# sine integral

The function (in Latin sinus integralis) from $\mathbb{R}$ to $\mathbb{R}$ is defined as

 $\displaystyle\mbox{Si }{x}\;:=\;\int_{0}^{x}\frac{\sin t}{t}\,dt=\int_{0}^{x}% \mbox{sinc}(t)\;dt,$ (1)

or alternatively as

 $\mbox{Si }{x}\,:=\,\int_{0}^{1}\frac{\sin{tx}}{t}\,dt.$

It isn’t an elementary function.  The equation (1) implies the Taylor series

 $\mbox{Si }{z}=z\!-\!\frac{z^{3}}{3\!\cdot\!3!}\!+\!\frac{z^{5}}{5\!\cdot\!5!}% \!-\!\frac{z^{7}}{7\!\cdot\!7!}\!+-\ldots,$

which converges for all complex values $z$ and thus defines an entire transcendental function.

$\mbox{Si }{x}$ satisfies the linear third differential equation

 $xf^{\prime\prime\prime}(x)\!+\!2f^{\prime\prime}(x)\!+\!xf^{\prime}(x)=0.$

Remark 1.$\lim_{x\to\infty}\mbox{Si }{x}=\frac{\pi}{2}$

Remark 2.  There is also another “sine integral”

 $\mbox{si }{x}\;:=\;\int_{\infty}^{x}\frac{\sin t}{t}\,dt\;=\;\mbox{Si }{x}-% \frac{\pi}{2}$

and the corresponding cosine integral

 $\mbox{ci }{x}\;:=\;\int_{\infty}^{x}\frac{\cos t}{t}\,dt=\gamma\!+\ln{x}+\!% \int_{0}^{x}\frac{\cos{t}\!-\!1}{t}\,dt$

where $\gamma$ is the Euler–Mascheroni constant (http://planetmath.org/EulerMascheroniConstant).

 Title sine integral Canonical name SineIntegral Date of creation 2015-02-04 12:58:26 Last modified on 2015-02-04 12:58:26 Owner pahio (2872) Last modified by pahio (2872) Numerical id 17 Author pahio (2872) Entry type Definition Classification msc 30A99 Synonym sinus integralis Synonym Si Related topic SincFunction Related topic SineIntegralInInfinity Related topic LogarithmicIntegral2 Related topic CurvatureOfNielsensSpiral Related topic LaplaceTransformOfIntegralSine Related topic FresnelIntegrals Related topic HyperbolicSineIntegral Defines sine integral Defines sinus integralis Defines cosine integral