I was surfing the net today and I fell on this cute integral
I have seen integrals of such kind before like for instance this .In fact something more general holds
The original integral does not fall into this category which is a real shame. Yet it does have a closed form and it does not contain an in its final answer. Strange, huh? So similar but so different at the same time these two integrals. A sign changes everything.
We begin by exploring the integral
Manipulating the integral ( substitutions and known Gaussian results) reveals that
where . Taking the imaginary part of the last expression we get that
and this is the final answer. See, no !. Of course we can also extract the real part and calculate the corresponding integral involving .
Let denote the floor function and denote the ceiling function. Evaluate the integral
We have successively
Let denote the usual inner product of . Evaluate the integral
where is a positive symmetric matrix and .
is a positive symmetric matrix , so is
. For a positive symmetric matrix
there exists an
positive symmetric matrix such that
. Applying this to
our integral becomes
where is the Euclidean norm. Applying a change of variables we have that
Since then by converting to polar coordinates we have that
Here denotes the surface area measure of the unit sphere and it is known to be
where denotes the Gamma Euler function for which it holds that
Let denote the Euler – Mascheroni constant. Prove that
Let us consider the following
It is known that . Also, the Mellin transform of is known to be
On the other hand
What we are seeking is .
and the result follows.
Let . Find a closed form for the integral
We are invoking a powerful theorem stating that:
Theorem: Let be a second degree polynomial (this implies ) . Then it holds that
Making use of the above theorem we have that