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Complex Integration Problem

  1. Mar 18, 2017 #1
    1. The problem statement, all variables and given/known data
    I need to evaluate the following integral using the antiderivative:
    $$\int log^2(z) \, dz$$
    I don't know how to make a subscript for the integral sign, there should be a "c" on the bottom part. C is any contour from ##π## to ##i##, not crossing the non-positive x-axis.
    2. Relevant equations
    Given above

    3. The attempt at a solution
    The only thing I can think of is to do a substitution, such as u=logz, like in the real case but I haven't officially learned if that's possible so I don't know if I can do it, nor if I even have to. And it specifically says to use the antiderivative so I can't parameterize.
     
  2. jcsd
  3. Mar 18, 2017 #2

    mfb

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    Did you try integration by parts?
     
  4. Mar 18, 2017 #3
    Integration by parts...
    u = (log(x))^2
    dv = 1dx
     
  5. Mar 18, 2017 #4

    mfb

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    If I understand that post correctly, that will make it worse. There is a better choice of the two parts.
     
  6. Mar 18, 2017 #5
    well, i don't want to post the solution but if you use u = log^2(x) than it will reduce the power on the log by 1 and leave you with an easier problem to integrate :)
     
  7. Mar 18, 2017 #6

    FactChecker

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    It works out fine with u=ln(z) and v'=ln(z), but you are right that your way is easier.
     
  8. Mar 18, 2017 #7

    mfb

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    Oh wait, for post 4 I was imagining logs in the denominator for some reason.
    Ignore post 4, both approaches work and the one from sunnnystrong is easier.
     
  9. Mar 18, 2017 #8

    Ray Vickson

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    You just put a '_C' next to your int instruction, to get ##\int_C \log^2 (z) \, dz##. Right-click on the formula and ask for a display of math as tex commands, to see how it is done.

    As for using antiderivatives: see, eg.,
    https://en.wikipedia.org/wiki/Antiderivative_(complex_analysis).
     
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