Periodic function

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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 length 2π. 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.

Contents

Definition

A function f is said to be periodic if

for all values of x. The least positive constant P with this property is called the period. A function with period P will repeat on intervals of length P, and these intervals are sometimes also 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.

Examples

For example, the sine function is periodic with period 2π, since

for all values of x. This function repeats on intervals of length 2π (see the graph to the right).

Everyday examples are seen when the variable is time; for instance the hands of a clock or the phases of the moon show periodic behaviour. Periodic motion is motion in which the position(s) of the system are expressible as periodic functions, all with the same period.

For a function on the real numbers or on the integers, that means that the entire graph can be formed from copies of one particular portion, repeated at regular intervals.

A simple example of a periodic function is the function f that gives the "fractional part" of its argument. Its period is 1. In particular,

The graph of the function f is the sawtooth wave.

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