Previously in this series, I have used two different techniques to show that
where and (and is a certain angle that is now irrelevant at this point in the calculation).
Earlier in this series, I used the magic substitution to evaluate this last integral. Now, I’ll instead use contour integration; see Wikipedia for more details. I will use Euler’s formula as a substitution (see here and here for more details):
so that the integral is transformed to a contour integral in the complex plane. Under this substitution, as discussed in yesterday’s post,
Employing this substitution, the region of integration changes from to a the unit circle , a closed counterclockwise contour in the complex plane:
While this looks integral in the complex plane looks a lot more complicated than a regular integral, it’s actually a lot easier to compute using residues. I’ll discuss the computation of this contour integral in tomorrow’s post.