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For a function on the Real Number s or on the Integer s, 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 ''P'' greater than zero if

: ''f''(''x'' + ''P'') = ''f''(''x'')

for ''all'' values of ''x'' in the domain of ''f''. An aperiodic function (non-periodic function) is one that has no such period ''P''.

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 ''P'' then for all ''x'' in the domain of ''f'' and all integers ''n'',

: ''f''( x + Pn ) = ''f'' ( x ).

In the above example, the value of ''P'' is 1, since ''f''( ''x'' ) = ''f''( ''x'' + 1 ) = ''f''( ''x'' + 2 ) = etc. The period of a function need not be the smallest value (least period) that satisfies the above equation, so ''P'' could also equal two.

Some named examples are Sawtooth Wave , Square Wave and Triangle Wave .

The Trigonometric Function s sine and cosine are common periodic functions, with period 2π. The subject of Fourier Series investigates the idea that an 'arbitrary' periodic function is a sum of trigonometric functions with matching periods.

A function whose domain is the Complex Number s can have two incommensurate periods without being constant. The Elliptic Function s 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 Sequence s 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.


TRANSLATIONAL SYMMETRY

If a function is used to describe an object, e.g. an infinite image is given by the color as function of position, the periodicity of the function corresponds to Translational Symmetry of the object.


SEE ALSO