Definition:Sine Integral Function

Definition
The sine integral function is the real function $\Si: \R \to \R$ defined as:


 * $\map \Si x = \ds \int_{t \mathop \to 0}^{t \mathop = x} \frac {\sin t} t \rd t$

Graph

 * SiFunction.png

Also defined as
Some sources consider the value of the sine integral function at $x = 0$ as a special case:


 * $\map \Si x = \begin{cases}

\ds \int_{t \mathop \to 0}^{t \mathop = x} \frac {\sin t} t \rd t & : x \ne 0 \\ \vphantom x \\ 0 & : x = 0 \\ \end{cases}$

but as $\map \Si 0 = 0$ as an emergent property of the sine integral function, this is not strictly necessary.

Also see

 * Limit of $\dfrac {\sin x} x$ at Zero
 * Sine Integral Function of Zero
 * Dirichlet Integral


 * Definition:Cosine Integral Function