Another Laurent Expansion Question

brianhawaiian
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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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Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...

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