30A99-ExamplesOfPeriodicFunctions.tex

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\pmtitle{examples of periodic functions}
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\pmtype{Example}
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\pmsynonym{common periodic functions}{ExamplesOfPeriodicFunctions}
\pmrelated{PeriodicityOfExponentialFunction}
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\begin{document}
We list common periodic functions. 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 \PMlinkname{imaginary}{ImaginaryNumber} 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 \PMlinkname{infinitely}{Infinite} 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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\end{document}