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

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) = Σ [(fj(0))÷(j!)] × zj

for b) laurent series formula?

3. The attempt at a solution

I'm fairly confident that the answer is f(z) = ([itex]\frac{1}{2}[/itex])_{0}[itex]\sum[/itex]^{[itex]\infty[/itex]}((z^{j}) + (-z)^{j})/(i^{j})

(sum from j=0 to infinity)

But don't understand how to calculate laurent series... I think I need to do it for the annulus centered at the origin with radius 1, and then again for the annulus centered at the origin but with arbitrarily large outer radius and inner radius of 1... or possibly just for the latter.

Do I need to change the format of this sum? split it into two parts? take out the first few terms? any ideas would be great!

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# Laurent Series Complex Analysis question

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