Trouble with an index change in Laurent series

In summary, the individual is struggling to understand how the index and argument were switched in a Laurent series example, as well as the change in upper limit to -infinity. They propose a solution involving rewriting the summation in terms of a new variable, k.
  • #1
saybrook1
101
4

Homework Statement


Hey guys, I'm just going through a Laurent series example and I'm having trouble understanding how they switched the index on a summation from n=0 to n=1 and then switched the argument from z^(-n-1) to z^n as well as changing the upper limit to -infinity. If anyone could shed some light on that for me I would really appreciate it.[/B]

Homework Equations


http://imgur.com/0PkgLim

The Attempt at a Solution


I thought that if you changed the index to n=1 then the power on z should just move up one erasing the ^(-1) and I'm not sure how switching the limit to negative infinity changes anything. Thanks for any help.
 

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  • #2
Try rewriting the summation in terms of ##k = -n-1##. What values does ##k## cover when ##n## goes from ##0## to ##\infty##?
 
  • #3
Thank you very much :)
 

Related to Trouble with an index change in Laurent series

1. What is an index change in Laurent series?

An index change in Laurent series is when the power of the variable in the series changes from positive to negative or vice versa. This can occur when the series has a singularity or pole at the origin.

2. How does an index change affect the convergence of a Laurent series?

An index change can impact the convergence of a Laurent series. If the index change occurs at a singularity or pole, then the series will not converge at that point. However, if the index change occurs at a point where the series is already convergent, then the convergence will not be affected.

3. How can I determine if a Laurent series has an index change?

An index change can be determined by looking at the powers of the variable in the series. If there is a sudden change from positive to negative or vice versa, then there is an index change. Additionally, if the series has a singularity or pole at the origin, there is likely an index change.

4. What are some common methods for dealing with an index change in Laurent series?

One common method for dealing with an index change is to use Cauchy's integral formula, which allows for the calculation of a series with an index change. Another method is to use partial fraction decomposition to simplify the series and make the index change more manageable.

5. Are there any real-world applications of Laurent series with index changes?

Yes, Laurent series with index changes have many real-world applications, particularly in physics and engineering. They are often used in the study of electromagnetic fields and in analyzing the behavior of circuits with capacitors and inductors. They are also used in the study of singularities and poles in complex analysis.

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