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).

In the previous solution, I used the “magic substitution” to convert the last integrand to a simple rational function. Starting today, I’ll use a completely different technique to compute this last integral.

The technique that I’ll use is 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,

Using these last two equations, I can solve for and in terms of and . I’ll begin with :

Though not necessary for this particular, let me solve for for completeness:

Finally, let me solve for the differential :

I’ll continue with this different method of evaluating this integral in tomorrow’s post.