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Homework Help: Slightly Harder Cauchy Integral

  1. Apr 28, 2012 #1
    1. The problem statement, all variables and given/known data

    Evaluate the integral [itex]I_1 = \int_0^{2\pi} \frac{d\theta}{(5-3sin\theta)^2}[/itex]

    2. Relevant equations

    3. The attempt at a solution

    I start off by switching the sine term for a complex exponential [itex]e^{i\theta}=cos\theta +isin\theta[/itex]
    I will consider only the Imaginary component of the solution.

    now make the substitution: [itex]z=e^{i\theta}[/itex]

    so we have:

    [itex]I_1 = Im\left(I_2\right)[/itex]

    [itex]I_2 = \int_0^{2\pi} \frac{1}{(5-3z)^2}\frac{dz}{iz}=\frac{1}{i}\int_0^{2\pi}\frac{dz}{z(5-3z)^2}[/itex]

    so we have 2 poles: a simple pole at z=0 and a second order pole at z=5/3

    I'm not sure wether i need to include both residues. the question does not indicate the contour to integrate. Should it be a unit circle? I'm not sure why it would be. any ideas?
  2. jcsd
  3. Apr 30, 2012 #2
    The original integral is over θ from 0 to 2π. When you change your integration variable to z=exp(iθ), what values can z take?
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