Laurent series: can calculate myself, just need a quick explanation how

To find the expansion in an annulus, you would use the same procedure as for a ball, except now you have to be careful about the two boundaries. In summary, The user is asking for clarification on how to find the three central terms of the Laurent series for 1/(cos(z)-1) in the region 2pi<|z|<4pi and how to calculate an expansion in an annulus. The term "central" refers to the terms around the z0 term and the procedure for finding the expansion in an annulus is similar to that for a ball, but with consideration for the two boundaries.
  • #1
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Homework Statement


Hi all,

I've just calculated the first three nonzero terms of the Laurent series of 1/(cos(z)-1) in the region |z|<2pi, and now I've been asked to 'find the three non-zero central terms of the Laurent expansion valid for 2pi<|z|<4pi' - firstly, what does it mean by 'central terms', and secondly, how do I calculate an expansion valid in an annulus? I have no idea how to expand except for in a ball |z-a|<r, unless I'm being slow here (very possible)! Perhaps the second question will answer the first for me anyway - thanks very much for the help,

M
 
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  • #2
"Central" probably means the terms around the z0 term. The Laurent series can have powers of z that go from -infinity to +infinity, so the "center" would be the n=0 term.
 

1. What is a Laurent series?

A Laurent series is a mathematical representation of a complex function, which is written as a power series with both positive and negative powers of the independent variable. It is useful in analyzing functions with singularities (points where the function is not defined).

2. How is a Laurent series calculated?

A Laurent series is calculated by expanding a function around a point, typically a singularity, into an infinite sum of terms with positive and negative powers of the independent variable. This is done using the Cauchy integral formula, which involves computing complex integrals.

3. What is the purpose of a Laurent series?

A Laurent series is used to approximate complex functions and to analyze their behavior near singularities. It can also be used to calculate residues, which are important in evaluating complex integrals.

4. Can I calculate a Laurent series myself?

Yes, with a good understanding of complex analysis and the Cauchy integral formula, you can calculate a Laurent series yourself. However, it can be a lengthy and complex process, so it is often more efficient to use computer software or consult a mathematical expert.

5. Can you give a quick explanation of how to calculate a Laurent series?

To calculate a Laurent series, choose a point around which you want to expand the function and determine the region of convergence. Then, use the Cauchy integral formula to compute the coefficients of the series. Finally, write out the series with the appropriate positive and negative powers of the independent variable. This process can be repeated for multiple singularities, if present.

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