(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

For f(z) = 1/(1+z^2)

a) find the taylor series centred at the origin and the radius of convergence.

b)find the laurent series for the annulus centred at the origin with inner radius given by the r.o.c. from part a), and an arbitrarily large outer radius.

2. Relevant equations

for a) (sum from j = 0 to infinity)

f(z) = [itex]\Sigma[/itex] [(f[itex]^{j}[/itex](0))[itex]\div[/itex](j!)] [itex]\times[/itex] z[itex]^{j}[/itex]

for b) laurent series formula?

3. The attempt at a solution

From what I understand, the radius of convergence is from Zo (in this case, the origin) to the closest point where f(z) isn't analytic. f(z) isn't analytic at i or -i. This function is a circle, discontinuous at i and -i. So, by inspection(?), the r.o.c. should be 1.

I don't get how to input the information I have into the formula for a). I think that in understanding this, finding the laurent series should be simplified.

Thanks

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# Complex analysis, taylor series, radius of convergence

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