?
 
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  • #2
You need to expand the other factor.
 

Related to Laurent series expansion help

1. What is a Laurent series expansion?

A Laurent series expansion is a representation of a complex function in terms of powers of (z-a), where z is a complex variable and a is a complex number. It is used to approximate a function in a region where it is not analytic, meaning it has singularities or poles.

2. When is a Laurent series expansion used?

A Laurent series expansion is used when a function has singularities or poles that prevent it from being represented by a Taylor series. It is also used to approximate a function in a region where it is not analytic.

3. How is a Laurent series expansion different from a Taylor series?

A Taylor series is used to approximate a function in a neighborhood of a point where it is analytic, while a Laurent series is used to approximate a function in a region where it is not analytic, including points with singularities or poles. Additionally, a Taylor series only contains positive powers of (z-a), while a Laurent series includes both positive and negative powers.

4. What is the process for finding a Laurent series expansion?

The process for finding a Laurent series expansion involves calculating the coefficients of the powers of (z-a) in the function. This can be done by using the formula for the coefficients or by manipulating the function to match a known Laurent series. It is important to determine the region of convergence for the series and to check for any singularities or poles that may affect the series.

5. Why is a Laurent series expansion useful?

A Laurent series expansion is useful because it allows us to approximate a function in a region where it is not analytic. This can help us understand the behavior of the function near singularities or poles and can also be used for numerical calculations. Additionally, it provides a way to represent a function in a different form, which may make it easier to manipulate or solve problems involving the function.

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