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March 26th, 2014, 02:08 AM  #1 
Newbie Joined: Mar 2014 Posts: 5 Thanks: 2  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^21)))^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? Thanks chanman 
March 26th, 2014, 07:03 AM  #2 
Math Team Joined: Sep 2007 Posts: 2,409 Thanks: 6  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 nonzero z. As an "annulus" that would be written as .

March 26th, 2014, 11:51 AM  #3 
Newbie Joined: Mar 2014 Posts: 5 Thanks: 2  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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analytic, detirmine, laurent, series 
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