Computing the sine integral

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

$P V \int_{- \infty}^{\infty} \frac{\sin x}{x} = π$

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:

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

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 $- i π$ , hence the integral along the real axis is $i π$ . Thus, its imaginary part is precisely $π$ .