How to calculate the Laurent series expansion of 1/(1-z)² in the region 1<|z|?

In summary, a Laurent series expansion is a mathematical representation of a function using both positive and negative powers of a variable. This allows for approximation of functions in regions where they cannot be expressed as a polynomial or regular power series. The key difference between a Taylor series and a Laurent series is that the former only uses positive powers while the latter includes both positive and negative powers. A Laurent series can be calculated using a specific formula, with coefficients found through integration or other methods. Singularities in a Laurent series represent points where the function being approximated is either infinite or undefined. These series are commonly used in mathematics and science to approximate complex functions, especially in fields like physics, engineering, and statistics. They also play an important role in the study of
  • #1
Eldonbetan
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1. I am trying to calculate the laurent series expansion of the function 1/(1-z)² in the region 1<|z|


2. None


3. I can get an answer informally by doing the polynomial division like in high school, but I don't know if this is the right answer and in case it is I cannot prove it. Any help would be largely appreciated.
 
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  • #2
Recall that if a function is analytic on a domain, then its Laurent series and Taylor series are identical. Do you know how to do the Taylor series expansion?
 

What is a Laurent series expansion?

A Laurent series expansion is a mathematical representation of a function in terms of powers of a variable, including both positive and negative powers. It is used to approximate a function in a region where it may not be expressed as a polynomial or a regular power series.

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

A Taylor series is a type of power series that represents a function using only positive powers of the variable, while a Laurent series includes both positive and negative powers. A Taylor series can only approximate a function in a small region around a single point, while a Laurent series can approximate a function in a larger region, including singularities.

How is a Laurent series calculated?

A Laurent series can be calculated by using the formula:
f(z) = ∑n=0∞ an(z-z0)n + ∑n=1∞ bn(z-z0)-n
where z0 is the center of the series and an and bn are coefficients that can be found using integration or other methods.

What is a singularity in a Laurent series?

A singularity in a Laurent series is a point in the complex plane where the function being approximated has an infinite value or is undefined. These points are represented by the negative powers in the series and can be either isolated or essential.

How is a Laurent series used in mathematics and science?

A Laurent series is used in mathematics and science to approximate complex functions, especially in areas such as physics, engineering, and statistics. It is also used in the study of complex analysis and in the development of numerical methods for solving differential equations and other problems.

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