Sine integral

This article is about a particular entire function: a holomorphic function defined on the whole of $\mathbb {C}$
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Definition

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

$\operatorname {Si} (z_{0}):=\int _{0}^{z_{0}}\operatorname {sinc} (z)\,dz$

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.