I am looking for a short proof that ∫0∞(sinxx)2dx=π2 What do you think?
I am looking for a short proof that
What do you think?
Answer & Explanation
Well, it's not hard to reduce this integral to . It is easy to calculate the Fourier transform
Taking the inverse Fourier transform, we get
and the result follows.
The second integral can be computed in a similar way. Just take (the indicator function of the interval
Now close the contour in the upper half plane, enclosing the pole at with residue , yielding
From squaring the identity
and integrating we get
as extends to a continuous function on . Hence as and