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Contour Integration used to solved real Integrals

  1. Oct 11, 2013 #1
    1. The problem statement, all variables and given/known data

    Essential Mathematical Methods for the Physics Sciences Problem 15.7

    Show that if f(z) has a simple zero at z0 then 1/ f(z) has a residue of 1/f'(z0). Then use this information to evaluate:

    ∫ sinθ/(a- sinθ) dθ, where the integral goes from -∏ to ∏.


    2. Relevant equations
    The book has further hints:
    The unit circle has only one pole at z= i*a-i(a2-1)^1/2 and therefore has a residue of -i/2 *(a2 -1)-1/2.


    3. The attempt at a solution

    The first part is:
    1/f(z) has a simple pole at z0 leaving the residue to be limz→z0 (z-z0) * 1/f(z) = 1/f'(z0) by definition of the derivative.

    But using this information to do the integral is escaping me. Can anyone just help me understand how to use the hint. I don't actually need the whole step by step guide to the integral.
     
  2. jcsd
  3. Oct 11, 2013 #2

    vela

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    I haven't worked it out, but I'd try using the substitution ##z=e^{i\theta}## and expressing ##\sin\theta## as ##\frac{z-1/z}{2i}##.
     
  4. Oct 12, 2013 #3
    What is a? That matters. For example if [itex]|a|<1[/itex] and real, the integral becomes a principal-valued integral.
     
  5. Oct 12, 2013 #4
    Sorry, a is real and a > 1. Rewriting sin as vela said came to the right answer, thanks.
     
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