PlanetPhysics/Examples of Periodic Functions

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

$$\begin{matrix} f\!:\; x\mapsto\! & \left\{ \begin {array}{ll} 1 & \mbox{when}\,\,x \in \mathbb{Q} \\ 0 & \mbox{when}\,\, x \in \mathbb{R}\!setminus\!\mathbb{Q} \end{array} \right. \end{matrix}$$ has any rational number as its period;\, a constant function has any number as its period.
 * 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