In this post I want to share this cute result that I learned recently:
Let us see how the first integral is derived and then generalised. The integrand has poles at in the complex plane and we may apply contour integration to proceed. Note that as is an odd function, and so
It therefore suffices to consider the integrand . We consider the semicircular contour of radius (to be traversed anticlockwise) in the upper half plane centred at 0. It encloses the pole at .
Along this closed contour we use the residue theorem to compute
On the semicircular arc for and so
where in the second last step we use the fact that distance to the origin of a circle of radius centred at -1 is at least . Then
It follows that and we are left with
To generalize this result to integrals of the form for non-negative integers , we choose the same contour as above and use the residue limit formula for poles of order :
To apply (7) we make use of the General Leibniz rule for the ‘th derivative of a product:
where is the ‘th order Bessel polynomial defined by
For example, for , the integral is given by
The general sum in (8) can be also be written as
where is the modified Bessel function of the second kind.
Proceeding similarly to (4), and so similar to (5) the integral on the arc converges to 0 as . Hence following the same argument as in (6) the desired line integral along the real axis is equal to the contour integral in (8).
Evaluating (8) for the first terms of are given by
In this sequence , the denominators are related to the previous ones by multiplication by , while curiously the numerators are related by the second order recurrence
This follows from the following recurrence relation for the Bessel polynomials:
This can be proved using (9). We have
Substituting this into (15),
thus verifying (14).