Elliptic Integrals¶
LaTeX source¶
Click here to download the TEX file.
\documentclass{article}
\usepackage[elliptic]{qed}
\begin{document}
\section*{Elliptic integrals}
The complete elliptic integrals of the first and second kinds are defined as
%
\begin{qed}
\ellipk(k^2) = \int_0^{\frac{\pi}{2}}
\frac{1}{\sqrt{1 - k^2 \sin^2\theta}} \, d\theta
\end{qed}
%
and
%
\begin{qed}
\ellipe(k^2) = \int_0^{\frac{\pi}{2}}
\sqrt{1 - k^2 \sin^2\theta} \, d\theta
\end{qed}
%
respectively. They have the following properties:
%
\begin{qed}
\ellipk(0) = \frac{\pi}{2}
\end{qed}
%
\begin{qed}
\ellipe(0) = \frac{\pi}{2}
\end{qed}
%
\begin{qed}
\ellipk(1) = \cinfty
\end{qed}
%
\begin{qed}
\ellipe(1) = 1
\end{qed}
%
\begin{qed}
\frac{d}{d k}\ellipk(k^2) = -\frac{\ellipk(k^2)
+ \frac{\ellipe(k^2)}{k^2 - 1}}{k}
\end{qed}
%
\begin{qed}
\frac{d}{d k}\ellipe(k^2) = \frac{\ellipe(k^2) - \ellipk(k^2)}{k}
\end{qed}
%
They also satisfy the Legendre relation:
%
\begin{qed}
\ellipe(k^2) \ellipk(1-k^2) + \ellipe(1 - k^2) \ellipk(k^2)
- \ellipk(k^2) \ellipk(1 - k^2) = \frac{\pi}{2}
\end{qed}
%
\end{document}
Building the PDF¶
To build the PDF, run
qed-setup
in the same directory as the tex file, then compile it by running
pdflatex elliptic_integrals.tex
qed
pdflatex elliptic_integrals.tex
if you have pdfLaTeX installed, or
tectonic elliptic_integrals.tex --keep-intermediates
qed
tectonic elliptic_integrals.tex
if you’re using tectonic.