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

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

• Results about the sine integral function can be found here.

Sources

• 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Some Special Functions: $\text V$. The Sine and Cosine Integrals
• 2005: Roland E. Larson, Robert P. Hostetler and Bruce H. Edwards: Calculus (8th ed.): $\S 4$ P.S.

• Weisstein, Eric W. "Sine Integral." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SineIntegral.html