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Periodic function

A periodic function is a function that repeats. 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 a + internal operation. Let f be a function from E to F.

f is said T-periodic (or periodic with period T) iff ∃ T in E such that ∀ x in E, f(x+T) = f(x)

Periodic sequences

Some naturally-occurring sequences are periodic, for example the decimal expansion of any rational number. 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.