# Online Encyclopedia

# Periodic function

In mathematics, a **periodic** function is a function that repeats its values, after adding some definite *period* to the variable. 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. More explicitly, a function *f* is **periodic with period t** if

*f*(*x*+*t*) =*f*(*x*)

for *all* values of *x* in the domain of *f*.

A simple example is the function *f* that gives the "fractional part" of its argument:

*f*( 0.5 ) =*f*( 1.5 ) =*f*( 2.5 ) = ... = 0.5.

If a function *f* is periodic with period *t* then for all *x* in the domain of *f* and all integers *n*,

*f*( x + nt ) =*f*( x ).

In the above example, the value of *t* is 1, since *f*( x ) = *f*( x + 1 ) = *f*( x + 2 ) ...

Some named examples are sawtooth wave, triangle wave.

Sine and cosine are periodic functions, with period 2π. The subject of Fourier series investigates the idea that an 'arbitrary' periodic function is a sum of trigometric functions with matching periods.

A function whose domain is the complex numbers can have two incommensurate periods without being constant. The elliptic functions are such functions. ("Incommensurate" in this context means not real multiples of each other.)

## General definition

Let *E* be a set with an internal operation + . A **T-periodic function**, or **function periodic with period T** on *E* is a function *f* on *E* to some set *F*, such that

- for all
*x*in*E*,*f(x+T) = f(x)*.

Note that unless + is assumed commutative this definition depends on writing *T* on the right.

## Periodic sequences

Some naturally-occurring sequences are periodic, for example (eventually) the decimal expansion of any rational number (see recurring decimal). We can therefore speak of the **period** or **period length** of a sequence. This is (if one insists) just a special case of the general definition.

## See also

frequency, definite pitch, almost periodic function