# A sine integral \int_0^{\infty} \left(\frac{\sin x }{x }\right)^n\,\mathrm{d}x\int_0^{\infty} \left(\frac{\sin x }{x }\right)^n\,\mathrm{d}x

The following question comes from Some integral with sine post

but now I’d be curious to know how to deal with it by methods of complex analysis.
Some suggestions, hints? Thanks!!!

Sis.

Here’s another approach.

We have

If $n-2k \ge 0$ we close the contour in the upper half-plane and pick up the residue at $x=i\epsilon$.
Otherwise we close the contour in the lower half-plane and pick up no residues.
The upper limit of the sum is thus $\lfloor n/2\rfloor$.
Therefore, using the Cauchy differentiation formula, we find

The sum can be written in terms of the hypergeometric function but the result is not particularly enlightening.