Computing the sine integral

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

$\operatorname {PV} \int _{-\infty }^{\infty }{\frac {\sin x}{x}}=\pi$

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 (when restricted to the real axis) is $x\mapsto (\sin x)/(x)$.

We now choose the "mousehole contour" method of integration (further information at mousehole contour integration method). The contour is given by semicircles centered at the origin and in the upper half-plane of radii $R_{1}$ and $R_{2}$, where $R_{1}$ tends to zero and $R_{2}$ tends to $\infty$.

By Jordan's lemma, the part of the integral on the outer circle tends to zero. The part of the integral on the inner circle tends to $-i\pi$, hence the integral along the real axis is $i\pi$. Thus, its imaginary part is precisely $\pi$.