Periodic function

In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of 2π radians. Periodic functions are used throughout science to describe oscillations, waves, and other phenomena that exhibit periodicity. Any function which is not periodic is called aperiodic.

A function $f$ is said to be periodic with period $P$ ( $P$ being a nonzero constant) if we have

for all values of $x$ in the domain. If there exists a least positive constant $P$ with this property, it is called the fundamental period (also primitive period, basic period, or prime period.) A function with period $P$ will repeat on intervals of length $P$ , and these intervals are referred to as periods.

Geometrically, a periodic function can be defined as a function whose graph exhibits translational symmetry. Specifically, a function $f$ is periodic with period $P$ if the graph of $f$ is invariant under translation in the $x$ -direction by a distance of $P$ . This definition of periodic can be extended to other geometric shapes and patterns, such as periodic tessellations of the plane.

A function that is not periodic is called aperiodic.

For example, the sine function is periodic with period $2\pi$ , since

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