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Homework Help: Integrals of trigonometric functions over [o,2pi]

  1. Jun 30, 2010 #1
    1. The problem statement, all variables and given/known data
    ∫dθ/(1+βcosθ)^2 ; -1<β<1
    θ=0 to 2pi

    2. Relevant equations

    3. The attempt at a solution
    attempt solution:

    1) make substitution:

    2) substitute:


    3) Next ?

    3a)Find the poles ?
    We dont know how....

    3b)Compute the residues

    3c)Calculate integral
    Last edited: Jun 30, 2010
  2. jcsd
  3. Jun 30, 2010 #2
    That's a tough one I think jwhite. Need to know how to find the poles to calculate the residue. First write it clearly:


    and doing the [itex]z=e^{it}[/itex] substitution, I get:


    Now, you can figure when that denominator is zero to find the poles and then figure which ones are in the unit circle when [itex]-1<b<1[/itex]. Note when you factor it (don't forget to factor out the b first), and the factors are squared, that means the poles are second order. You'll need to know how to compute the residue of a second-order pole. For example, if it were:


    then the residue at for example z2 would be:

  4. Jul 1, 2010 #3
    Thank you very much Jackmell, really appreciate it.
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