PlanetPhysics/Examples of Periodic Functions

We list common periodic [[../Bijective/|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

$\begin{array}{cc}f:x\mapsto & \{\begin{array}{cc}1& \text{when}x\in \mathbb{Q}\\ 0& \text{when}x\in ℝsetminus\mathbb{Q}\end{array}\end{array}$

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