Integrate the following:
(sin(x)/x)^4 between negative infinity and infinity.
The residue theorem, contour integral techniques.
The answer should be 2pi/3
The Attempt at a Solution
I'm not even sure where to start honestly. I define a function f(z)=(sin(z)/z)^4. I'm not quite sure what to make of the point z=0, but I make a contour integral the shape of half a donut in the upper half plane with a little half-circle above z=0. So, I have 3 integrals to consider, the principal value integral on the x-axis, the one on the little half-circle and the big half-circle.
According to the residue theorem, the sum of these integrals should give me 0. I'm pretty sure that using Jordan's lemma, we can prove that the integral on the big half-circle is 0. Also, the principal value integral on the x-axis is the original function. What do I do with the last part now?
I define z=εe^(iθ) there and insert in my function. The integral is between pi and 0, and I need to take the limit of ε as it goes to 0.
I'm honestly lost, is there any chance someone could help me at least start this problem? I don't know if what I've written above is correct or not. Just a little help please =(, this problem has been a great a source of stress for me recently.