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jtleafs33
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Homework Statement
Evaluate the integral:
[itex]\int[/itex][itex]^{2\pi}_{0}[/itex][itex]\frac{d\theta}{(A+Bcos(\theta))^2}[/itex]
[itex]a^2>b^2[/itex]
[itex]a>0[/itex]
The Attempt at a Solution
First, I convert this to contour integration along a full sphere in the complex plane.
I let:
[itex]z=e^(i\theta)[/itex]
[itex]dz=ie^(i\theta)[/itex]
[itex]d\theta=-idz/z[/itex]
[itex]cos(\theta)=(z+z^-1)/2[/itex]
Now, substituting back into the integral, I drop the integral sign for now and just work on the integrand:
=[itex]\frac{-idz}{z(a+\frac{bz+bz^-1}{2})}[/itex]
=[itex]\frac{-idz}{z(\frac{2a+bz+bz^-1}{2})}[/itex]
=[itex]\frac{-4idz}{z(2a+bz+bz^-1)^2}[/itex]
=[itex]\frac{-4idz}{b^2z^3+4abz^2+4a^2z+2b^2z+4ab+\frac{b^2}{z}}[/itex]
And from here I am stuck on how to find the poles of this function. I want to use the residue theorem to evaluate this integral, but like I said I'm stuck here.
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