Another Laurent Expansion Question

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SUMMARY

The discussion focuses on finding the Laurent expansions for the function (z - 1) / (z + 1) centered at 0 and at z = -1, specifically determining the convergence at z = 1/2. The breakdown of the function into two parts, [z / (z + 1)] and [-1 / (z + 1)], is essential for deriving the series. The first part can be expressed as 1 / (1 + (1/z)), while the second part requires further analysis for proper expansion. The largest open set for convergence is also a critical aspect of the problem.

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Homework Statement


Find all possible Laurent expansions centered at 0 for
(z - 1) / (z + 1)

Find the Laurent Expansion centerd at z = -1 that converages at z = 1/2 and determine the largest opens et on which
(z - 1) / (z + 1) converges



Homework Equations





The Attempt at a Solution



(z - 1) / (z + 1) breaks down into [z / (z+1)] - [1 / (z+1)]

For the first one divide out the z to obtain 1 / 1 + (1/z) I think? However not being in the form 1 / 1 - (1/z) would this be the same series but negative? That doens't seem right.

For breaking down -1 / (z + 1) I didn't know how to attack that one.
 
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