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examples of periodic functions (Topic)

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

  • 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).

  • 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.

  • Two-periodic functions:  elliptic functions.
  • Functions with infinitely many periods:

    the Dirichlet's function

    $\displaystyle f\!:\; x\mapsto\!$   $\displaystyle \left\{ \begin {array}{ll} 1 & \mbox{when}\,\,x \in \mathbb{Q}\\ ... ...mbox{when}\,\, x \in \mathbb{R}\!\smallsetminus\!\mathbb{Q} \end{array} \right.$  

    has any rational number as its period;  a constant function has any number as its period.



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Classification:
Physics Classification: 02.30.-f (Function theory, analysis)
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