sine integralSineIntegral2005-03-04 13:07:352008-10-04 09:58:18Definitioncomplex functionsine integralsinus integraliscosine integral% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
\DeclareMathOperator{\Si}{Si}
\DeclareMathOperator{\si}{si}
\DeclareMathOperator{\ci}{ci}
\DeclareMathOperator{\sinc}{sinc}The function {\em sine integral} (in Latin {\em sinus integralis}) from $\mathbb{R}$ to $\mathbb{R}$ is defined as
\begin{align}
\Si{x} \,:=\, \int_0^x\frac{\sin t}{t}\,dt = \int_0^x\sinc(t)\,dt,
\end{align}
or alternatively as
$$\Si{x} \,:=\, \int_0^1\frac{\sin{tx}}{t}\,dt.$$
It isn't an elementary function.\, The equation (1) implies the Taylor series \PMlinkescapetext{expansion}
$$\Si{z} = z\!-\!\frac{z^3}{3\!\cdot\!3!}\!+\!\frac{z^5}{5\!\cdot\!5!}
\!-\!\frac{z^7}{7\!\cdot\!7!}\!+-\ldots,$$
which converges for all complex values $z$ and thus defines an entire transcendental function.\\
$\Si{x}$ satisfies the linear third \PMlinkescapetext{order} differential equation
$$xf'''(x)\!+\!2f''(x)\!+\!xf'(x) = 0.$$
\begin{center}
\includegraphics[scale=0.4]{sinint}
\end{center}
\textbf{Remark 1.} \quad$\lim_{x\to\infty}\Si{x} = \frac{\pi}{2}$\\
\textbf{Remark 2.}\, There is also another ``sine integral''
$$\si{x}\; :=\; \int_\infty^x\frac{\sin t}{t}\,dt\; =\; \Si{x}-\frac{\pi}{2}$$
and the corresponding {\em cosine integral}
$$\ci{x} := \int_\infty^x\frac{\cos t}{t}\,dt = \gamma\!+\ln{x}+\!\int_0^x\frac{\cos{t}\!-\!1}{t}\,dt$$
where $\gamma$ is the \PMlinkname{Euler--Mascheroni constant}{EulerMascheroniConstant}.