Integral of sin(x)/x

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

Integral of sin(x)/x

Posted onJuly 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

.