Talk:PlanetPhysics/Examples of Periodic Functions

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%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: examples of periodic functions %%% Primary Category Code: 02.30.-f %%% Filename: ExamplesOfPeriodicFunctions.tex %%% Version: 1 %%% Owner: pahio %%% Author(s): pahio %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}

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We list common periodic \htmladdnormallink{functions}{http://planetphysics.us/encyclopedia/Bijective.html}. In the parentheses, there are given their period with least modulus.

\begin{itemize}

\item One-periodic functions with a real period:

sine ($2\pi$), cosine ($2\pi$), tangent ($\pi$), cotangent ($\pi$), secant ($2\pi$), cosecant ($2\pi$), and functions depending on them -- especially the triangular-wave function ($2\pi$); \,the mantissa function $x\!-\!\lfloor{x}\rfloor$ (1).

\item One-periodic functions with an imaginary period:

exponential function ($2i\pi$), hyperbolic sine ($2i\pi$), hyperbolic cosine ($2i\pi$), hyperbolic tangent ($i\pi$), hyperbolic cotangent ($i\pi$), and functions depending on them.

\item Two-periodic functions:\, elliptic functions.

\item Functions with infinitely many periods:

the Dirichlet's function \begin{eqnarray*} f\!:\; x\mapsto\! & \left\{ \begin {array}{ll} 1 & \mbox{when}\,\,x \in \mathbb{Q}\\ 0 & \mbox{when}\,\, x \in \mathbb{R}\!\smallsetminus\!\mathbb{Q} \end{array} \right. \end{eqnarray*} has any rational number as its period;\, a constant function has any number as its period.

\end{itemize}

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