Computing the sine integral

This article studies the computation of the following improper real integral:

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{\sin x}{x}=\pi$

Here, the value at $0$ is assigned to be 1. (The function being integrated is termed the sinc function and its indefinite integral is termed the sine integral.

Computation

We first consider the function:

$z\mapsto \frac{e^{iz}}{z}$

This is a holomorphic function and its imaginary part is $(\sin x)/(x)$.

By the Jordan's lemma, we see that the integral