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Homework Help: Power series expansion

  1. May 30, 2010 #1
    1. The problem statement, all variables and given/known data
    Find a power series expansion for log(1-z) about z = 0. Find the residue at 0 of 1/-log(1-z) by manipulation of series, residue theorem and L'Hopitals rule.


    2. Relevant equations



    3. The attempt at a solution
    Is this power series the same as the case for real numbers.
     
  2. jcsd
  3. May 30, 2010 #2
    I have the power series expansion about z = 0 for log(1-z).
    -z - z2/2 - z3/3 - ...
    But how do I find the residues with the methods mentioned
     
  4. May 30, 2010 #3
    When I manipulate do I use the power series for log(1-z)
     
  5. May 31, 2010 #4
    My series for -1/log(1-z) is:
    1/z - 1/2 - z/12 - z2/24 - ...
    So my residue is a-1 = -1/2.
    Is that right?
     
  6. May 31, 2010 #5
    How do I do it by the residue theorem and L'Hopitals rule.
     
  7. Jun 1, 2010 #6
    Make me think of the power series of log(1+z)

    Recall Sir,

    [tex]log(1+z) = \sum_{j=1}^\infty} \frac{(-1)^{j+1}}{j}z^{j} = z - \frac{z^2}{2} + \frac{z^3}{3}-\cdots[/tex]

    so by very very simply replacing z with -z

    you get [tex]-z - \frac{(-z)^2}{2} + \frac{(-z)^3}{3}-\cdots[/tex]

    So the power series expansion of log(1-z)

    Is [tex]P_{n} = -\sum_{j=1}^{n} \frac{z^n}{n}[/tex]
     
    Last edited: Jun 1, 2010
  8. Jun 1, 2010 #7

    vela

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    No. Where'd you get -1/2 from?
     
  9. Jun 1, 2010 #8
    The second term in the series.
     
  10. Jun 1, 2010 #9
    I think I got it now. It is 1 beacuse this is the constant for the z-1 term (the term 1/z)
     
  11. Jun 1, 2010 #10
    Using the formula for the residue at a simple pole (Residue theorem) I also get 1 as my residue.
    res0 = 1
     
  12. Jun 1, 2010 #11

    vela

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  13. Jun 1, 2010 #12
    True.
    Also got 1 using L'Hopitals rule.
    Didn't realise it was so easy.
    Cheers.
     
  14. Oct 25, 2011 #13

    Hi Every body!

    I wan to compute the power series expansion of dedekind eta function. Specifically, I want to know the power series expansion of η(τ)/η(3τ)? How could I expand this function? I would be happy if you could help me as I am stuck at this state when I am computing the modular polynomial of prime number 3.
     
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