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Complex Variables: Missing Step in Example

  1. May 20, 2008 #1
    1. The problem statement, all variables and given/known data
    Integral from 0 to 2*pi of [(Cos(3*theta)) / (5 - 4Cos(theta))] (d*theta)

    2. Relevant equations
    Cos(theta) = (z + z^(-1)) / 2,
    Cos(3*theta) = ((exp(3*i*theta)+exp(-3*i*theta)) / 2 = (z^(3) + z^(-3)) / 2,
    dz = (i*z) (d*theta)

    3. The attempt at a solution
    This was an example in my book that didnt show all the steps T_T
    Heres what they did:
    =>(integral of...){[(z^(3) + z^(-3))/2] / [5 - 4(z + z^(-1))} (dz/iz)
    ==>-(1/2i)(integral sign)[(z^(6) + 1) / [(z^(3))(2z - 1)(z - 2)]] (dz)

    i dont know what voodoo magic they pulled there, but i would like to find out!
    thanks for your time!
  2. jcsd
  3. May 20, 2008 #2
    Factor out [tex]z^{-3}[/tex] on the top, pull the [tex]2/i[/tex] right out of the integral, multiply the denominator through with that [tex]z[/tex] that was dividing [tex]dz[/tex]. You are now almost there...

    Multiply the denominator by -1, and put a -1 outside of the integral so that you haven't changed the expression. The denominator should now look like [tex]z^3(2z^2-5z+2)[/tex] now you must factor that quadratic [tex]2z^2-5z+2 = (2z - 1)(z - 2)[/tex].
  4. May 20, 2008 #3
    Huzzah! i can see clearly now(!)...
    thanks a bunch! now (hopefully) i can apply that stuff in my homework problems!
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