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All Laurent series expansion around 1.

  1. Jul 14, 2010 #1
    1. The problem statement, all variables and given/known data

    Question is= Find all Laurent series expansion of f(z)=z^4/(3+z^2) around 1. I will be very very thankful if someone can help me to do this question.

    2. Relevant equations



    3. The attempt at a solution

    can I assume (z-1=u) here and change the function in terms of $u$. then i will have

    f(u+1)=(u+1)^4/(u+2)^2

    It should have singularity at $u=-2$. now I m big confuse about in what regions I should compute Laurent series.

    I have trouble of thinking that how many laurent series I will have and what will be the conditions on $z$.

    Thanks !!!!
     
  2. jcsd
  3. Jul 14, 2010 #2

    lanedance

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    first, subbing in for z = u+1
    g(u) = f(u+1)=(u+1)^4/(3+(u+1)^2) =(u+1)^4/(u^2+ 2u +4)
     
  4. Jul 14, 2010 #3

    lanedance

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    there will be a laurent series for each disk about z-1 = u = 0 defined by the poles of the function
     
  5. Jul 14, 2010 #4

    lanedance

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  6. Jul 14, 2010 #5

    Gib Z

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    You've done the substitution to center the Laurent expansion around u=0, now draw a diagram, plotting the poles of the new function and draw the annuli these poles split the plane into. For each annuli, decide what the rule is for that region (eg 8< |u| < 23 or something).

    The next part is a bit harder: Try to manipulate the expression you have ( [tex]\frac{ (u+1)^4}{(u+1)^2+3}[/tex] ) into the product of two terms, 1 a polynomial, and another that you can interpret as the sum of a geometric series. Then you can expand the geometric series into a sum, and multiply the polynomial into it to get the Laurent series.
     
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