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'sine integral'
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| Title of object: |
sine integral |
| Canonical Name: |
SineIntegral |
| Type: |
Definition |
| Created on: |
2005-03-04 13:07:35 |
| Modified on: |
2005-03-04 19:37:51 |
| Classification: |
msc:30A99 |
| Synonyms: |
sine integral=sinus integralis |
Preamble:
% 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{\sinc}{sinc} |
Content:
The function {\em sine integral} (in Latin {\em sinus integralis}) from $\mathbb{R}$ to $\mathbb{R}$ is defined as
$$\Si(x) := \int_0^x\frac{\sin t}{t}dt = \int_0^x\sinc(t)\,dt.$$
So it has the Taylor series \PMlinkescapetext{expansion}
$$\Si(z) = z-\frac{z^3}{3\cdot 3!}+\frac{z^5}{5\cdot 5!}
-\frac{z^7}{7\cdot 7!}+-...,$$
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{sinint}
\end{center} |
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