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Laurent Series for a Complex-Valued Function

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Consider f(z) = [(z-i)(z+4)(z-3)]^(-1)

restricted to the domain of definition 0 < |z|< infinity

How many different Laurent series centered at z_0 = 0 does it have? Explain.
Discuss the convergence and divergence sets of each of those Laurent series.
Find two non-zero terms of the Laurent series which represents this f for all z outside some circle |z| = R but diverges inside the circle and find the numerical value of R. Is this function f(z) analytic at 0? at infinity? Explain.

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A Laurent series for a complex-valued function is found. The solution is detailed and well presented. The response received a rating of "5" from the student who originally posted the question.