Sine integral

This article is about a particular entire function: a holomorphic function defined on the whole of

$\mathbb{C}$

View a complete list of entire functions

Definition

The sine integral is defined as the antiderivative of the sinc function taking the value 0 at 0. Denoted $\mathrm{S}\mathrm{i}$, it is defined as:

$\mathrm{S}\mathrm{i}(z_0):=\underset{0}{\overset{z_0}{\int }}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}(z)dz$

The integration could be done over any piecewise smooth path from $0$ to $z_0$: all such integrals yield the same value because of the homotopy-invariance formulation of Cauchy's theorem, and the fact that $\mathbb{C}$ is simply connected.