This series was inspired by a question that my wife asked me: calculate
Originally, I multiplied the top and bottom of the integrand by
and performed a substitution. However, as I’ve discussed in this series, there are four different ways that this integral can be evaluated.
Starting with today’s post, I’ll begin a fifth method. I really like this integral, as it illustrates so many different techniques of integration as well as the trigonometric tricks necessary for computing some integrals.
where is the contour in the complex plane shown below (graphic courtesy of Mathworld).
Amazingly, contour integrals can be simply computed by evaluating the residues at every pole located inside of the contour. (See Wikipedia and Mathworld for more details.) I handled the case of in yesterday’s post. Today, I’ll begin the case of .
To find the poles of the integrand, I use the quadratic formula to set the denominator equal to zero:
As shown earlier in this series, the right-hand side is negative if . So, for the sake of simplicity, I’ll define
so that the four poles of the integrand are , , , and . Of these, only two ( and ) lie within the contour for sufficiently large , and so I’ll need to compute the residues for these two poles.
Before starting that task, I notice that
Matching coefficients, I see that
These will become very handy later in the calculation.
The integrand has the form , and each pole has order one. As shown earlier in this series, the residue at such pole is equal to
In this case, and so that , and so the residue at and are given by
Finally, to evaluate the contour integral, I simply the sum of the residues within the contour and then multiply the sum by :
So, to complete the evaluation of , I’m left with the small matter of simplifying the right-hand side. I’ll tackle this in tomorrow’s post.