examples of periodic functions

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 (http://planetmath.org/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.

• Two-periodic functions:  elliptic functions.

• Functions with infinitely (http://planetmath.org/Infinite) 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.

 Title examples of periodic functions Canonical name ExamplesOfPeriodicFunctions Date of creation 2013-03-22 17:57:29 Last modified on 2013-03-22 17:57:29 Owner pahio (2872) Last modified by pahio (2872) Numerical id 10 Author pahio (2872) Entry type Example Classification msc 30A99 Classification msc 26A09 Synonym common periodic functions Related topic PeriodicityOfExponentialFunction Related topic HyperbolicIdentities Related topic RationalAndIrrational Related topic PeriodicFunctions Related topic FloorFunction Related topic Floor