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examples of periodic functions
We list common periodic functions. In the parentheses, there are given their period with least modulus.

Oneperiodic 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 triangularwave function ($2\pi$); the mantissa function $x\!\!\lfloor{x}\rfloor$ (1).

Oneperiodic 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.

Twoperiodic functions: elliptic functions.

Functions with infinitely many periods:
the Dirichlet’s function
$\displaystyle f\!:\;x\mapsto$ $\displaystyle\left\{\begin{array}[]{ll}1&\mbox{when}\,\,x\in\mathbb{Q}\\ 0&\mbox{when}\,\,x\in\mathbb{R}\!\smallsetminus\!\mathbb{Q}\end{array}\right.$ has any rational number as its period; a constant function has any number as its period.
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