Home > Periodic function

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.

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.)

1 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.

2 Periodic sequences

Some naturally-occurring sequenceThis is a page about mathematics. For other usages of "sequence", see: sequence (non-mathematical). In mathematics, a sequence is a list of objects (or events) which have been ordered in a sequential fashion; such that each member either comes before, ors are periodic, for example (eventually) the decimalDecimal also called denary is the base 10 numeral system, which uses the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 (called digits) together with the decimal point and the sign symbols + (plus) and − (minus) to represent numbers. Decimal is the princip expansion of any rational numberIn mathematics, a rational number (or informally fraction is a ratio of two integers, usually written as the vulgar fraction a ''b where b is not zero. The set of all rational numbers is denoted by Q or in blackboard bold. Using the set-builder notation i (see recurring decimalRecurring decimals are a way of representing as decimals certain fractions which are not, in lowest terms, of the form k (2''n''5''m . These decimal representations include an infinitely repeated pattern at the end of the fraction (this repeated pattern m). 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.