sine integral

The function sine integral (in Latin sinus integralis) from $ℝ$ to $ℝ$ is defined as

$\text{Si }x:={\displaystyle {\int }_0^x}{\displaystyle \frac{\sin t}{t}}𝑑t={\displaystyle {\int }_0^x}\text{sinc}(t)𝑑t,$ (1)

or alternatively as

$$\text{Si }x:={\int }_0^1\frac{\sin tx}{t}𝑑t.$$

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

$$\text{Si }z=z-\frac{z^3}{3\cdot 3!}+\frac{z^5}{5\cdot 5!}-\frac{z^7}{7\cdot 7!}+-\mathrm{\dots },$$

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

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

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

Remark 1.  ${lim}_{x\to \mathrm{\infty }}\text{Si }x=\frac{\pi }{2}$

Remark 2.  There is also another “sine integral”

$$\text{si }x:={\int }_{\mathrm{\infty }}^x\frac{\sin t}{t}𝑑t=\text{Si }x-\frac{\pi }{2}$$

and the corresponding cosine integral

$$\text{ci }x:={\int }_{\mathrm{\infty }}^x\frac{\cos t}{t}𝑑t=\gamma +\ln x+{\int }_0^x\frac{\cos t-1}{t}𝑑t$$

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