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

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SUMMARY

The discussion focuses on calculating the central terms of the Laurent series for the function 1/(cos(z)-1) in the annulus defined by 2π < |z| < 4π. The user has already computed the first three nonzero terms for |z| < 2π and seeks clarification on what "central terms" means and how to perform the expansion in an annular region. It is established that "central terms" refer to the terms surrounding the z0 term, specifically the n=0 term in the series.

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  • Understanding of Laurent series and their properties
  • Familiarity with complex analysis concepts, particularly annular regions
  • Knowledge of Taylor series for comparison
  • Basic skills in manipulating trigonometric functions in complex variables
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Students and professionals in mathematics, particularly those studying complex analysis, as well as anyone involved in advanced calculus or mathematical physics who needs to understand series expansions in annular regions.

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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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"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.
 

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