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).