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Complex analysis problems

  1. Feb 19, 2013 #1
    1. The problem statement, all variables and given/known data
    a) Find the radius of convergence of the following complex series and the complex point, where the center of the disk of convergence is located:

    [tex] \sum_{n=1}^{inf} 4^n (z-i-5)^{2n} [/tex]



    b) Find the Laurent series of the following function, f(z), about the singularity, z = 2, and find the residue of f(z)

    [tex] f(z) = \frac{1}{z(z-2)^3} [/tex]


    c) Evaluate the following integral:

    [tex] \int_{0}^{inf} \frac{dx}{(x^2 + a^2)^4} [/tex]

    2. Relevant equations

    Given


    3. The attempt at a solution

    a) I gather that 5+i is the center of the disk of convergence? Doing the ratio test I get,

    |4(z-(5+i))^2| < 1

    I'm a bit lost how to solve this from here.

    b) I don't know how to go about expanding this as a Laurent series. If it were a Taylor series, I would factor out a 1/-2^3 from 1/(z-2)^3 and then expand the remaining 1/(1-z/2) and cube it. But this gives me the expansion about z = 0.


    c) You can extend this integral to the complex plane and write

    ∫(closed) 1/(z^2+a^2)^4 dz
    where singularities would be z = +or- i a
    And choosing the upper half of the semi circle contour, I only have to deal with the +'ve i a

    Then using the Residue equation for poles of higher order,
    I find that the integral is 2∏(0) = 0.

    But I'm not sure its correct.
     
    Last edited: Feb 19, 2013
  2. jcsd
  3. Feb 19, 2013 #2

    haruspex

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    What would the relationship be between |z2| and |z|?
    Since you need to expand about z=2, I would substitute w=z-2. This should make it more obvious.
    I don't. Pls post your working.
     
  4. Feb 19, 2013 #3
    a) The relationship would be,

    [tex] |z| = \sqrt{x^2 + y^2} = r [/tex]
    &
    [tex] |z^2| = |z|^2 = r^2 [/tex]


    b) Okay, I got the series expanded by using the substitution.

    [tex] f(z) = \frac{1}{2w^3} - \frac{1}{4w^2} + \frac{1}{8w} - ... [/tex]
    So, 1/8 is the residue.


    c)

    I found my mistake on part c)

    I took the limit before taking the derivative in the formula for the residues of higher order poles.

    Now I get

    Res(a i) = 5/(32a^7i)

    And setting the integral from -inf to inf equal to 2PI i * Res(a i)
    I get 5PI/16a^7

    which becomes 5PI/32a^7 since I'm taking integral from 0 to inf instead of -inf to inf and can multiply by 1/2 since its an even function in the integrand.
     
    Last edited: Feb 19, 2013
  5. Feb 20, 2013 #4

    haruspex

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    Right, so apply that to |4(z-(5+i))2|
    Your b) and c) answers look right.
     
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