Integral of sin(x)/x

https://raghumahajan.wordpress.com/2010/07/26/integral-of-sinxx/

Integral of sin(x)/x

Posted on July 26, 2010 byRaghu Mahajan

There is cute formula that reads

.

To derive this, note that is the imaginary part of. Thus we consider the integral of instead. But this has a pole on the real axis. We would like to convert this into a contour integral in the complex plane. For that, we would need an exponential suppression at , which can be obtained only in the upper half plane for . Thus we push the pole into the upper half-plane, to.

Now, a straightforward contour integral yields

.

Taking the imaginary part of this, we get

.

We now take the limit . To establish our formula, we must show that the second term on the left hand side of the above equation equals .  This is true because in the limit, becomes; and thus, the second term above yields.