PlanetPhysics/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 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

${\begin{matrix}f\!:\;x\mapsto \!&\left\{{\begin{array}{ll}1&{\mbox{when}}\,\,x\in \mathbb {Q} \\0&{\mbox{when}}\,\,x\in \mathbb {R} \!setminus\!\mathbb {Q} \end{array}}\right.\end{matrix}}$

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