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Complex Variables: Missing Step in Example

  • Thread starter MadCow999
  • Start date
  • #1
5
0

Homework Statement


Integral from 0 to 2*pi of [(Cos(3*theta)) / (5 - 4Cos(theta))] (d*theta)


Homework Equations


z=exp(i*theta),
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)


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!
 

Answers and Replies

  • #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].
 
  • #3
5
0
Huzzah! i can see clearly now(!)...
thanks a bunch! now (hopefully) i can apply that stuff in my homework problems!
 

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