Previously in this series, I have used two different techniques to show that
where this last integral is taken over the complex plane on the unit circle, a closed contour oriented counterclockwise. Also,
and
,
are the two distinct roots of the denominator (as long as ). In these formulas,
and
. (Also,
is a certain angle that is now irrelevant at this point in the calculation).
This contour integral looks complicated; however, it’s an amazing fact that integrals over closed contours can be easily evaluated by only looking at the poles of the integrand. In yesterday’s post, I established that lies inside the contour, but
lies outside of the contour.
The next step of the calculation is finding the residue at ; see Wikipedia and Mathworld for more information. This means rewriting the rational function
as a power series (technically, a Laurent series) about the point . This can be done by using the formula for an infinite geometric series (see here, here, and here):
The residue of the function at is defined to be the constant multiplying the
term in the above series. Therefore,
The residue at is
From the definitions of and
above,
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