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March 26th, 2014, 02:08 AM   #1
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Detirmine where a laurent series is analytic

I have the following series and have to discuss where it is analytic
g(z)= sum_{n=1}^infinity ( i( n^2+3)/(3*(2n^2-1)))^n * (z^n + z^-n)

This is a laurent series. I found that without the negative powers of z, the series has a radius of converge of 1/6.
g(z) is abviously not analytic at z=0. But I thought the idea was to find some anulus where the whole thing is analytic.
How can I do this with z^-n part?
or is the answer as easy as we need |z|<1/6 and z not equal to zero, and in this case, how can I show this?

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chanman
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March 26th, 2014, 07:03 AM   #2
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Re: Detirmine where a laurent series is analytic

Since the is the only "problem" and then only at z= 0, this is clearly analytic for all non-zero z. As an "annulus" that would be written as .
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March 26th, 2014, 11:51 AM   #3
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Re: Detirmine where a laurent series is analytic

That is wrong to say because if the laurent series does not converge, then the function g(z) is not analytic
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